Multiscale Wavelet Methods for Partial Differential Equations
Multiscale Wavelet Methods for Partial Differential Equations
Wavelet Analysis and Its Applications The subject of wavelet analysis has recently drawn a great deal of attention from mathematical scientists in various disciplines. It is creating a common link between mathematicians, physicists, and electrical engineers. This book series will consist of both monographs and edited volumes on the theory and applications of this rapidly developing subject. Its objective is to meet the needs of academic, industrial, and governmental researchers, as well as to provide instructional material for teaching at both the undergraduate and graduate levels. This is the sixth book of the series. While wavelet techniques and algorithms have proved to be very powerful in several areas of applications such as signal processing and image compression, the wavelet approach to solving complex problems governed by physical models such as partial differential equations (PDEs) has to compete with the well-established and already very effective methods. There is, however, a common link between the multiresolution approximation structure in wavelet analysis and multilevel and multigrid techniques in numerical solutions of P DEs. This volume is aimed at bridging these two most fruitful approaches and is designed to update current developments in PDEs that are somewhat related to the wavelet approach. The series editor wishes to congratulate the editors of this volume for an outstanding job in selecting the most relevant chapters and in carefully editing the volume. He would also like to thank the authors for their very fine contributions.
Multiscale Wavelet Methods for Partial Differential Equations Edited by Wolfgang Dahmen Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen, Germany
Andrew J. Kurdila Department of Aerospace Engineering Texas A&M University College Station, Texas, U.S.A.
Peter Oswald Institute for Algorithms and Scientific Computing, GMD Sankt Augustin, Germany
ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto
Copyright
9 1997 by Academic Press 0
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Library of Congress Cataloging-in-Publication Data Multiscale wavelet methods for partial differential equations / edited by Wolfgang Dahmen, Andrew Kurdila, Peter Oswald. p. cm. m (Wavelet analysis and its applications ; v. 6) Includes bibliographical references and index. ISBN 0-12-200675-5 (alk. paper) 1. Differential equations, PartialmNumerical solutions. 2. Wavelets (Mathematics) I. Dahmen, Wolfgang. II. Kurdila, Andrew. III. Oswald, Peter. IV. Series. 97-12672 QC20.7.D5M83 1997 CIP 515'.2433mDC21
Printed in the United States of America 97 98 99 00 01 IP 9 8 7 6 5 4 3 2 1
Contents Preface .............................................................. Contributors ........................................................ I. F E M - L i k e
Multilevel
Preconditioning
.........................
Multilevel Solvers for Elliptic Problems on Domains . . . . . . . . . . . . . . . . . . . Peter Oswald Wavelet-Like Methods in the Design of Efficient Multilevel Preconditioners for Elliptic P DEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii xi 1
3
59
Panayot S. Vassilevski and Junping Wang II. F a s t W a v e l e t A l g o r i t h m s : C o m p r e s s i o n a n d A d a p t i v i t y
.. 107
An Adaptive Collocation Method based on Interpolating Wavelets ... 109
Silvia Bertoluzza An Adaptive Pseudo-Wavelet Approach for Solving Nonlinear Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gregory Beylkin and James M. Keiser A Dynamical Adaptive Concept Based on Wavelet Packet Best Bases: Application to Convection Diffusion Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pascal Joly, Yvon Maday, and Valdrie Pettier Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
137
199
237
Stephan Dahlke, Wolfgang Dahmen, and Ronald A. De Vore I I I . W a v e l e t S o l v e r s for I n t e g r a l E q u a t i o n s
...................
Fully Discrete Multiscale Galerkin BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
285 287
Tobias yon Petersdorff and Christoph Schwab Wavelet Multilevel Solvers for Linear Ill-Posed Problems Stabilized by Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Andreas Rieder
347
Contents
vi IV. S o f t w a r e T o o l s a n d N u m e r i c a l E x p e r i m e n t s
..............
381
Towards Object Oriented Software Tools for Numerical Multiscale Methods for PDEs using Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Titus Barsch, Angela Kunoth, and Karsten Urban
383
Scaling Function and Wavelet Preconditioners for Second Order Elliptic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jeonghwan Ko, Andrew J. Kurdila, and Peter Oswald
413
V. M u l t i s c a l e I n t e r a c t i o n a n d A p p l i c a t i o n s t o T u r b u l e n c e . . . 4 3 9 Local Models and Large Scale Statistics of the Kuramoto-Sivashinsky Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
Juan Elezgaray, Cal Berkooz, Harry Dankowicz, Philip Holmes, and Mark Myers Theoretical Dimension and the Complexity of Simulated Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
473
Mladen V. Wickerhauser, Marie Farge, and Eric Goirand V I . W a v e l e t A n a l y s i s of P a r t i a l D i f f e r e n t i a l O p e r a t o r s . . . . . . . 4 9 3 Analysis of Second Order Elliptic Operators Without Boundary Conditions and With VMO or H61derian Coefficients . . . . . . . . . . . . . . . .
495
Jean-Marc Angeletti, Sylvain Mazet, and Philippe Tchamitchian Some Directional Elliptic Regularity For Domains With Cusps . . . . . . . 541
Matthias Holschneider Subject Index .....................................................
567
Preface Wavelet methods are by now well-established as a novel and successful mathematical tool with applications in Signal Analysis and Image Processing, Theoretical Physics and Mathematics itself. Meanwhile several books and edited volumes have appeared, in particular, in this series Wavelet Analysis and Its Applications, addressing this fast growing field of reasearch. More recently the intriguing aspects of these concepts have stirred considerable interest in applying them also to more traditional areas of numerical analysis and engineering applications. The high-level contributions to the Special Session on Wavelet Galerkin Methods at the 31st Annual Meeting of the Society of Engineering Science held in College Station, TX, in October 1994 as well as the Special Session on Meshless and Wavelet Methods at the 3rd U.S. National Congress in Computational Mechanics held in Dallas in June 1995 and the partly very controversial panel discussions on these meetings indicated a strong need of an up-to-date publication on current developments and further prospects of this methodology, specifically in the area of partial differential equations (PDEs). It is fair to say that wavelet analysis as it stands has by far not yet reached a steady state in this field. As promising as many of the underlying concepts are, it would be naive to expect their immediate practical success in complex, real-life applications governed by PDEs. Current wavelet methods and corresponding software tools are still primarily confined to model problems. While these new ideas may also actually trigger new developments in the context of well-established multilevel techniques, they will ultimately have to compete with the discretization methods and existing software packages for PDEs. This volume aims at contributing to the progress in this direction. We do not claim to reflect an exhaustive account of the state of the art. However, we have made an effort to address several aspects which we feel are important and typical in connection with solving partial differential equations and which bear potential for further progress. Key ideas such as sparsity of wavelet representations of operators, economic vii
viii
Preface
representations of functions with localized singularities or strongly scale dependent behavior, the availability of new libraries of flexible localized basis functions (wavelets and wavelet packets) as well as resulting fast multiscale algorithms will be presented in the context of linear and nonlinear operator equations. To support a first orientation we have grouped the material into six chapters which are, however, interrelated in many respects. The following comments might serve as a brief guide. The first chapter may be viewed as a bridge to finite element based multilevel preconditioning and multigrid techniques. Recent development has shown that the multiscale space decomposition framework typical for wavelets provides significant insight into the understanding of algorithms like hierarchical bases or BPX schemes. On the other hand, multigrid technology adds further algorithmical variety and fuels intertwining of these concepts. Oswald focusses on frame based multilevel Schwarz preconditioners for elliptic boundary value problems on bounded domains in IRd. The main goal is to preserve as much as possible the algorithmic advantages of scale- and shiff-invariant discretizations typical for the classical wavelet setting by introducing local modifications only near the boundary. As for finite element schemes, adaptive nested refinement can be incorporated. Vassilevski and Wang start from the hierachical basis method by Yserentant for linear finite element discretizations (which is asymptotically nonoptimal), and propose an improvement based on approximately computing wavelet-like complement basis functions. They also discuss the multiplicative algorithms corresponding to these space decompositions which provides a link to multigrid V-cycle solvers. Chapter 2 offers information on principal features of wavelet based discretizations centering upon sparse representations of operators and functions as well as resulting adaptivity concepts. Bertoluzza highlights the algorithmic benefits of interpolatory Deslaurier-Dubuc wavelets for adaptive collocation methods. The method is illustrated on several linear problems including dominating convection, but might be attractive for nonlinear problems as well. Beylkin and Keiser present a systematic algorithmic study of a class of periodic nonlinear evolution equations covering, for instance, the viscous Burgers equation and Korteweg-de-Vries equation. Their main objective is to produce a scheme which solves such problems for a given tolerance at a cost which remains proportional to the number of significant wavelet coefficients of the solution. Essential ingredients are sparse operator representations in the so-called nonstandard form and the fast evaluation of nonlinear terms. Joly, Maday, and Perrier apply wavelet packets and compression techniques to the adaptive treatment of nonlinear evolution equations. In particular, a best basis concept based on cardinal entropy is introduced and its practical implementation
Preface
ix
for time-dependent PDEs is discussed. The numerical examples are concerned again with the Burgers equation with small viscosity, and simple convection-diffusion problems. Dahlke, Dahmen, and DeVore address the issue of adaptivity from a primary analytical viewpoint. The goal is to interrelate the concepts of nonlinear approximation, Besov regularity, and wavelet based adaptive techniques for stationary elliptic problems covering integral as well as differential operators. In particular adaptive refinement strategies are shown to converge without additional a priori assumptions on the solution. In Chapter 3, two papers on integral equations closely related to the material of the previous chapters are included. Von Petersdorff and Schwab propose a wavelet based, fully discrete Galerkin scheme for a zero-order elliptic boundary integral equation. They combine the wavelet compression of the intergral operator with a carefully designed adaptive quadrature scheme which ensures the same asymptotical complexity in the computation of the compressed matrix as the solver. This is a central step towards efficient practical implementations. The paper by Rieder is devoted to additive and multiplicative wavelet solvers of Tikhonov regularized ill-posed problems. The techniques are similar in spirit to those in Chapter 1. The paper by Barsch, Kunoth and Urban in Chapter 4 surveys a software toolbox under development which aims at providing an experimental platform for wavelet discretizations of PDEs and integral equations. Some emphasis is put on the use of C + + for treating multidimensional multiscale data structures and algorithms. Ko, Kurdila and Oswald present a comparative study of several multilevel preconditioners for a second order model problem on simple domains. In particular, schemes based on finite elements, Daubechies and AFIF scaling functions and wavelets are compared. Chapter 5 is different in nature. Rather than the efficiency of algorithms, the main concern here is to employ wavelets as an adequate tool for analyzing and simulating multiscale interaction/separation in flows. In Elezgaray, Berkooz, Dankowicz, Holmes and Myers, local models for the study of coherent structures of solutions of the Kuramoto-Sivashinsky equation on large intervals are investigated. The use of periodic wavelets is proposed and compared with the traditional Fourier approximations. In the paper by Wickerhauser, Farge and Goirand, the complexity of fully developed turbulent flows is investigated with the aid of concepts like theoretical dimension, best bases and wavelet packets. Numerical experiments for the Burgers equation and two-dimensional Navier-Stokes flow are presented which suggest a strong interrelation of the theoretical dimension and the number of coherent structures in two-dimensional viscous turbulent flows. Chapter 6 is devoted to the use of wavelets primarily as a tool for analysing differential operators. Angeletti, Mazet and Tchamitchian study
x
Preface
second order differential operators in divergence and non-divergence forms with diffusion tensors of rather weak regularity on ]Rd. One of the main results is concerned with the boundedness of Galerkin projection operators in Lp-Sobolev norms. Holschneider presents a wavelet based microlocal analysis of local regularity spaces. As an application, the regularity of elliptic differential operators on domains with cusps is treated. This volume is comprised of both invited and contributed chapters. All contributions, whether of survey character or containing primarily original material, were refereed according to their respective goal. We wish to thank all authors and reviewers for their most valuable contributions, in particular, for their patience and cooperation. We are indebted to Margaret and Charles Chui for their diligent and kind assistance during the editorial process.
Aachen, Germany College Station, Texas Sankt Augustin, Germany June, 1997
Wolfgang Dahmen Andrew J. Kurdila Peter Oswald
Contributors Numbers in parentheses indicate where the authors' contributions begin.
J. M. ANGELETTI (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de Saint-Jdrdme, 13397 Marseille Cedex 20, France, et LATP, CNRS, URA 225 [jean-marc. angeletti@math, u-3mrs.fr] TITUS B ARSCH (383), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [barsch @igpm. rwt h-aachen, de] GAL BERKOOZ (441), BEAM Technologies, Ithaca, N Y 15850 [gal@cam. cornell.edu] SILVIA BERTOLUZZA (109), I.A.N.-C.N.R., v. Abbiategrasso 209, 27100 Pavia, Italy [
[email protected]. pv. cnr.it] GREGORY BEYLKIN (137), Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, U.S.A. [
[email protected]] WOLFGANG DAHMEN (237), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [dahmen @igprn. rwt h-aachen, de] STEPHAN DAHLKE (237), Institut fffr Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [dahlke@igprn. rwth-aachen, de]
xi
xii
Contributors
HARRY DANKOWICZ (441), Department of Mechanics, Royal Institute of Technology, S-100 44 Stockholm, Sweden [
[email protected]] RONALD A. DEVORE (237), Department of Mathematics, University of South Carolina, Columbia, S.C. 29208, U.S.A. [
[email protected]] JUAN ELEZGARAY (441), CRPP-CNRS, Av. Schwietzer, 33600 Pessac, F~ance [
[email protected]. fr] PHILIP HOLMES (441), PACM, Fine Hall, Princeton University, Princeton, NJ 08544-1000, U.S.A. [
[email protected]] MARIE FARGE (473), LMD-CNRS, Ecole Normal Superieure, 24 Rue Lhomond, F-75231 Paris, France [
[email protected]] ERIC GOIRAND (473), LMD-CNRS, Ecole Normal Superieure, 24 Rue Lhomond, F-75231 Paris, France [goirand~lmd.ens.fr] MATTHIAS HOLSCHNEIDER(541), CNRS CPT Luminy, Case 907, F-13288 Marseille, France [
[email protected]] PASCAL JOLY (199), Laboratoire d'Analyse Numdrique, Tour 55-65, 5~me dtage, Universitd Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France
[email protected]] JAMES M. KEISER (137), 659 Main St., Apt. B, Laurel, MD 20707-4067, U.S.A. [
[email protected]] JEONGHWAN KO (413), Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, U.S.A. [
[email protected]] ANDREW J. KURDILA (413), Department of Aerospace Engineering, Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. [
[email protected]] ANGELA KUNOTH (383), Institut fffr Geometrie und Praktische Mathematik, RWTH Aachen, 52056 Aachen, Germany [kunot h @igpm. rwt h-aachen, de]
Contributors
xiii
YVON MADAY (199), ASCI, UPR 9029, Bat. 506, Universitd Paris Sud, 91405 Orsay Cedex, France g~ Laboratoire d'Analyse Numdrique, Tour 55-65, 5@me dtage, Universitd Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, Prance [
[email protected] ussieu, fr] S. MAZET (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de Saint-Jdr6me, 13397 Marseille Cedex 20, Prance, et LATP, CNRS, URA 225 [
[email protected]] MARK MYERS (441) PETER OSWALD (3, 413), Institute for Algorithms and Scientific Computing, G M D - German National Research Center for Information Technology, D-53754 Sankt Augustin, Germany [
[email protected]] VALt~RIE PERRIER (199), Laboratoire d'Analyse, Gdomdtrie et Applications, URA 742, Universitd Paris Nord, 93430 Villetaneuse, & Laboratoire de Mdtdorologie Dynamique, Ecole Normale Supdrieure, 24, rue Lhomond, 75231 Paris Cedex 05, Prance [
[email protected]] TOBIAS
PETERSDORFF (287), Department of Mathematics, University of Maryland College Park, College Park, MD 20742, U.S.A. [
[email protected]]
VON
ANDREAS RIEDER (347), Fachbereich Mathematik, Universita't des Saar1andes, Postfach 15 11 50, 66041 Saarbrubken, Germany [andreas @num. uni-sb, de] CHRISTOPH SCHWAB (287), Seminar ffir Angewandte Mathematik, Eidgeno'ssische Technische Hochschule Zffrich, Ra'mistrasse 101, CH8092 Zffrich, Switzerland [
[email protected]] P. TCHAMITCHIAN (495), Laboratoire de Mathdmatiques Fondamentales et Appliqudes, Facultd des Sciences et Techniques de SaintJdr6me, 13397 Marseille Cedex 20, Prance, et LATP, CNRS, URA 225 [tch am p hi @mat h. u-3mrs, fr] KARSTEN URBAN (383), Institut ffir Geometrie und Praktische Mathematik, R W T H Aachen, Templergraben 55, 52056 Aachen, Germany [urban@igpm. rwth-aachen, de]
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Contributors
PANAYOT S. VASSILEVSKI (59), Center of Informatics and Computing Technology, Bulgarian Academy of Sciences, "Acad. G. Bontchev" street, Block 25 A, 1113 Sofia, Bulgaria [
[email protected]] JUNPING WANG (59), Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071, U.S.A. [
[email protected]] MLADEN VICTOR WICKERHAUSER (473), Department of Mathematics, Campus Box 1146, One Brookings Drive, Washington University, Saint Louis, Missouri 63130, U.S.A. [
[email protected]]
I0 FEM-Like Multilevel Preconditioning
This Page Intentionally Left Blank
M u l t i l e v e l Solvers for Elliptic P r o b l e m s
on Domains
Peter Oswald
A b s t r a c t . We study to which extent the geometric multilevel approach based on dyadic scales of shift-invariant subspaces on ]Rd can be used to produce accurate discrete solutions of elliptic boundary value problems of positive order on nonrectangular domains. We also deal with the construction of optimal preconditioners, including the case of nested refinement. Sufficient geometric conditions on a domain are given such that a robust and asymptotically optimal algorithm can be expected. In contrast to other approaches which emphasize biorthogonal wavelet decompositions, we are satisfied with a simpler frame concept which incorporates recent experience with finite element multilevel solvers.
w
Introduction
For the numerical solution of elliptic boundary value problems for partial differential equations, multilevel methods have gained popularity over the last decade. This is mainly due to their nearly optimal complexity for a number of model problems. In m a n y practical cases, they are based on multiresolution scale of nested finite-dimensional subspaces
Vo cv~ c . . . c ~
c...
(1.1)
of a Hilbert space V serving as the energy space for the given variational problem. The scale (1.1) is used to produce stable subspace splittings v j - v0 + v~ + . . . + v j M u l t i s c ~ l e W a v e l e t M e t h o d s for P D E s Wolfg~ng D~hmen,
Andrew
J. K u r d i l ~ , ~ n d P e t e r O s w M d ( e d s . ) , p p . 3 - 5 8 .
C o p y r i g h t ( ~ 1 9 9 7 by A c a d e m i c P r e s s , I n c . All r i g h t s of r e p r o d u c t i o n in ~ny f o r m reserved. ISBN 0-12-200675-5
(1.2)
4
P. Oswald
and to design fast iterative solvers related to such splittings for the discretized variational problem associated with a computational discretization space Vj (or a properly defined subspace V] C Vj). For 2mth order elliptic boundary value problems in Sobolev spaces, considerable progress has been achieved in the theoretical understanding of multilevel and multigrid methods as well as of other subspace correction methods for finite element discretizations (see [65, 67, 11, 60]). We survey some of these results in Section 2. The underlying theory also applies to various wavelet discretizations; see, e.g., [26, 44, 45, 29] for some papers that deal with wavelet solvers for elliptic problems and are related to our approach. Roughly speaking, in these algorithms suitable "detail spaces" Wj C t~ are constructed, together with their algebraic bases, such that t~ - t~_l+Wj provides a stable splitting of V) into "low frequency" ( ~ - 1 ) and "high frequency" (Wj) parts. Using this two-level decomposition recursively, (1.2) is replaced by vj
= VoSW~ $ . . .4wj
.
(1.3)
One can use this splitting indirectly (i.e., the original problem is discretized with respect to a standard basis in Vj, and the wavelet decomposition behind (1.3) implicitly defines the structure of the multilevel preconditioner) or directly. In the latter case, the discretization is performed with respect to the wavelet basis and is automatically well conditioned. To achieve asymptotically optimal work estimates one has to use compression arguments. For elliptic problems of order 2m, and k~ with locally supported basis functions, the first approach is often preferred since the Vj-discretization matrix is automatically sparse and available from standard engineering codes. The direct use of the wavelet decomposition (1.3) is promising for those situations in which the Vj-discretization is not a priori sparse, e.g., for integral equations. In both cases, the explicit introduction of detail spaces Wj is the crucial step, and may add some theoretical and practical difficulties. For example, most of the popular examples of wavelet spaces (see [30, 16]) are derived in a one-dimensional, shift-invariant setting on IR. Multivariate examples on IRd are mostly obtained by tensor-product techniques. Adaptations to bounded intervals and domains have been studied in, e.g., [2, 17, 19, 18]. However, up to now there has been no comprehensive study of the practical potential of discretizations using multilevel structures based on shift-invariance and dyadic dilation (modulo boundary modifications) in the case of general, nonrectangular geometries. It is not completely clear to the author what will be left from the powerful wavelet machinery if the basic algebraic assumptions (invariance with respect to integer shifts and (dyadic) dilation) are significantly relaxed.
Multilevel Solvers for Elliptic Problems
5
In our opinion, the departure from these assumptions is unavoidable for many engineering problems. A simple example is 2ruth order elliptic PDEs with rapidly varying coefficients that exhibit a large ratio of ellipticity constants and are therefore far from a generic Hm-problem. Principal difficulties are to be expected if nonsymmetric equations with dominant low order hyperbolic parts, such as convection-diffusion equations, or nonlinear problems are to be studied. In our opinion, this robustness aspect is one of the target problems for future investigation, i.e., the adaptation of the multilevel concept to a class of operator equations (or even to an individual equation) still remains a decisive issue for practical implementations and engineering applications. An indicator of this tendency is analogous efforts within the FEM community and the renewed interest in the algebraic multigrid method and related algorithms. There is another observation that dampens expectations concerning the practical use of wavelet solvers, even for standard symmetric elliptic boundary value problems. Numerical experiments [39, 49] show that for generic HI-problems, i.e., for second order elliptic equations, wavelet and prewavelet discretizations perform slightly worse than traditional finite element preconditioners associated with (1.2). While condition numbers of L2-problems (and sometimes also of HS-problems with negative s) are usually improved and become uniformly bounded if J --+ c~, the preconditioning effect in the Hi-norm is reduced by a constant factor. This poses the problem of determining more carefully problem classes in which the use of wavelet preconditioners based on (1.3) is justified compared to simpler methods based only on the use of scaling functions and related to (1.2). Also, one may argue whether new wavelet families (Daubechies orthogonal wavelets [30], AFIF elements [51], etc.) are generally useful and are able to compete with traditional finite element and spline constructions from this more practical viewpoint. The reader should not expect an answer to these more philosophical questions. In Sections 3-5, we concentrate on studying the influence of the domain geometry on the optimality of multilevel preconditioners resulting from a standard multiresolution analysis (these sections are sometimes rather technical and represent the original part of this paper). The concept is based on sequences {Vj} of subspaces of a fixed sequence {Vj} of subspaces in some HS(IR d) which "live" on a uniform structure generated by shift-invariance principles and dyadic dilation as usual. These auxiliary subspaces Vj are spans of scaling functions (or, in the finite element terminology, nodal basis functions) of levels _< j such that a canonical HS(lRd)-elliptic Galerkin discretization can be solved efficiently, e.g., by preconditioned iterative methods, with the preconditioner inherited from the generating system, or frame, consisting of scaling functions (see Section 4). Thus, we essentially stay with splittings of the type (1.2).
6
P. Oswald
The connection of the auxiliary problems in ~. with the originally given HS(f~)-elliptic problem, with natural boundary conditions, on a generic bounded domain f~ C lRd is established in Section 3 where sufficiently rich subspaces ~ , n C V)In will be constructed. The construction consists of a local boundary modification which is similar to constructions outlined in other papers on wavelets on intervals and domains, too, but is relatively simple. A drawback is that, in contrast to { ~ } , the sequence V),n is not monotone, which requires additional considerations when adaptivity is an issue. Information between V),~ and k~ is exchanged by appropriate restriction (R j) and extension (E j) operators. To obtain uniform condition number and work estimates, certain geometric conditions on f~ arise in a natural way. Asymptotically, they hold for domains with boundaries satisfying a uniform Lipschitz condition. For details, see Sections 3 and 4. In Section 5 several extensions will be considered in less depth. We have decided to detail the exposition in Sections 3 and 4 to tensor-product spline spaces t~. In Subsection 5.1, we discuss the conditions on generating functions r r and the necessary modifications such that the conclusions of the theory of the previous sections still hold. In Subsection 5.2 modifications for problems with essential boundary conditions will be considered. This case is somewhat more difficult to handle (compare [44]), and we do not have a satisfactory proposal for d _> 3 at present. For d - 2, the basic idea is to enforce additional refinement near the boundary which leads to a modified construction of ~ , n (resp.
yj).
Adaptivity by nested refinement (or, in our terminology, nested basis function selection) is dealt with in Subsection 5.3. We share the more naive viewpoint of most of the adaptive finite element codes, and use local a posteriori estimators based on a sort of local higher regularity or superconvergence assumption (which is hard to justify theoretically but leads to reasonable results in practical computations). The author admits that there is a lot of closely related work, and that some of the ideas are straightforward and have appeared, in one or another form, in other papers, too. For example, the construction of Cohen, Dahmen, DeVore [18] of a biorthogonal wavelet system for Sobolev spaces, though complicated in technical details, looks theoretically much more powerful and is based on a clever boundary modification as well. There is a lot of activity on solving large linear systems arising from finite element discretizations on engineering (so-called unstructured) grids or of obstacle problems, where an embedding into a regular structure has been one of the options. See the recent papers [54, 55, 66, 5, 42] as well as [43, 35, 48]. Some of these investigations originate from the domain embedding or fictitious domain methods for finite difference discretizations, where boundary modifications, extension, and restriction operators have
Multilevel Solvers for Elliptic Problems
7
been used for a long time, also in connection with multigrid techniques. We refer to [40, 52, 36, 8, 54]. Finally, we wish to mention one more time that this is a paper on geometric multilevel methods; i.e., the approximating subspaces as well as their multilevel splittings are constructed for a generic linear, symmetric, uniformly elliptic HS-problem (s > 0). No anisotropies, behavior of coefficient functions, physical background, or other specifics of the boundary value problem have been used. It is tempting but rather difficult to further extend this work to a more operator-adapted setting (see, e.g., [46, 22, 23]), and to attack the robustness aspect at large. w
Stable subspace splittings and iterative methods
This is a short introduction to a class of approximation and solution methods for operator equations in Hilbert spaces which is based on the concepts of multilevel scales and stable subspace splittings. The solvers for the approximate problems fall into the class of iterative subspace correction methods. Particular examples are domain decomposition and multigrid algorithms, orthogonal, Riesz basis, and frame decomposition and reconstruction techniques, and others. The framework typically incorporates adaptivity (with respect to individual features of the solution of the operator equation) in a natural way. We refer to the books and surveys by Bramble [11], Chan and Matthew [15], Dahmen [25], Hackbusch [41], Oswald [60], Xu [65], and Yserentant [67], where different aspects and the history of the subject have been illuminated. We start with the notion of stable subspace splittings for Hilbert spaces. Assume that V is a (finite-dimensional or separable) Hilbert space, with scalar product (-, ")v and norm I1" IIv. Let {Vj } be an at most countable collection of closed subspaces of V such that V - E j Tv~, in the sense that for each u E V there is at least one V-converging representation
J Let us further assume that the symmetric bilinear forms a(., .) (resp. bj(., .)) are scalar products on V (resp. t~ for all j), such that the spaces are Hilbert when equipped with these alternative scalar products. We use the notation {V; a} (resp. {1~; bj}) to indicate this assumption. In particular,
cllull,5 _< Ilulla < Cilull,5
VuEV,
(2.1)
where I[ulla = v/a(u, u) denotes the energy norm in {V; a}. The two positive constants 0 < c,C < ~ in (2.1), are sometimes called ellipticity constants of a(.,.) with respect to V, and do not depend on u E V. In
8
P. Oswald
the following, we use the symbol ~ for two-sided inequalities such as (2.1), while A _ B (resp. A _ B) stands for a one-sided inequality A < C . B (resp. A >_ c. B). Thus, A ~. B is the same as A ~ B ~ A. The constants 0 < c, C < oe, are assumed to be generic, if not stated otherwise; they do not depend on the arguments of the expressions A and B. The subspace splitting {V; a} - E { V j ; bj}
(2.2)
J
is called stable if IIlulll~bj -
inf E bj(uj uj) x a(u, u) ,,jevj . , , = ~ j ,,j j
(2.3)
One of the simple consequences of this definition is the fact that the operator equation 7:'u-r
P - ERj~
'
r162
J
(2.4)
J
where Rj 9 ~ ~ V denotes the natural injection, and :I) 9 V ----, I~ t5
9 bj(Tju, vj) - a(u, vj) 9
V Yj E
(vj)
t5
(2.5)
for all j, is an equivalent formulation of the variational problem of determining u E V such that a(u, v) - r
Vv E V
(0 E V*) .
(2.6)
The problem (2.4) is called the additive Schwarz formulation of the variational problem (2.6) associated with the subspace splitting (2.2). Moreover, the additive Schwarz operator P acting in V is symmetric, positive definite with respect to a(., .), with spectral condition number defined by
~('P)- )~max(~O) )~min(~) '
(2.7)
where )~max('P) - -
sup
a(Pu, u) -
inf
a(Pu, u) -
uEV "a(u,u)=l
sup
O#uEV
a(u, u)
[]]U[l[~bi}
and ~min(~O)
--
u6_V "a(u,u)=l
inf
O~uEV
a(u, u)
[[[U[[[~bj}
Multilevel Solvers for Elliptic Problems
9
As a consequence, the operator equation (2.4) is well conditioned if the norm equivalence (2.3) holds with tight constants. This is the key to preconditioning using subspace splittings. Before we discuss the typical algorithms associated with finite-dimensional versions of (2.2) we wish to add a few comments. First of all, the inclusion of infinite-dimensional spaces, and countable splittings is very appropriate if we come to our multiscale applications. As a rule, if Cx9
j=O
is stable then the finite splittings J
{Vj; a} - ~--~{Vj ; bj}
j=o
are uniformly stable for J ~ cr and vice versa. This is a desirable feature for algorithmical reasons" the convergence rates of the solvers do not degenerate if better resolution is needed. In addition, this "asymptotical" viewpoint links us with the rich mathematical theory of Fourier analysis and function space decompositions. For instance, any complete orthogonal system but also any Riesz basis or frame in V leads to stable subspace splittings. On the other hand, the practical performance of the approach depends on a "nonasymptotical" range of small to moderate J, and is heavily influenced by the constants in (2.3). Thus, constructions which do not lead to stable splittings in the infinite-dimensional setting may well be successful in some applications. The hierarchical basis method of Yserentant designed for HI-elliptic problems on two-dimensional domains may serve as an illustration. However, especially for methods working in an adaptive refinement environment where larger J are much more likely, the stability of the multiscale splitting in V becomes crucial. A second remark is on the typical method of proof of the stability condition (2.3) which has been put into its present abstract form by J. Xu in his thesis (see [65]). The lower bound, i.e., I]lu]]]{bj} ~ a(u, u), is usually formulated as a separate condition; proofs combine (depending on the context) methods of approximation theory and elliptic regularity arguments. The upper bound is often replaced by assuming so-called slrengthened Cauchy-Schwarz inequalities controlling the interaction between the different subspaces in the splitting. One formulation is to require
a(uj, uk) 2 < 7y,kbj(uj, uj)bk(uk, uk)
V uj E t~, uk E Vk .
(2.8)
Then the finiteness of the largest eigenvalue )~m~x(F) of the matrix F = ((Tj,k)) is a sufficient condition for the upper estimate. It is convenient to
P. Oswald
10 assume that
7j,j = 1
(2.9)
in (2.8) for j - k, which amounts to an appropriate scaling. The inequalities (2.8) and the condition on F are also important for the more sophisticated multiplicative algorithms for which stability alone will not give the desired optimal result; see below. With slight modifications, this concept is also surveyed in [67, 38]. A refined theory serving the needs of multigrid applications, and including a discussion of all kinds of perturbations typical for practical implementations, is given in [11]. See also [41, Sections 10-11] for an introduction and the link to multigrid. Finally, the approach of [60, 25] emphasizes the close connection of the stability assumption for multiscale splittings with results on scales of approximation spaces. For a survey on applications to domain decomposition methods (not necessarily of multilevel type), see [15]. Thirdly, it is possible to further generalize the concept by removing the assumption V~ C V. This is important for several reasons. In the multilevel context, we have in mind situations in which the monotonicity condition Vj C Vj+I is violated. The formal cure is the introduction of a suitable set of mappings Rj : t~ ---+V (replacing the natural injections) such that R - E j ~ j 9~)vj ~ y is onto, and ]][ull] 2
-
inf
{bj,Rj} -- ujEVj " u = E j R j u j
y~. bj(uj, uj) ~ a(u, u) j
Vu e V
The latter condition replaces (2.3), and guarantees that V' - E j RjTj' preserves the above-mentioned properties of P. Condition (2.5) is replaced by
bj(Tj'u, vj)
-
-
a(u, njvj)
V vj E t~ ,
where Tj' : V ---, }). All these modifications can be subsumed in a simple Hilbert space lemma, called fictitious space lemma, which was first used in connection with fictitious domain and domain decomposition methods by Nepomnyaschikh [54]; see [60, Theorem 17] or [66]. As a last remark we note that the abstract formulation of the stability concept in the form of the two-sided inequality (2.3) allows us to better understand simple transformations of one splitting into another, which are useful especially for practical purposes and add flexibility. In [60, Section 4.1] and [38], refinement, clustering, and selection have been discussed. Just to give an example, selection is typical for adaptivity applications and can be characterized as follows" For each j we select a subspace Vj* C (both extremes Vj* - ~ and Vj" - {0} are allowed!), and form the new, selected space V* - ~ j Vj*. Due to the definition of the triple-bar norm,
Multilevel Solvers for Elliptic Problems
11
establishing the stability of the new splitting {V*; a } - ~-'~.{Vj*;bj} , J
requires only the verification of the lower estimate, while the upper estimate is preserved from (2.3), with the same or a better constant. A further simplification comes for the case of splittings into direct sums of subspaces where the infimum in the definition of the triple-bar norm can be removed: in this case any selection leads again to a stable splitting, with the same or a better condition. Splittings into one-dimensional subspaces are of particular interest. Let Vj be generated by the (nontrivial) element fj E V. It turns out that then the stability condition (2.3) is equivalent to the frame property of the normalized system 1
v/bj(fi, fi) fj in {V; a} (or, equivalently, in V equipped with the original scalar product). Recall that a system {gj } in a Hilbert space V is called frame if
J
(see [30, Section 3.2] for the definition and properties of frames). Indeed, one easily computes that
a(u, fj) ~ u - ~_ _ ~j-(]j : -f; ) f j ,
(2.10)
which together with the stability of (2.2) implies
a(u, u) "~ a(7)u, u) -- ~ la(u, ]j)l 2 . J Alternatively, look at [30, Proposition 3.2.4] and compare with (2.3). Riesz bases, which are, by definition, minimal frames, are particularly attractive since they lead to stable direct sum splittings of V. Another reason for the interest in frames or Riesz bases is that, based on their stability property, one can derive many other computationally interesting stable splittings. Examples are provided in [60, Section 4] and [58] for finite element applications. The explicit formulas for 7) (2.10) resp. for the subspace mappings 7) and the ej associated with the given functional (I) in the right-hand side of
Tj " u C V,
, Tju-
a(u, fj) vJtJJ,JJC-/-7:)-)fJ e Vj
O(fj) ( r = bj77-7 )
)
(2.11)
12
P. Oswald
are useful to derive matrix representations suitable for the implementation of the algorithms explained next. Let us now briefly outline the basic algorithms associated with a subspace splitting (2.2). For this purpose we make the natural assumption that all spaces involved are finite-dimensional and that their number is finite: ./
(2.12)
(V;a}j=O
Note that stability itself is obviously guaranteed; the question is the size of ~(P). The iteration step of the additive algorithm A is given by J
un+l -- un -~- ~ E
rj(un) ~
= Rj(r
(2.13)
- Tju).
j=0
The same amount of work is formally needed to perform one iteration of the multiplicative algorithm M V0
---
It n
vJ+ 1
=
~tn+l
--
Vj .+. w r j _ j ( v j ) , vJ+l
j=0,...,J
.
(2.14)
The role of the relaxation parameter w is analogous to the classical iterative methods (as a matter of fact, (2.13) corresponds to the extrapolated Jacobi (resp. . Richardson) iteration while (2.14) generalizes SOR). Note that these are stationary linear iterative schemes (in the usual terminology; see [41]); the iteration operators for the two algorithms are MA = I d - w P
,
MM = ( I d - w R o T o ) ( I d - w R i T 1 ) . . . ( I d - w T j )
.
The convergence theory (which is trivial for the additive algorithm) is covered in [65, 67, 11, 38, 60]. We quote the following result from [38], or [60, Theorem 18]. T h e o r e m 1. Assume that V is finite-dimensional and that the algorithms A and M are defined with respect to (2.12). (i) The additive algorithm A converges for 0 < w < 2/Amax(P). The optimal convergence rate is achieved for ~* = 2/(Amax(P)-bAmin (P)), and equals PA *
-
2 min I [ M A l l . - 1-. 0<~<2/~=~. 1 + to(P)
(2.15)
13
Multilevel Solvers for Elliptic Problems
(ii) A s s u m e (2.8) and (2.9). Then the multiplicative algorithm M converges for 0 < w < 2. The (analogously defined) optimal convergence rate can be estimated by
(p~)2 < 1 )Imin(~D) -2~m~x(r) + 1 "
(2.16)
W i t h o u t A s s u m p t i o n (2.8), one still gets 1
(p~u) 2 < 1 - l o g 2 ( 4 ( J + 1)). ~(P) "
(2.17)
Intuitively, it might seem that the multiplicative algorithm M should perform better than A (and this is indeed the case for many standard applications, and parallels the experience with Jacobi- and Gauss-Seidel methods for specific classes of linear systems); however, it was shown for some exotic Toeplitz systems [59] that, in general, the logarithmic factor in (2.17) cannot be removed. Note that the choice of w is only sensitive for the multiplicative algorithm. If A is replaced by the conjugate gradient iteration applied to (2.4), one essentially has the same iteration step (2.13), with an automatic choice of w = w(u n) (the latter requires some additonal storage and computation of scalar products). Moreover, the cg-iteration results in an even better estimate for the average convergence rate: paVer
~.
~ 1-
2 1 + V/~(P)
.
(2.18)
There are numerous modifications of the multiplicative algorithm, and refined theories which serve some applications better and lead to sharper estimates under special circumstances. We mention a symmetrized version of M, the s y m m e t r i c multiplicative iteration SM, which is the abstract counterpart of the SSOR method. It is approximately twice as expensive as M and combines two steps of (2.14) (the second performed in the opposite order) into one, see [65]. The iteration operator takes the form MSM = ( I d - w T j ) . . . ( I d - w R 1 T 1 ) ( i d - w R o T o ) ( I d - w R 1 T 1 )
. . . (Id-wTj).
A more general version of SM, the variable symmetric multiplicative algorithm, has been popularized by Bramble et al. (see [11, Algorithm III] for a general multigrid exposition). In a multiscale environment, the general recommendation is to allow for more subspace correction steps corresponding to small j (which are the low-dimensional subspaces and, therefore, the unexpensive subproblems in (2.5)). The benefit is that weaker assumptions suffice to state optimal convergence estimates without increasing the arithmetic complexity of the iteration in the asymptotical range (J ---+oo). For details, we refer to [11].
14
P. Oswald
We finish this section by discussing the matrix representations of the above algorithms for the multiscale setting. This also leads us to an understanding of some implementational issues. Assume that the ~ are an increasing multiresolution scale (1.1) of finite-dimensional subspaces of dimension nj, with a designated algebraic basis, which we denote by 2V'j -- { r .... ,n i. In wavelet (resp. finite element)discretizations, the Cj,i are dilates and translates of the scaling functions resp. nodal basis functions. All matrix representations will be with respect to these bases. According to (1.1), there are representations nj
aj ;i,i , Cj,i ,
C j - l,i' --
i'-
1 , . . . , nj - 1 ,
i-1
the coefficients of which enter the matrix representations Ij of the natural embeddings Vj-1 ~ Vj. More precisely, Ij is the nj x nj_l matrix given by Ij = ((aj;i,i'))i=l,...,nj;i'=l,...,nj_l which transforms the coefficient vector of u E ~ - 1 with respect to Afj-1 into the coefficient vector of the same u with respect to Afj. We first discuss algorithms associated with (1.2). We denote the stiffness matrices of the bilinear forms a(., .) (restricted to V)) and bj(., .) with respect to Afj by Aj and Bj, respectively. Thus, the entries of Aj are a(r Cj,i,), i, i t : 1 , . . . , n j , analogously for Bj. It is now obvious that the matrix representations of the operators 7) : Vj ---. V) are given by B?I/?+I
j - O,...,J ,
"''ITAJ, J
and that the additive operator P j - }-~j=0 RJTi acting in Vj leads to the matrix expression J
Z Ij'." j=o
Ij+IB;II?+I
... ITAj
- CCAj.
(2.19)
Note that the preconditioning matrix C A is formally independent of A j; it depends only indirectly on a(., .) via the choice of the Bj. The intergrid transfer operations behind Ij and I T can be interpreted as prolongations and restrictions, respectively. Finally we mention that the matrix I d j -~oCAAj corresponds to the iteration operator MA. Here and in the following, Idj stands for the identity matrix of dimension nj. The recursive structure behind (2.19)should be emphasized. The preconditioning matrix CJ t can be defined recursively as C A - No I ,
Cr - IjCr
? -Jr-N; 1 , j -
1,...,J.
(2.20)
Multilevel Solvers f o r Elliptic Problems
15
An analogous structure is exhibited by the multiplicative method. induction, one verifies that C M - B o l
By
- B? 1 , j-
C f ff - I j C ~ I _ l l f ( Z d j - o o A j B j - 1 ) - {
1,...,J,
(2.21)
leading to the matrix expression Ida - r for the iteration operator M M . While (2.20) can be implemented directly for use in a preconditioned cg-iteration, the implementation of the multiplicative algorithm is usually performed in a different fashion, as a multigrid V(1,~ as explained in [11, 41]. What we wish to show with the formal recursion (2.21) is that the two methods look similar, with the difference that the multiplicative method M contains an additional residual evaluation. The latter is therefore slightly more expensive, and needs the so-called Galerkin coarse grid matrices
fi~j - I T + l ' ' ' I T A , I.1 ' ' ' I j + l .
(2.22)
Provided that all entries of Aj are computed exactly, one easily verifies that flj - A j . Still these matrices have to be precomputed and stored. However,
in many other cases (when using quadrature rules, for perturbations of the nestedness condition, etc.) the matrices Aj are different from A j , and may change if J is increased. Then additional considerations are required (see, e.g., [11, Sections 4-7]). In the remainder of this paper we will be concerned only with the properties of additive preconditioners as defined in (2.20). We do not further discuss the multiplicative (and more general multigrid) algorithms which are a valuable and flexible tool in practice. In order to justify this "negligence", let us just recall that Theorem 1, (2.17), guarantees that knowledge about the additive iteration A, and thus about the characteristics of the underlying stable splitting, also leads to sufficiently good convergence statements for some simple multiplicative algorithm. Since our examples below are mainly frame-based, i.e.we typically consider the refinement j
{Vj;a} - Z
nj
Z{VJ, i;a}'
l~,, - span{Oj,i},
(2.23)
j=O i=l
of the splitting (1.2), it is worth mentioning that in this case By-1 is just a diagonal matrix" By - diag{a(r
r
9
This follows from the explicit formula for the additive Schwarz operator P associated with (2.23)which can be derived from (2.10). More involved approximate solvers on the subspaces ~ are possible but will not be considered. The above choice is extremely simple but incorporates at least a
16
P. Oswald
minimum of information from Aj into the subspace corrections. Note that the only components of the CA-recursion which change if (1.2) is replaced by (1.3) are the operations B~-1. Assume that we have a Riesz basis in V (resp. in Vj) associated with the splitting (1.3). The basis functions that span Wj are denoted by Cj,i, i - 1 , . . . , mj, where mj - nj - nj_l, j > 1. For the preconditioner which results from the Riesz basis H0 u
u...u
one would need to put
B/1
-
-
IjBTllf
,
[~j
-
-
diag{a(r
~)j,i)}i=l,...,mj
, j > 1,
in the recursion (2.20). Here, the nj • mj matrix Ij describes the natural embedding Wj ~ t~, and contains as entries the mask coefficients in the expressions nj
Cj,i' -- ~ aj,iCj,i 9 i=1 This, and the precomputation of the diagonal matrix/~j-1, are the places where the choice of the Riesz basis adds to the arithmetical complexity of the preconditioning operation. Other advantages (e.g., better stability estimates or robustness properties) should compensate for this drawback. Some examples of "cheap" finite element Riesz bases are available, see [50] for a survey. w 3.1
S u b s p a c e s for b o u n d e d d o m a i n s
C o n s t r u c t i o n of Vj,~
Throughout the paper, we use the following notation. Let ~ E IRd be a bounded open d-dimensional domain, and 0fl its boundary. We assume that ~ possesses the extension property for the scale of Sobolev spaces H s (for it to hold, the uniform cone condition would be sufficient, see [64, 1]). Let the Euclidean space I~ d be partitioned into cubes of sidelength 1 such that the origin is a vertex of one of the cubes. The collection of all these socalled 0-cubes will be denoted by TO0 (partition of level 0). The partitions Tdj of level j >_ 1 into j-cubes of sidelength 2 -j will be obtained from TO0 by dyadic dilation. Let
t~ -
S~ (Tij ) N L2(IR d)
(3.1.1)
be the L2-subspaces of tensor-product splines of degree k and smoothness r with respect to 7~j where 0 < r < k - 1. Obviously, { ~ } is an increasing
17
Multilevel Solvers for Elliptic Problems
sequence of subspaces of the Sobolev spaces Hs(]Rd), 0 < s < r + 3/2. Alternatively, the V) could be defined by dyadic dilation from V0" Vj = {u(2J.) 9 u E V0}. Note that ~ locally contains all algebraic polynomials of degree < k. We fix the local and L2-stable basis of tensor-product B-splines {r in ~ , see [63]. This basis has the remarkable property of local linear independence" If uj E V~ vanishes on a j-cube [] then cj,i - 0 for coefficients in the B-spline representation corresponding to all basis functions Cj,i which do not vanish identically on []. For our convenience, we introduce the notation wQ for the set of indices i such that Cj,i does not vanish on the j-cube []. Thus, local linear independence is equivalent to cj,iCj,i(x) - O, x E []
==~
cj,i - 0, i E w 9 9
(3.1.2)
i
A consequence of the local linear independence property is the existence of well-localized biorthogonal functions" For any j-cube [] C supp Cj,i (or, in other words, for any i E wo) there is a function r/j,i E L ~ (IRd) supported on [] such that V i, i' .
o r]j,iCj,i, dx -
(3.1.3)
As is obvious from the translation-dilation invariance of all constructions, the rlj,i can be obtained as scaled translates of dilates of a finite number of functions associated with the unit cube u0 - [0, 1]d. The j-cube associated with rlj,i will be denoted by 9 It will be fixed depending on the specific setting. If no explicit choice is made, then any j-cube in supp Cj,i will serve. We introduce some modified basis functions which will be used for the boundary modification below. Consider the finite-dimensional space X0 - V0iDo (which in this specific case coincides with all polynomials of coordinate-wise degree < k). It contains all monomials x ~, ic~l < k, the set of which can be complemented by some other functions to yield a basis in X0. Let {r (i E WOo) denote this basis, and {r/ 9149the corresponding biorthogonal system in X0, i.e., /o
dx - hi,i, ,
r
i,i' E WOo 9
(3.1.4)
o
The same notation r will be used for the extensions to S[: (JRd) obtained as follows: For the monomials x a, lal < k, there is a unique representation x ~ -
el i
x
18
P. Oswald
while for the complementing basis functions the minimal extension is used, i.e., the B-spline coefficients of the extension vanish for r with i ~ Who (coefficients with i E WOo are uniquely determined by the spline values on Do as follows from (3.1.2)). This construction is illustrated for the bilinear case ( d - 2, k - 1, r - 0) in Figure la-d. The upper row shows the nodal values at the integer points near Do of the extended Cno,i corresponding to the monomials 1, Xl, x2, and one arbitrarily fixed complementing basis function, respectively, while the second row depicts the nodal values of the bilinear functions on Do defining r/no,i. By translation and dyadic dilation we obtain systems {r and {r/n,~} (i E wn) for any j-cube D and all j > 0. To be definite, and in order to preserve the biorthogonality relation (3.1.4), we apply scaling (by a factor 2jd) only to the y-functions. In the final construction, suitable restrictions of the extended functions Cn,i will be used, see below. .0 .
.
.
1
1
1
-I
0
1
1
1
I
0
0
1
1
1
I
-1
0
I
0
0
0
0
1
0
0
I
1
1
-I
0
I
-1
-1
-1
0
0
0
0
.
0
0 0
"0
1
i 0
0
_0
o
I0
-14
-6
6
6
-18
-2
I0
-6
6
-6
18
a)
b)
c)
d)
181 -18
-8
4
16
-8 e)
Figure 1. Bilinear elements: Nodal values for r and y-functions. From now on, we assume that generic constants C, c , . . . (also those occurring in ~ and ~ relations) may depend on k, r, s, and ~ but are independent of other parameters, especially of j, i, l, and the functions involved. For each j > 0, we define the sets F~j C ~ C ~ as unions of j-cubes: ~j-U{DET~j
9[ ] C ~ } ,
~--U{DET~j
9DNgt#O}.
(3.1.5)
We require the following geometric property of ~" (G1) For each j-cube D E fl~ and any l - 0, 1 , . . . , j, there is at least one /-cube [:1~ C ~j at a distance < C2 -z from D. The constant C is assumed to be independent of D, j > 0, and 1. Roughly speaking, this condition means that the domain has a sufficiently "fat" interior and a regular boundary. In particular, (G1) implies F~0 ~ 0. Though restrictive, (G1) seems to be rather natural if robustness of a
Multilevel Solvers for Elliptic Problems
19
geometric multilevel method is expected. Asymptotically (i.e., if required only for j0 _ 1 _< j and some sufficiently large j0), the above condition is satisfied for domains with a Lipschitz boundary (resp. the uniform cone condition). By wj (resp. cOwj) we denote the sets of all indices i such that the support of the tensor-product B-spline ej,i intersects ~j (resp. intersects f~ but not ~j). These are the sets of interior and boundary indices of level j. Let Cj be a family of j-cubes rn C ~j near the boundary of f~ satisfying the following properties: 9 No function
ej,i
contains two different cubes from Cj in its support.
There is a constant C such that for each i E cOwj there exists a cube []i E Cj at a distance _~ C2 -j from the support of ej,i. The second property implies the existence of a partition of the set of boundary indices cOwj into small sets cOwo, [] E Cj, such that for i E cOwo the distance condition is satisfied with this particular [] (we will assume that 0wo is nonempty, otherwise the corresponding cube [] can be excluded from Cj). The existence of the families Cj easily follows from (G1) for l - j; the constant C depends on the constant in (G1) and k, r. Analogously, the set of interior indices wj decomposes into the pairwise disjoint sets w 9 (• C Cj), and a possibly larger "remainder" set ~j - ~j \ uoecj ~o .
Figure 2 schematically illustrates the definitions of ~j, ~$, and Cj for the bilinear case. The j-cubes in Cj are given by hatching. The numbers at nodal points associated with w 9U cgwo indicate the number of the corresponding [] C Cj, and the unnumbered points correspond to indices from wj. Since for i C wj, the support of ej,i contains at least one j-cube [] C ~j (which is not in Cj !) we can fix ~j,i such that its support clj,i is in supp ej,i n ~j. Figure le shows the nodal values of the biorthogonal function for j - 0. The general case follows by scaling with a factor 2jd. We come to the description of the boundary modification. Roughly speaking, only basis functions ej,i with i E w 9 for j-cubes from Cj will be changed. For notational convenience, define the restriction operation M~ associated to an arbitrarily given set w of indices of level j by
uj - ~
cj,iej,i E S~(Tij), i
~ M~ouj - ~
cj,iej,i e S~(Tij) .
(3.1.6)
iEw
If w is finite, the mapping is clearly into I~. The restriction of M~ onto Vj is an L2-bounded operator, due to the L2-stability of the B-spline basis. For each [] E Cj, we replace the associated set {r 9 i E r by a set
P. Oswald
20
10
10
10 ~_ _
I
2
10
9
10 1
1
2
3 2
3 3
8
L
7
8
~
~
8
8
3
7
7
~
~
7
6
6
~
5
6
f
7
4
, , 4N4 N / N /
6 7
3
3
9
N9 9 N N
8
2
6
5
~ ~ "
-
55
6
6
4
5
Figure 2. Boundary cubes and nodes for bilinear elements. of "new" basis functions {r 3,$ 9 i E wo} which coincides up to index ordering with {M~ou0~0or (i.e., with the restrictions of the dilated and translated extensions r defined above to some neighborhood of n, compare Figures 1 and 2). The biorthogonal functions r/~,i are identified with rio,i, accordingly. Thus, our construction ensures that
IICj,~IIL~(R~) "< 1, II~j,illL~(Oj,,) ~ IIr
___ 1,
2jd ,
II~/~,illL~(O) ~ 2jd ,
!
i E wj,
(3.1.7)
i E wo,
hold for all o E Cj, and j > 0. In addition, the biorthogonality conditions are preserved. The main advantage of our construction is the relatively simple, cube-oriented local basis exchange which still guarantees local reproduction of polynomials of (total) degree < k in the spaces Vj,a to be defined next. Let (3.1.8) ~ , a - span{r " i E wj } , where
r
I Cj,ila r
if
if
iEwj
!
i Ewo,
oer
21
Multilevel Solvers for Elliptic Problems
The space Vj,a is a subspace of Vj la, the restriction of ~ to f~ (as a rule, the inclusion is proper). More precisely, functions from ~ , a are uniquely determined by their values on fly, and extended to f~ in a specific way. This can be seen from the definition of the biorthogonal system by r/j,i,~which
ensures
I
rlj,i r/~,i
if if
!
i E wj , iEwm, []ECj,
Dj, i -- supp rlj,i C f2j to hold, and
u - Qju - E
Aj,i(u)r
f Aj,i(u) - 1=
iEwj
~Tj,i,audx,
(3.1.9)
doj,,
for all u E Vj,a since o
rlj,i,flCj,i,,a dx - hi,i, ,
i, i' E wj .
(3.1.10)
j,i
The spaces Vj,~ still have the same approximation power (with respect to the Sobolev scale) as V)I~ since the boundary modification allows for local reproduction of polynomials of total degree < k. Indeed, the quasiinterpolant operator (3.1.9) is well defined for functions u E L I ( ~ j ) and maps into ~ , a . By construction, Qj is a projector onto V),a. Also, if u coincides on fly with a spline function from V) or from S~(Tij) then its values are preserved on ~j. By a standard argument, we can prove
[IQjulIL,(~)
d
II~IIL~(~j),
1 _ p _< ~ .
(3.1.11)
To this end, observe that on each ra N f~ at most a fixed number of terms in (3.1.9) does not vanish. This gives
Ilqj l'IL~(Ona)
I
-
i E w j 9O n s u p p C j , , , n # 0
for all j-cubes D. The generic term in this sum is the Lp-norm of a function of the form fo, r/u dx. r where r 7/satisfy (3.1.7), and D' C f~j is a j-cube at distance < c2 - j from D. This implies
II/ , and
IIQjull (on )-<
E O ' e f l ' 9d i s t ( O , O ' ) < c 2 - J
I1 11[ (o,) 9
P. Oswald
22
Now summing over all j-cubes D from Ft~ and observing that each Lp (D')norm term is repeated only finitely many times, we conclude (3.1.11). We gave the proof only for the sake of completeness. On later occasions, analogous considerations are left to the reader. To show approximation estimates in H ' - n o r m s (0 _< s _< r + 1) for u E H t, s < t < k + 1, it suffices to consider integers s - m - 0, 1 , . . . , r + 1, t - k + 1, and to prove the norm estimates V u E Hk+l(D) .
(3.1.12)
The general case (as well as estimates in terms of moduli of smoothness) can be concluded from (3.1.11) and (3.1.12) by interpolation methods. To prove (3.1.12), let E : HS(~) ---. H s ( R d) denote the bounded extension operator. Actually, here we need only s - k + 1. Note that each
uj - iEwj
has a natural extension to ~ (which is different from Euj) given by
u,...,j - oECj iEwo
iEw~
Moreover, Qju - QjEu, and as a by-product of the proof of (3.1.11), we have for the natural extension of Qju
]]QjU]]L~(l~) '< ]]U]]L:(nj) . Next, the obvious inequality
I]u - Qju]]2H.,(n) <_ ~ ]]Eu - Q~---Eu]]2H,,,(o) OC~ reduces the global estimation to local estimations on each D C ~ . By construction r C w~ for all D C fl~. Therefore, to each D E ~ and i E wo we can associate a j-cube Di C flj according to the following rules" If i ~ wjI then i E To. U COwo. for some j-cube D* E Cj and Di -- D*. Otherwise, if i E w j ' then set Di -- Dj,i (C suppCj,i). In all cases, these cubes are at a distance < C2 -j from D. We choose U(D) to be the smallest cube (say, a union of j-cubes) containing D and all Di, i E To, simultaneously. The diameter of U(D) is bounded by C2 -j, with an absolute constant C which depends on the constant from assumption (G1). Let p denote an arbitrary polynomial of total degree < k. By the above construction it is clear that Qjp - p on ~ (to see the coefficient
Multilevel Solvers for Elliptic Problems
23
reproduction in the representation of p with respect t o {r for indices i E ~on U &on, look at the definition of the modified boundary functions Cn,i (part of which coincide on rn E Cj with the monomials of degree < k), and their biorthogonal counterparts). Hence
lieu Qj'--EulI2Hm(~)<_2 ( l i E u -
-
pll#,,,(=) + ilQj(E-~- p)ll~..(,-,))
-'<
lieu -
pll~(~) +
~_
lieu
pll~.,(o)+
--z, -
LIEu
-
-
22r
-
2 P)IIL~(~)
~-'~ 22JmllEu _ pl 21L2(n) iEwa
p]! H 2 -, (U(O)) §
22Jml[ Eu
-
Pl IL~(U(=)) 2
9
Taking the infimum with respect to all polynomials of degree _< k and taking into account the size of the cube U(EI), by the Bramble-Hilbert lemma we arrive at l i e u - Qj'--Eu{12H,.(m) ~- 2-2(k+~-m)JllEull2
H"+~(U(O))
9
After summing up with respect to o C f~, this yields (3.1.12). (Note that by the extension property the Hk+l(IRd)-norm of Eu is bounded by the Hk+l(f~) norm of u. The norm of the extension operator E enters the constants.) The outlined argument establishes Lemma 1. Let f~ have the extension property and satisfy the above geometric assumption (G1). Then the quasi-interpolant operator Qj defined in (3.1.9) provides good approximation order in Sobolev norms: For all O < s < r + l and s < t < k § l we have
I]u- QjUllH,(a ) <_ c2(t-~)JllullH,(a )
V u E Hk+l(~) .
(3.1.13)
Moreover, the operators Qj are projectors onto t~,a. They also preserve the values on f~j of spline functions from ~ (resp. S[:(Ttj)). By Cea's lemma and the Aubin-Nitsche trick, this lemma ensures optimal approximation rates for solutions of discretized (with respect to {Vj,a}) elliptic variational problems in Sobolev spaces on f~ which are comparable with the rates for traditional finite element schemes. Let a(., .) denote a generic HS(f~)-elliptic bilinear form, in particular, we have [[u[[~,(a) v a(u, u) ,
V u E HS(f~) .
(3.1.14)
We fix s satisfying 0 < s _< r + 1, i.e., we consider only elliptic problems of positive order which is a restriction of our approach. For these s, we
P. Oswald
24
have V),~ C Vf~- HS(f~) (resp. k) C V - H'(IRd)). There is a possibility to extend the results by additional considerations to the range r + 1 < s < r + 3/2. We, however, do not know of any practical needs. Cases of practical interest for the bilinear case, namely s = 1 (second order elliptic boundary value problems with Neumann or Robin boundary conditions) and s = 1/2 (hypersingular integral equations) are still covered. Let ff be a bounded linear functional on H-S(f2). Then, by the LaxMilgram result and Cea's lemma, the variational problem Find
u e Va
a ( u , v ) - ,~(v)
such that
V v E Va,
(3.1.15)
V vj e t~,a
(3.1.16)
as well as the finite-dimensional problems Find uj E ~ , a such that
a(uj, vj) - r
possess unique solutions u - ur E Vf~ (resp. uj - uj,r E Vj,a) for which Ilu-
ujllv
-< vjEVj,~ inf
vjllv .
(3.1.17)
This estimate in the energy norm I1" IIY~ leads, in conjunction with Lemma 1, to asymptotical a priori error estimates if additional regularity of u (e.g.u E Hi(f2) for some t > s) is known. A general treatment of this topic is beyond the scope of this paper. 3.2
C o n s t r u c t i o n of Vj
Since usually k~,f~ C ~+l,f~ does not hold, we have difficulties to use the sequence { ~ , n } directly for a multilevel scheme (or a multiresolution analysis) in HS(~). Instead, we construct an auxiliary sequence {Vj } of subspaces of ~ , where each Vj is defined as the sum of one-dimensional subspaces V~,i spanned by the B-spline basis function r J (3.2.1) l--O iECoj,!
The choice of the index sets ~j,z, 1 < j, depends only on f2j, and the splitting (3.2.1) defines automatically an additive Schwarz preconditioner for potential variational problems on Vj. This preconditioner will be investigated in the next section, together with a brief description of the algorithmical switch between t~,f~ and t~. Here, besides the construction of k), we establish the theoretical properties of restriction and extension operators 9
--, v ) , n ,
acting between the sequences (3.1.8) and (3.2.1).
,
Multilevel Solvers for Elliptic Problems
25
F i g u r e a. The domains {~}, I _< j, and the associated cube partitions. Let D > 1 be an integer which will be fixed later. We temporarily fix j and define a monotone sequence of sets
~ j + l -- ~j C ~j C "'" C ~0
(a.2.2)
according to the following rule" ~z (l - j , j - 1 , . . . , 0) is the union of all (1- 1)-cubes with a / I - d i s t a n c e < D . 2 -z from ~Z+l. Figure 3 shows the construction of {(~z} for j - 2, D - 1, and a fictive ~3 - f~2 (which is the hatched region). After this we define &t - {i " supp Ct,i C ~z},
/-0,1,...,j,
(3.2.3)
and assume that the above D was chosen sufficiently large such that ~oo C ~l
(3.2.4)
for all/-cubes rq in ~/+1 and l - 0, 1 , . . . , j . In the bilinear case, D - 1 would be enough to guarantee (3.2.4), compare Figure 3. The construction
P. Oswald
26
of I~ described above on the basis of (3.2.2), (3.2.3) is an example of nested basis function selection from the infinite splitting ~j~176 ~-,i l~,i as discussed in [60, 4.2.2]. As we will see below, the corresponding systems of basis functions are frames in their spans with respect to HS-norms, 0 < s < r + 3/2. From (3.2.4)for l - j, Lemma 1, and the geometric assumption ( G 1 ) we see that and dim V),a _< dim l)j <_ C dim V),a (for sufficiently large D one could also enforce Vj,a C I?jla; compare the geometric conditions for the boundary modification required in Subsection 3.1). We define Rj " ~tj e (/j ---, Qjftj e ~ , a (3.2.5) as the restriction of the quasi-interpolant operator (3.1.9) from Subsection 3.1 to IYj. Obviously, Rjfij - fij on f~j , (3.2.6) and the boundedness of Rj in the HS-norm follow from Lemma 1 (set 0 < s - t <_ r + 1). For the extension operator Ej we need some more preparations. The following lemma has a long history; it follows from the basic direct and inverse inequalities in spline approximation theory in a straightforward way (see, e.g., [34], [56], [60, 3.6]). L e m m a 2. For the above defined spline spaces and 0 < s < r + 3/2, we have the following norm equivalencies:
V u E H~(IRd),
IlUlIH,(R~) • II1~111, • Illulll(P,),, where
(3.2.7)
O0
IIi~111~ --
inf U l E Ti/'I " ~t ~ E
~ I~ o U l
2~'" I1~,1~IL2(R~),
l "-- O
and for any family { Pj ) of uniformly L2-bounded quasi-interpolant operators Pj 9 L2(IR e) ~ ~ (preserving locally polynomials of degree _ [s], at least), CX3
IL~(R~) + Y~ 22*~llP*u- PZ-lUlIL~(R~) 9 /=1
We set
J Ej " uj E V~,a ~ Ejuj - Z Mc~,(Pluj - P l - l u j ) E Vj /=0
(3.2.8)
Multilevel Solvers for Elliptic Problems
27
where {Pz} (P-lu - 0) will be a particular family of quasi-interpolant operators (to be formally correct, in (3.2.8) we apply these operators to the natural extension fij of uj to ~ ) . To define Pz we again make use of (G1). The construction is very similar to the definition of the quasiinterpolant operators Qj from (3.1.9). To each Cz,i we associate an/-cube [--]l,i inside or of minimal distance from supp el,i, such that for 1 < j if
["]l,i C ~j
supp r
N ~z r O.
This definition guarantees that []l,i belongs to the support of the basis function r whenever possible. The exceptions are functions (for I <_ j) whose support overlaps Qz\flj. Due to (G1) this set is a thin corridor of "thickness" < D(2 -z + . . . + 2-J) <_ 2D2 -l near cOQ. This implies that the distance between Dl,i and supp Cz,i is either 0 or does not exceed C2 -l. As above we define functions r/l,i with support on []z,i such that the functional
Az,i(u)- /[]
rlz,iud x - cl,i
(3.2.9)
Z,z
reproduces the coefficient of r in the B-spline representation for any u - ~ i cz,ir E S~(TCz) if [::]l,i C suppCz,i, resp. only for all polynomials of total degree _< k if [:]l,i ~ suppCz,/. Since, in both cases, the coefficient is uniquely determined by u[oz,,, the construction of these r/t,i reduces to a local problem which will be first solved for the unit cube and then transferred to [::]z,i. Due to the restriction of the distances between Ol,i and supp Cz,i, we can again assume II'Jz,~liL=
_~ 2 zd 9
The above properties automatically yield uniform local Lp-boundedness, and local reproduction of polynomials of degree _ k of the quasi-interpolant operators
Pl " u E LI(]Rd) '
) Pin - E
~z,i(u)r
,
1 - 0, 1, . . . .
(3.2.10)
i Under these circumstances, Lemma 2 is applicable (the proof is outlined in [60, Chapters 2 and 3]).' Moreover, for l = j, if u coincides with a spline from S~(Tiz) on ~j then Pju = u on ~j. Indeed, according to the above rules for all Cj,i with support intersecting ~j the j-cube [::]j,i will be chosen inside ~j ('i supp Cj,i, in which case Aj,/ reproduces the B-spline coefficient corresponding to Cz,/. These observations, together with the definition of {~t} and &z, l _< j, ensure that the extension operator Ej from (3.2.8) is well defined on ~ , ~ (only information from ujiuj is needed after the application of the
28
P. Oswald
restriction operators M~,), and preserves the values of uj on f~j. As a consequence, we have Rj Ej uj - uj ,
(3.2.11)
V uj C ~ , a .
Moreover, if E" HS(f~) ~ H'(]R d) is again the bounded extension operator then by Lemma 2 and the L2-stability of the B-spline bases
IIE ujll z,(R )
=
[]EjEuiII2H.(R~)
J "< ~ 22J'I]M~,(PI
-
2 Pl_l)Euj[In~(n~ )
I=0 OO
~
22J'[]( P'
-
P ' - l l Z u j [ 2[L,(R,)
1=0
_-< IIEu
_-< Ilu II=H,(a) 9
Subsumming these results, we have Lemma 3. Under the above assumptions for the construction of ~ , ~ and k~ (which include (G1)) and the HS(f~)-elliptic form a(., .), we have for the extension and restriction operators (3.2.8), (3.2.5) the property (3.2.11) and the estimates a(Rjfij,Rjfij) < C[[fij[[~/o(R~) , and
IIECur
< Ca(uj, uj) ,
V ffj
C k~
,
V uj E Y~,n.
(3.2.12)
(3.2.13)
The constants in (3.2.12), (3.2.13) depend on the constants in (G1), on D, k, r, s, and the ellipticity constants of a(., .), see (3.1.14). w 4.1
Multilevel p r e c o n d i t i o n e r s of B P X t y p e
H S - p r e c o n d i t i o n e r for Vj
In this section we first deal with preconditioning a generic H ' (IRd)-elliptic problem on the finite-dimensional subspaces Vj given by the index sets ~l (resp. the domains ill, l _< j), as indicated in (3.2.1) and (3.2.3). However, we slightly generalize the construction of the subspaces to include some other interesting situations such as discussed in Subsection 5.2 (boundary refinement) and 5.3 (adaptive nested refinement). Consider an arbitrary but fixed j > 0. In contrast with the above construction in the particular setting of Subsection 3.2, we start now with
29
Multilevel Solvers f o r Elliptic P r o b l e m s
a sequence of pairwise disjoint sets ft~ which should be unions of/-cubes (or empty), 0 _< 1 _< j, and set ~ l -- I..Jl<_l'<_j ~l* ,
~l
-- I'-JO<_l'<_l ~l*
9
After this, define as before 05z by (3.2.3), and Vj by (3.2.1). Roughly speaking, ~0 is "computational" domain covered by a cube refinement structure, where f~7, f~, f~l are the regions covered by cubes of level exactly l, _ l, and >_ l, resp. (see Figure 4a for an example). An algebraic basis in Vj is given as follows. Define ~o~* as the set of all indices i E 05t for which the support of the corresponding B-spline intersects the set of/-cubes f~' 9 supp r N f~' :/: 0. We claim that /~j - {r
" i E w~ , l - 0 , . . . , j }
(4.1.1)
is an algebraic basis in ~ . This can be shown by induction. Recall that by (3.2.1) ~ j - {r " i C &z , l - 0 , . . . , j } (4.1.2) is a generating system for @. Set 1 - 0 and take any i E 050\w;. By definition of the index sets, supp r
C ~1 ,
which shows that this r can be expressed by a linear combination of basis functions r with i t E 051. Hence all these r can be neglected. If i C a~ then there is at least one 0-cube in the support of r which also belongs to f~. Recall that ~ restricted to this cube contains only linear combinations of basis functions from V0. By the local linear independence property, we conclude that this r cannot be dropped from the generating system. After deleting all unnecessary r we proceed with i E o51\w~ and so on. This proves that the system/~j is a basis in l)j. We need the following geometric condition. (G2) For each 1 < j and each i such that supp r r fl0, there is an/-cube Again, C is Ql,i outside f~0 at a distance _< C2 -z from the support of r assumed to be independent of l, i, and j. The subspaces I)j defined in Subsection 3.2 fit these definitions if one takes f~7 - Dz\Dl+l. Then (G2) is trivially satisfied since ~0 is the union of (-1)-cubes (and, therefore, of/-cubes for any l >_ 0). A more general example for the bilinear case and j - 3 is given in Figure 4a while 4b shows a domain with a slit where condition (G2) would be violated if the construction were continued for j ~ exp.
30
P. Oswald
II II
,
I
I
E I
I
I
ii I
I
iii
ii'i'
m
a)
b) Figure 4. Illustrations for (G2).
Now we define appropriate/-cubes t3z,i for all possible 1 and i which enter the construction of the quasi-interpolant operators {Pz}, as explained in Subsection 3.2 (see (3.2.10), (3.2.9)). If/>__ j, the only condition is [::ll,i E supp Cz,i 9
(4.1.3)
If I < j we have three cases9 if supp Cz,i C (2z+1 then only (4.1.3) has to be observed, 9 for i with supp Cz,i f)f~* r 0 there is at least one l-cube (denoted by t::lz,i) in the intersection of supp Cz,i and f~*, 9 finally, for the remaining i ~ wz we fix as [::iz,i an/-cube outside f~0 which satisfies (4.1.3) or, if this is impossible, is closest to supp Cz,i (according to the first two cases and (G2), this distance cannot exceed
c2-z).
By Lemma 2, for 0 < s < r + 3/2, the second norm equivalence in (3.2.7) holds. In particular, it applies to all fij E ~ C ~. By construction, we have Pzuj - uj for all l >_j and uj E ~ . Hence, J
" llP0 - -
l
IL~(R') + ~ 22t'll(Pz - P'-~)aJ 121L,(rt') /=1
9
(4.1.4)
31
M u l t i l e v e l Solvers f o r Elliptic P r o b l e m s
Now we observe that according to our specific choices of the cubes rnL,i, for l < j the functionals At,i(fij) given by (3.2.9) vanish if the index i corresponds to the last case. Moreover, on the set f2~* we have Pzfij - fij. Indeed, let us fix I. By definition of the index sets ahz,, each fij E Vj can be split into two parts J uj = i "supp Cz,iClrlo
11=I+1 iEt~l
9
,j
~,
r
where Wl+l has support in QI+I- For i corresponding to the second case (where Dz,i C supp Cz,i N Q~* -f~0\f~l+l), we get by examining the values of the linear functionals Al,i(') Az,i(vz) - ( cz,i 0
if supp Ct,i C f20 otherwise,
)~l,i(Wl+l) -- 0
Since there is no contribution from i corresponding to the third case, we have UJ -- Pl~tJ =
E
Cl,ir
"Jr"Wl+l 9
i "supp r
This shows the coincidence of Plfij and fij on f~}**. In turn, this implies that the difference Pl?-lj -- Vl- l Uj
--
( U j -- P l - 1 u j ) -
:
( W l - W l + l ) -t-
(ttj -
Pl ttj )
E
Cl-l,ir
i 9supp Cz- t, i C ~z
E
Cl,ir
i "supp r
i ECoz
belongs to the subspace -span{r
" i E Cot} .
(4.1.5)
The latter produce, according to (3.2.1), a subspace splitting of I)j, for which we have just proved the following analog of Lemma 2.
32
P. Oswald ..,,,
Lemma 4. Assume that the construction of {az}, {&t} (see (3.2.3)) from the sequence f~ satisfies the above conditions, especially (G2). Let the spaces f/j and Q be defined by (3.2.1) and (4.1.5), respectively. Then, for 0 < s < r + 3/2, we have the norm equivalence J
II~Jll~.(~,,) ~ (111~.~111;)~ -
2 Z 2~J'll'i'llz,~(R") (4.1.6)
inf
'-';'~, '~,=~-o ~, z=o
for all fij E Vj. The constants in (4.1.6) depend on the constants in the norm equivalencies of Lemma 2 and in (G2).
One direction of the two-sided estimate follows from
II~jlI~.(R,) _-_
IIl,X.~lll; < (llla.~lll;) ~ , which is obvious by the infimum definition of the III. Ill-norms. The other direction is the consequence of our construction of the family {Pz} of quasi-interpolants as given above, specifically, of (4.1.4) and (Pz -
Pt-~)~ e v?.
J + ~ 22"11( P, - P~-,)ai il L2(R 2 ~) -< (lll,Z~lll;) ~ < IIPo~.~l Ir_.~(R.,) ~ -- !1,~.~11~.(R e) --
I""
1
The result of Lemma 4 is the final preparation for the construction of preconditioners. On its own, according to the theory of subspace correction methods outlined in Section 2, it yields a frame-based multilevel preconditioner for the ~.-discretization for any symmetric HS(IRd)-elliptic variational problem (0 < s < r + 3/2). To be precise, let 5(., .) be a symmetric positive definite bilinear form on Hs(IRd). In particular,
a(u,
u)x
II,-,11.~.(~,)
V u E H'(IRd).
Consider the variational problem Find
fij e lYj such that
5 ( f i j , ~ j ) - ~(~j)
V 9j e ~ ' ,
(4.1.7)
where q) is a linear functional on ~ . Using/~j defined in (4.1.1) as the standard basis in ~ , the problem (4.1.7) turns into a linear system
Aj~j- )~',
(4.1.8)
9
Multilevel Solvers for Elliptic Problems
33
where the v e c t o r xj - {~gl,i " i E ;o~, l - 0 , . . . , j } coefficients of the basis representation ~j-
contains the unknown
J
E E /=0 i~.to~
of the solution ?~j of (4.1.7). The symmetric positive definite m a t r i x .,4j (resp. the vector fj) are given by their elements
5j;(l,i),(l',i') -- 5(r
CZ',i')
resp.
)~"(Z,i) -- ~(r
9
If no additional precaution is taken it may happen even for integer s that fi,j is rather dense, and contains significantly more than O(nj) nonzero elements where nj - dim l)j. We will postpone presenting a cure for this undesired feature of our above construction until Subsection 5.3, and concentrate first on preconditioning .Aj. What is certain is that the spectral condition number of Aj satisfies (2j)
c2
,
which is typically attained in the asymptotic range, i.e., for j ---, oe. To construct the preconditioner, consider the additive Schwarz formulation (2.4) associated with the splitting (3.2.1) (compare also (2.10))" J 1=0 iE~Sl
J
~l'i -- ~J -- E
E
I=0 iE~z
~ r-d-l:i
9
(4.1.9)
For the scalings dl,i, any choice satisfying
dt,i ~ 221s[lr
2
)
(4.1.10)
will be appropriate. For practical implementations, a good choice is
dz,i - 5(r
r
(4.1.11)
for which (4.1.10) usually holds (for spline basis functions and integers s _< r + 1 this can be verified directly). As was shown in Section 2, the matrix representation of the equation (4.1.9) takes the form Cj ftj xj - Cj fj , (4.1.12) where the multiplication by (~j corresponds to a sparse matrix multiplication (the recursive structure of Cj is slightly more complicated than shown in (2.20), due to the more complicated structure of the basis/~j defining
P. Oswald
34
the stiffness matrix Aj). This becomes obvious if one considers how the coefficient vector ~j of Pj~j is computed from the coefficient vector ~j of fly. To describe the details, let mz (resp. m~ < mz) denote the number of indices in ~z (resp. w~), l - 0, 1 , . . . , j . Note that nj - ~ l m~, and set dz - m z - m~. As a first step, Aj~cj yields the values of 5(fj, Cz,i) for all i E w~ and l - 0 , . . . , j. After this, to compute all remaining coefficients in (4.1.9) and to get ~j by suitable summation, one has to implement a V-cycle algorithm which corresponds to Cj. Since ~hj - wj by construction, we have already all necessary 5(fij, Cz,i) in (4.1.9) for l - j. For any iEr 1, the support of Cj-l,i is completely contained in ~j, i.e., the unique basis representation of Cj_l,l in the B-splint basis of Vj needs only basis functions Cj,i, with i ~ E ~hj. Using the linearity of 5(., .) in the second argument, we can compute the value 5(fij, Cj_ 1,i) from the available values 5(fij, Cj,i,). Thus, after this operation, which can be described by a rectangular matrix Sj of dimension (nj + dj_l) x nj, we get a vector containing all fi(fij, Cz,i), with i E ~l for l > j - 1 and i E w~ for l < j - 1. It is easy to see that a multiplication by Sj can be performed in O(dj_~) arithmetical operations, and that there is no principal need for storing this matrix since its entries come (by using the translation-dilation invariance of the bases) from a finite number of coefficient sets describing the expressions of the few types of different B-splints corresponding to V0 in the basis of V1. This process can now be repeated leading after j steps to the vector S 1 . . . SjAjxj of length nj + dl-1 -Jr-... + do which contains the values fi(fij, r for all i E ~z and 1 < j. Applying a diagonal m a t r i x / ) serves for the scaling by dz,i (which can be precomputed and stored, if necessary). Finally, going the inverse direction from l - 1 to l - j, the vector ~j can be computed by using again the expressions for Cz-l,i with i E wt-l\W~_l in terms of el,i, with i' E Col, to eliminate all terms in (4.1.9) involving Cz,i q~ Bj. A close look to this process yields -
.
.
.
b
&
.
.
.
,
where the multiplication by the symmetric preconditioning matrix (4.1.13) can be performed by O(nj + do + dl + . . . + dj_l) operations. It is easy to see that under the conditions of Subsection 3.2, this operation count can be bounded by O(nj). In general, one has to take care of the behavior of {dz} (and the complexity of a matrix-vector multiplication with .4j) by additional assumptions (see Subsection 5.3). The general theory of subspace correction methods leads in conjunction with Lemma 4 to the following
35
Multilevel Solvers for Elliptic Problems
Theorem 2. Let the assumptions of Lemma 4 hold. Then, for any symmetric HS(IRd)-elliptic~bilinear form 5(., .) (0 < s < r + 3/2), the additive Schwarz operator Pj in (4.1.9), (4.1.10), corresponding to the splitting (3.2.1) of k) is symmetric with respect to 5(., .) and has uniformly bounded spectral condition number
< c, where the constant C depends on k, r, s, the ellipticity constants of 5(., .), and the constants in (G2) and (4.1.10). As a consequence, the number of iterations to reach a fixed error reduction in a Richardson iteration for (4.1.12) (algorithm A) or preconditioned conjugate gradient algorithm with preconditioning matrix Cj for (4.1.8) is bounded independently of j and other specifics (except for the constant in (G2)) of the construction of the subspaces ~ .
To prove Theorem 2, we need to establish the stability of the splitting J /=0 iE&z
expressed by the norm equivalence
5(fij, fij)~
J
~ ~ 22J"ltu+,yii~(n~)
inf
(4.1.14)
ut,ievl,, ~ s = ~ l ~-~iul,, t=o iecoz
which has to be verified for all ttl -
Uj C ~"
~
Cl,ir
iE&z
(compare Section 2). Since any -
~
Ul,i
iECvz
from ~ satisfies 2 v Ilti,llL~(rt+)-
~
2- Idc2,,~ • ~
iEtbz
2 Ilu,,~IIL~(,~,),
iE~z
we get, using the L~-stability of the B-spline basis, J inf ~ ~ uz,,~Vz,,'ai=~l ~ i uz,, l=0ie~z
22J'lluz,jlIL~(rt~) ~ J
inf f+zEVz'(tJ=2zaz
22J'11 ,112
/=0
2
36
P. Oswald
Since by the HS-ellipticity, 5(fij,fij) • Ilfijll~/.(Ra), Lemma 4 gives the result. Theorem 2 can be interpreted as a result about sub]tames generated from the set of all B-splines {r } (see Section 2 for the definition of frames and their connection with stable subspace splittings). Proposition 1. Under the assumptions of Theorem 1, the system
(
1
r
" i E &l , 1 - O, . . . , j }
is a frame in ~. equipped with the HS-elliptic scalar product 5(., .). The frame constants are bounded independently os the specific Vj. They depend on the same quantities as the condition number of 75j. The statement holds also for infinite-dimensional spaces V produced by the above construction if j --+ oe.
Note that the frames fi'] are obtained from the subset ~'j of the set of all B-splines ~" - (r by suitable scaling, and that the scaled version fi-s of the infinite set ~" forms a frame in HS(]Rd), 0 < s < r + 3/2. This is a consequence of the general theory and Lemma 2. Thus, the conditions formulated for the construction of Vj are sufficient conditions for a subset of 9TM to form a frame in its linear span considered as a subspace of H ~(IRd), without disturbing the frame constants too much. Thus, it is easy to construct simple frames with nice properties in subspaces of HS(]Rd). A last comment" If the finite sequences { ~ 9 1 < j}, which appear in the beginning of this subsection, are defined (for different j) as sections of a fixed infinite sequence { ~ }, then the resulting sequence of subspaces Vj from (3.2.1) as well as the index sets &j are increasing in j. Another obvious case where the subspaces Vj as well as the frames ~'~, are monotone in j is given in Subsection 3.2. More general situations arising, e.g., in practical adaptive refinement applications (see [62]), where the resulting frames are obtained by adding and possibly deleting, new functions. These have not yet been studied in a rigorous way. 4.2
H S - p r e c o n d i t i o n e r for Vj,n
We present now a preconditioner for the linear system A j x j - fj
(4.2.1)
which is the discretization of the variational problem (3.1.16) with respect to the basis /3j,a - {r
" i ~ ~oj }
Multilevel Solvers for Elliptic Problems
37
of the discretization space ~ , a described in Subsection 3.1. Recall that we have two types of basis functions: the "boundary-adapted" basis functions Cj,i,a = Mo~oou~ooCn,ila i f / C ~0n for some [] E Cj, and "unmodified" basis functions Cj,i,a - Cj,i]a if i E co}. The solution vector xj of (4.2.1) represents the coefficient vector of the solution uj of (3.1.16):
uj -
xj,iCj,i,fl . iEwj
We do not detail the assembly process of the matrix Aj and the vector fj which involves the typical integrals over f~ (resp. 0f~) for products of basis functions Cj,i (resp. input functions from the variational problem) plus some local transformations corresponding to the newly introduced functions r i E w o . Since the partitions T~j are not adapted to the boundary 0f~, it seems to be necessary to modify existing codes for uniform rectangular grids by some "boundary integration" rules. But this might be the only serious change in this part compared to the situation of a rectangular domain. Now we put together the results of Subsections 3.1, 3.2, and 4.1. Roughly speaking, our preconditioner for (4.2.1) is the result of switching from the Vj,a-discretization to the associated V)-discretization as described in Subsection 3.2. The advantage is that for the latter an asymptotically optimal multilevel preconditioner Cj is already available (see Subsection 4.1), and that the "switch" is just a two-level method and easy to understand. Let us comment that the idea of switching from a given discretization to a closeby discretization of similar complexity for which fast solvers are available is by now standard in the field, and can be successfully used for theoretical and implementational purposes (see [14, 61, 13]). Let us preserve the notation Rj for the matrix representation (with respect to the bases /~j in I)j resp. Bj,a in l ~ , a ) o f the restriction operator defined by (3.2.5)and (3.1.9). This matrix is rather sparse and can be implemented by half of a V-cycle (to compute the B-spline-coefficients in V) corresponding to fijlaj from the given ~j) and some local transformations involving the values of the biorthogonal functions r/o,i for i E ~0n, [] E Cj. The first part can be avoided if the control parameter D is chosen sufficently large; compare the construction of {f~t} in Subsection 3.2.
Theorem 3. Let the bilinear form a(-, .) be symmetric and H'(f~)-elliptic (0 < s <_ r+ 1), and let the remaining assumptions of L e m m a 3 be satisfied. Then
P. Oswald
38
is a symmetric preconditioning matrix for Aj which satisfies
)~max(CjAj ) < C tc(CjAj) =_ ~min(CjAj) '
(4.2.2)
where the constant C depends on k, r, s, the ellipticity constants of a(., .), and the constant in (G1). Thus, applying the pcg-algorithm to (4.2.1) yields uniformly bounded iteration numbers for a t~xed error reduction if the constant in (G1) is independent of j --, oc. The operation count for a matrix-vector multiplication with Cj is O(dim ~,a). Proof: This is a simple consequence of the fictitious space lemmaof Nepomnyaschikh, see, e.g., [38, 66], or [60, Theorem 17], and the norm estimates of the previous subsections. We give the full argument. Let us fix any d 9 H s (IR)-elhptic bilinear form 5(., .). For example, take the scalar product of HS(]R~d). Since the construction of Subsection 3.2 automatically yields (G2), we have the result of Theorem 2:
((Ajd~A~, ~j))- a(~3~, ~j)• a~(~j,~ ) - ((~~, ~)) which yields (by substituting .~j-l~j instead of ~j)
((5~e~,~)) • ((~i71~, ~ ) ) ,
v ~.
Here and in the sequel, ((., .)) denotes the Euclidean product of lRn-vectors where the dimension n should be clear from the context. It is easy to observe that X - Rj A-(1RTAj is invertible (to this end, show the positivity of ~t
((AjXxj,xj))-
~t
( ( f t ; 1 R T A j x j , R T A j x j ) ) > IIRTAjxjll 2
by using the surjectivity of Rj). By the above spectral equivalence of Cj and .~-1, we have 2
((Aj RjCj RTAjxj, xj)) - ((Cjflj, flj)) ~_. ((~j-l~)j, ~)j))_ ( ( A j X x j , x j ) ) ,
(4.2.3)
where ~lj - R TAjxj. Since Rj is onto, for each xj there is at least one s such that xj = Rj ~j. Thus,
( ( A j X - l xj, xj)) - ((7tjflj, ~j)) <_ r
flj ) ) r
~j)) ,
where f/j - A 7 1 R T Aj X -I Rj ~j . Since
( f4j flj , flj ) ) - ( ( R j A ~ I R j A j X
-1 x j , A j X -1 xj)) - ( ( A j X - l x j , xj)) ,
Multilevel Solvers for Elliptic Problems
39
we obtain ( ( A j X - l x j , x j ) ) <_ ((.Aj~j,2j)) for any of these ~j. The particular choice ~j - ~j which indeed satisfies Rjflj - Rjf4-~IRTAj X - l x j - xj shows that equality is attained. This implies
((AjX-lxj,xj))
inf
-
((Aj~j,~j))
~j : x j = R j ~ j
.
(4.2.4)
Now Lemma 3 and the assumed HS-ellipticity of the bilinear forms come into play. Consider all fij E ~' such that uj - Rjftj. By (3.2.12) we have
a(uj, uj) = a(Rjftj, Rjftj) < Cfi(ftj, fij), while the particular choice fij (3.2.11), and (3.2.13) gives
Ejuj
=
satisfies uj -- R j ~ t j according to
5(~j, ~j) = 5(Ejuj, Ejuj) <_ Ca(uj, uj) . Altogether, we arrive at
a(uj, uj) ~
inf
fij : u j = R j f i j
5(fij, ~j),
which by (4.2.3)and (4.2.4)yields tc(CjAj) x n ( X ) = ~(X -1) <_ C . Theorem 3 is established. w 5.1
Extensions
General multiresolution analyses
When analyzing the considerations of the above sections, we see that not too many "spline-specific" properties have been used. A generalization to other types of scaling functions and multiresolution analyses on IRd is straightforward under the following conditions. (A) The scaling functions are refinable:
(~1,...,
L
(~L E
1I
L2(IR) d possess local support and
l = 1,...,L.
P = I flEZ a
(B) We have tz E Ht(1Rd), 1 - 1,..., L, for some t > 1.
P. Oswald
40
(C) The integer translates of the scaling functions form a Riesz basis No in the L~-closure V0 of their span, i.e., L
L
U I=1 /3EZ ~
I=1 /3EZ ~
for all (12(zd))L-sequences. (D) Polynomials of total degree < k can be represented by A/0, i.e., there exist coefficient sequences ~c~a~ such that I.
a
L
l=l
~[fl
j
l,ar ,8EZ ,t
(E) The functions in Af0 are locally linearly independent, i.e., if L
E 4r
I=1 /3EZ ~
z) = o
on the unit cube Do then c~ - 0 for all index pairs (l, fl) such that supp Cz(._ ~) N Do ~: O. Under these sufficient (and partly overlapping) conditions, the above results on approximating and solving HS(f~)-elliptic problems remain valid under the same geometric assumptions on f2, at least, for integer 1 < s < so, where so is the largest integer less than both t and k + 1. The practical implementation is restricted to examples where the support of the scaling functions is reasonably small, and the control parameter D is not too large. Otherwise, boundary modifications and the geometric set-up will become increasingly difficult. The crucial assumption which makes our geometric, cube-partitionoriented approach possible is (E) which is stronger than (C). For results on (E), see [47, 9]. Assumptions (A), (C), (D) are standard, and can be studied on the basis of the refinement equation form (A) (see [16, 30]). The calculation of the optimal smoothness parameter t in (B) is a delicate issue, compare [30, 31, 50]. Most of the papers on multiresolution analyses study the case L - 1, for the so-called multiwavelet case L > 1, see [28]. Assumptions (A)-(D) yield characterizations of Sobolev spaces HS(IRd), 0 < s _ so, analogous to Lemma 2 (see [25] for a survey on multilevel approximations and related function spaces). (E) is essentially used in the definition of the spaces on domains and for the corresponding quasiinterpolants. It is also advisable to assume (E) in order to avoid a very
Multilevel Solvers for Elliptic Problems
41
complicated implementation. We leave it to the reader to fill in the details (some minor changes are necessary such as replacing the Loo-estimates like in (3.1.7) by L2-bounds). 5.2
E s s e n t i a l boundary conditions
For applications to boundary value problems for elliptic partial differential equations of second or fourth order, the case of essential boundary conditions is of importance. This leads to subspaces of HS-spaces characterized by trace conditions on part of 0Q. For simplicity, we only discuss the case of a H~(ft)-problem (pure homogeneous Dirichlet boundary conditions) and integer s - 1 , . . . , r + 1. Also, we avoid discussing regularity problems arising from the geometry of the domain by assuming that Ft is a C ~ domain. If we wish to stay within the framework of conforming discretizations, we need to modify the construction of V),a to make it a subspace of Sg (~). A natural approach is to consider only scaling functions with support in as generating functions for ~ , a . Then, in order to preserve the approximation order, we need to include in the definition (3.2.1)standard basis functions of level > j near the boundary. The wavelet counterpart of this approach has been considered in [44, Section 3] (however, Jaffard [44] did not elaborate on solving the resulting linear systems in practice and on other efficiency and robustness issues). Unfortunately, the construction leads to satisfactory results only for d - 2 while for d >_ 3 the newly constructed subspaces have either significantly larger dimension or still reduced approximation power. This coincides with analogous results of [44]. The alternative would be a more subtle boundary modification or a hybrid construction which uses more flexible finite element partitions and functions to better approximate the problem in a boundary strip. As a whole, the problem of general boundary conditions requires additional thought. Let us again consider the sets f~j C ~ C f~ as defined in Subsection 3.1. Fix integers j, J > j, D >_ 1, and define f~, l <_ J as follows. For l - J, set
~r~ -- (Uoc~..~j\~.~j_I i--1)U (Uoc~'~j_I
.dist(O,O~j_l)
rl) .
Now, if ~l*+l is already constructed ( l - J - 1 , . . . ,j + 1), set
~ -- (Uoc~z\(~z_lu~2hl) F'I) U (Uoc~z_l "dist(O,0~+l)
\a;+l,
f~-O,
l<j.
42
P. Oswald
~ J
a)
b)
Figure 5. Boundary refinement: a) {f~l} and b) {f~t}.
The idea of this construction is to introduce a layer of regular refinement up to level J near the boundary of f~j. The choice of J is dictated by complexity considerations and will be discussed later. Figure 5 shows the construction for j - 2, J - 5, and D - 1. Note that in this hypothetical example f~z - 0 for l - 0, 1 and f17 - 0 for l < 2, respectively. The conforming subspace t~,0 - Vj C H ~ ( ~ ) i s now defined via (3.2.1). In contrast to Subsection 3.1, a boundary modification by introducing new types of basis functions is avoided, at the price of enlarging the subspace by standard basis functions from levels l > j. What we claim is that for d - 2, under appropriate conditions on fl, this subspace ~,0 provides good approximation rates under the usual regularity assumptions on u E H~(f~), satisfies the theory of Subsection 4.1, and still has dimension comparable with the dimension of a (simplicial and boundary-adapted) finite element subspace of the same approximation power. We will assume (G2) from Subsection 4.1. Since, by construction, f~0 ~ j , this is actually a condition on the thickness of the discrete boundaries ~ \ ~ z , l _< J. It is asymptotically satisfied for Lipschitz domains but the constant C in (G2) might be very large (due to the macroshape of the
Multilevel Solvers for Elliptic Problems
43
domain) for the computationally interesting range of small j and J. With this assumption, we can use Theorem 2 to get optimal condition number estimates for the additive Schwarz operator associated with the splitting from (3.2.1) for the Vj,0 discretization of a generic symmetric H~(~)-elliptic problem. All one has to do is to formally extend the symmetric bilinear form from H~ (~) to H ~(IRd). Obviously, the dimension of t~,0 will depend on the number of additional boundary layers j < 1 ~_ J and on the size and shape of ~ itself. We assume again that Ft0 :/: 0. This automatically yields dim t~,0 ~ 2jd. As a by-product of (G2), it can easily be seen that the number of/-cubes in ~ is proportional to 2 t(d-1). The constants may depend on D, the constant in (G2), and the ( d - 1)-dimensional measure of 0~. This yields J
2Jd___dim~,o~ (2Ja-q- E
2z(d-1) ) .
(5.2.1)
/=j+l
The upper bound, which asymptotically represents the correct dimension if D is large enough, yields the desired dim ~,0 ~ 2jd only if J ~ j d / ( d - 1). For larger J, the bound is ~ 2 J(d-1). Now we come to the more difficult part, the estimation of the approximation power of Vj,0. Under the above assumptions (recall that s - 1 , . . . , r + 1 is an integer) we have for m - 0, 1 , . . . , s the estimate
Ilullnm(~,) _< C ~ m - ~ + l / 2 1 1 u l l n . + ~ ( a )
,
u E H~(a) A HS+l(a) , (5.2.2)
in a boundary strip ~6 = {x E ~ : dist(x, 0~t) _~ 6). This estimate does not improve if u is more regular. Since functions from Vj,0 may vanish or are of very limited flexibility in a boundary strip of size x 2 - J , we cannot expect to have error estimates better than O(2 -J/2) in the energy norm (i.e., in the HS(~)-norm), even for very smooth solutions of a boundary value problem in H~(~). L e m m a 5. Let the solution u of a H~(~)-elliptic variational problem belong to H~(~) A gs+l(~). Suppose that uj solves the variational problem discretized with respect to the subspace t~,o C H~(~) as constructed above. Then we have, under the above assumptions and for sufficiently large D, the error estimate
I1 - ujllH,( )
c(2
+ 2-J)llullH,+ ( ) ,
(5.2.3)
analogous estimates holding in other norms. Proof." Again we construct a specific quasi-interpolant operator. To do this we enlarge ~ by adding the set IRd\~j. After this enlargement, the union
P. Oswald
44
of pairwise disjoint sets gt}*, I - j , . . . , J, is IRd. This leads, via (3.2.1), to a space Y~,0 C Vj which contains both ~ and ~,0. The algebraic bases for ~,0 (resp. t~,0) are introduced along the lines of Subsection 4.2, and will be denoted by Bj,o (resp. /~j,0). Furthermore, by taking a sufficiently large D in the above construction, we may ensure the following condition to hold: belong t o ~j,0, and if their supports have a nontrivial (G3) If Ct,i and r intersection (with positive d-dimensionM measure), then necessarily I I l' I <_ 1 ( l , l ' - j , j + 1 , . . . , J ) . Note that one cannot expect local linear independence of the basis Bj,o on all /-cubes [::] E ~ . However, any polynomial of degree _< k has a unique representation with respect to Bj,o where, according to (G3), the coefficients can be determined from local information. This will be used below. The condition (G3) also guarantees a good sparsity of the stiffness matrix with respect to Bj,0, and is important to handle the implementation of the preconditioner in a simple way. With each Ct,i E Bj,0, we associate an/-cube [::]l,i C fl~ inside or at the closest distance to supp Ct,i, and such that [:]t,i C Qj whenever supp Cz,i intersects with ft. By the construction of {fl~'} and Bj,0, the latter condition may lead to a choice for Dt,i outside supp Ct,i only if 1 - J (and near the boundary of f~). By (G2) we therefore have dist(suppCt,i, [::lt,i) <_ C2 -z for all indices of interest. We define J
0
,o
u c
,
-
c
(s.2.4)
l=j iEw~
where the functionals At,i are defined via (3.2.9). The definition of r/t,i in (3.2.9) is such that ~z,i(p) reproduces the coefficient in front of Ct,~ in the basis representation with respect to /~j,0 for any polynomial p of degree _ k, and satisfies the Lob-bound 2 'a
uniformly in i, 1. The following considerations show that this is possible. Fix any of the cubes ~t,i, and the affine transformation that transfers it into the unit cube E30. The function Ct,i corresponds now to a certain function r the support of which either contains o0 or is at distance < C from the origin. Each monomial z ~ with [a[ < k possesses a unique representation
9~
J-I
/
l'=j-I
i
E
Multilevel Solvers for Elliptic Problems
45
where the summation has to be carried out with respect to the basis obtained from/~j,0 by the fixed affine transformation. By assumption (G3) and the local linear independence property (compare the construction of the basis (4.1.1)), the coefficients c'~,io, lal < k, of interest depend only on the local geometry of the transferred cube partition associated with f2z and f2z-1 and, if I = J, from the above distance bound. Since there are only finitely many different possibilities for these local cube partitions, we have an overall bound [caO,iol <- C. By the same arguments as at the beginning of Subsection 3.1, we find bounded functions rl~ such that
P-- E
lal
c~176
fo
~
o
Prl'~dz- c'~
(l~l___k).
If we set rlo,io ' - ~l~l_
/o r/~176 o
E
ca CO,io ~
I~l<_k
coincides with the coefficient of r in the basis representation of p with respect to the transferred basis. Moreover, []r/~,io[[Loo(ao) -< 1. Transferring back, we see that for some/3 6 Z d the function r/t,i(x) - 2Zdr/o,io(21x --/3) has the desired properties. Now, the usual machinery of estimating the approximation error of quasi-interpolant operators can be applied to demonstrate Lemma 5. We omit the details; compare Subsection 3.1 for the main steps of the reasoning in a similar situation. A closer look at the above arguments shows that only for d = 2 and minimal extra regularity u 6 H'+l(f2) do we get the desired, optimal result when using Vj,0 as discretization space for an H~(Q)-elliptic problem: T h e o r e m 4. Let d = 2, and s be a positive integer. In the above construction of Vj,o, set J = 2j, assume (G2), Qo # 0, and choose D sufficiently large to guarantee (G3) and the desired "thickness" of the boundary strip
Qj. * Then
a) The dimension of l~,o satisfies dim ~,o x 2 2j . b) Let u E H~(f~) (rasp. uj,o 6 I'~,o) denote the solutions of a symmetric
H~-elliptic problem (rasp. of its discretization with respect to Vj,o). If u 6 Hs+I(Q) then -
.y
IIH,(n) _< c2-J II llH,+ (n)
9
c) The frame-based splitting (3.2.1) associated with ~,o leads to
ditioner Cj,o for the linear system Aj,oxj,o = fj,o
(5.2.5)
a precon-
P. Oswald
46
representing the ~,o-discretization of the variational problem which satisfies g(Cj,oAj,o) <_ C .
(5.2.6)
The multiplications by Aj,o and Cd,o can be performed in 0(2 2j) arithmetica• operations. For d > 2, the choice J - 2j would still lead to (5.2.5) and (5.2.6), at the expense of a dimension of Vj,0 and work estimates which are larger than the desired ~ 2jd bound by an exponentially growing factor. Alternatively, the choice J - [ d j / ( d - 1)] would asymptotically lead to the correct dimension, but then the convergence rate in (5.2.5) is expected to deteriorate from 2 -j to 2 -jd/(2(d-1)). 5.3
Adaptivity
A formal advantage of multilevel Riesz basis and frame decompositions is that they provide isomorphisms to coefficient spaces for a number of BesovSobolev spaces on domains. This can be used for optimal approximation (resp. compression purposes). See [53, 32, 33, 20] for some information in this direction. Roughly speaking, adaptivity strategies based on coefficient information are successful (and can be justified theoretically) if a suitable decomposition of the function under consideration is available at low cost. This is the case for many applications in signal and image processing but much harder to implement for the solution of operator equations (see, however, [21, 20] for a possible strategy). As a compromise, we outline here an approach which is partly based on heuristic arguments. It is implemented in several adaptive finite element codes [3, 6] (see also [62] for a somewhat different variant), and has led to satisfactory results in standard applications. The ideas will be presented within the framework of Sections 3 and 4, i.e., for approximations to a symmetric HS(f2)-elliptic problem by tensor-product splines, and integers s = 1 , . . . , r + 1. Due to the lack of space, we do not give full proofs. In order to have the flexibility necessary for an adaptive solver based on local refinement, we present the following construction. Let us start with an auxiliary family {f2z, l - j 0 , . . . , J} of disjoint sets where each f21 is a union of/-cubes from f2z. Here, j0 is the number of the "coarsest" level, from where the adaptive algorithm starts (this number is fixed throughout the following exposition), while J indicates the number of the current "finest" level of refinement. Let {r l - j 0 , . . . , J} be another sequence, where f2~ is obtained from f21 by adding some/-cubes from the boundary strip f2~\f~z. Thus, h,C~C~,
h~\hzcf~\f~,,
l-jo,...,J.
Multilevel Solvers for Elliptic Problems
47
As before, we produce the sets
fij - u/_~fi~,
fi; - u~_j fiT,
j - j0,..., J,
which stand for the regions of j-th level refinement, and we require that ft is covered by ft 30" ~. Note that by these definitions ftjo\ft j is a union of (j - 1)-cubes from ~j_ 1. To each ~j we associate the index set d2j - {i 9 s u p p t j , i n ~j r O, supp tj,i N ~jo\~tj -- ~}
(5.3.1)
and define sets of basis functions
l)j - {r
" i E &j } ,
13 - U]=jo/) j .
(5.3.2)
Analogously we introduce wj, 13~, and/)~ For an illustration we refer the reader to Figure 6 where an example with J - 2, j0 - 0 is shown (the meaning of the gray squares will be explained below). ^ e
12-%! ! I//"%1
1
,----4--~ I I11 II I
l"k m !11 I!i
m
/i ....
.
.
.
.
Figure 6. Cube partition for adaptive refinement. It can be shown by induction (see the beginning of Subsection 4.1 for analogous considerations) t h a t / 9 (resp. /)e) are linearly independent sets of functions, and form algebraic bases in l) - span/)l~Sj ~
(resp.
9 e - span/~e Off,o).
P. Oswald
48
However, to guarantee the existence of suitable biorthogonal functions flj,i, with supports near supp Cj,i (i E &j, j = j 0 , . . . , J), and reasonable Loobounds, we need to assume another geometric condition which plays a role similar to (G3) in Subsection 5.2: ((33)' If the supports of two basis functions Cj,i and Cj,,i, from/~e intersect at interior points, then I J - J~l < 1. To make possible a boundary modification in analogy with Subsection 3.1, we introduce a stronger substitute for (G1)" ^
(G1)' For each /-cube o C f2~ and any j0 < I < J there is an /-cube t::l~ C ~t such that a) wo, C ~bt, and no function from/~z-1 contains El' in its support, b) dist([=l, El') _< C2 -t . The above conditions are satisfied in the example of Figure 6 if the bilinear case is considered. Since our sequences {~,} (resp. {~2~)) describe the actual refinement process, the requirement b in (G1)' follows from (G1) if the refinement near the boundary is properly implemented. Assumption a in (G1)' is related to the implementation of (G3)' near the boundary. We do not know whether it can be neglected. Together, (G1)' and (G3)' imply that the corridors Qj\Qj+I - Qj are sufficiently "thick" which is a certain restriction for the refinement process (rapid, local changes of the refinement level seem to be impossible). With (G1)' and (G3)' at hand, we can now finish the construction. According to (G1)', we can select families (~j of/-cubes [] C ~l near the domain boundary such that * assumption a in (G1)' (with o ~ replaced by El) is satisfied, 9 the index sets ~o are pairwise disjoint, 9 for all i E &bz - &~\~bl, there is a cube [] E Cz at distance < C2 -z from supp Ct,i. The gray squares in Figure 6 indicate a possible choice for (~t, l - 0, 1, 2 in the bilinear case. As in Subsection 3.1 we now modify the basis functions Cz,i associated with D E (~t by replacing them by certain locally extended
r
E Ci'i~r i~E~o
The union of the sets wo C ~bo C wo U 0~bz, [] E (~t, which are defined by using the third of the above properties of (~z, cover 0r To guarantee local polynomial reproduction, one has to change some of the ci,i, in comparison to the construction of Subsection 3.1. This is the case if the basis function
Multilevel Solvers for Elliptic Problems
49
Cn,i corresponds to some (shifted and dilated) monomial z a, and if CL,i' is in the support of some (unchanged) basis function Ct-l,i- from/~z-1. Note that, by our above assumptions, no interference with basis functions of levels < 1 - 2 and with modified basis functions of levels < l - 1 is possible. Figure 7 shows the values of the locally extended functions corresponding to the monomials z ~, lal _< 1, for the bilinear case. For simplicity, we assume that l - 0 and that the shadowed cube rn coincides with rn0. The nodal points in r are indicated by circles; the modified values at P are due to the influence of the basis function from/~z-1 corresponding to Q.
(~
~
11 ~
~
Q
,.
,,)
3/4
1
k..J
"//
it, 1r
5/4
(~p
k.
,..1
)1 Or
.1
_
,
,.
~
lc,
"i
Or
7/4 P
,, J
,0
Figure 7. Boundary modification for an adaptive refinement example. Now, we introduce the subspace tYr~ (which is the counterpart to ~ , a from Subsection 3.1) spanned by the basis/~a - {q~t,i,a}, where the basis functions Cz,i,a are defined as the restrictions to f~ of all previous basis functions (the modified r D E dt and the remaining, unmodified Cz,i E /)t), from all levels 1 - j 0 , . . . , J. Again, functions from l)a are uniquely
P. Oswald
50
determined by their values on ~jo, and the restrictions of the spaces IYa and l) to ~j0 coincide. The space 1) can be used to derive preconditioners for a lYa-discretization along the lines of Sections 3 and 4. We state the approximation result generalizing Lemma 1 for s = 0. The proof which is omitted shows implicitly that refinement in regions of low regularity of a function u can reduce the overall error, here with respect to the L2-norm. L e m m a 6. Assume that all assumptions (especially (G1)' and (G3)') are satisfied for the construction outlined above. Then, there exists a quasiinterpolant operator 0 ~" L2(fijo)' ' l)~ such that t'or any u E Hm(f~), m = 1 , . . . , k + 1, and for its bounded extension i t - Eu E Hm(lRd), we have the estimate J
~ 22'mllfi-- C)~ull~(o) _< CllullHm(a) ~
Z=jo oc~
9
(5.3 .3)
Moreover, the restricted operator Of/," u e n2(fl) : , (~eula de/ines a projection onto l)'a, and satis/ies an analogous estimate. This result is complemented by other standard inequalities which easily follow from the construction (the properties of the locally defined biorthogonal functions 7)l,i,a are of special importance), such as the inverse inequality J
(5.3.4)
II~all~m(a)-< ~ ~ 2~'mll~all~L~(o) l=jo oC~l
and the L2-stability of the corresponding basis/)a expressed by J
J
2'( l=jo oC~z
")c
,
(5. a.5)
l=jo iE~ol
which are valid for all J
/tal=jo i~.d~z
and all m - 0 , . . . , r + 1. These inequalities also provide us with a heuristic a posleriori error estimator built from local error indicators which are based on conforming higher-order local approximations. This concept has widely been used
Multilevel Solvers for Elliptic Problems
51
within the finite-element community (see the recent paper [4] for some references). In our context, let W~ be the subspace constructed in the same way as V~ but for tensor-product splines of degree k + 1 (and the same smoothness order r). We do not change the "geometrical part" of the above construction. It can be assumed that ^
(5.3.6)
~ c w~,
and that the basis/~w of I/~/a can be obtained in hierarchical manner from /~v - Ba by adding new functions Cz,i,w, i E &z,w, l - j o , . . . , J, i.e., J
~ - [_J {~,,~,~
9 i e ~,} u {~,,,,~
9i e
~,,~ }.
(5.3.7)
l--jo
Indeed, this can be achieved by switching at the very beginning of the whole construction from the B-spline basis to another, hierarchical basis in S~+I(Tt j) which contains as a subset the B-spline basis functions Cj,i of S~(Ttj). For the bilinear case, the additional functions of such a "hierarchical" basis would simply consist of tensor products of a quadratic bubble function in one direction with quadratic bubble (resp. piecewise linear) hat functions in the other direction. The error estimator is based on the following, heuristic assumption: Let fly (resp. ~w) be the solutions of the V~ (resp. l/~V~)-discretization of a symmetric HS(~)-elliptic variational problem given by the bilinear form a(., .) and a linear functional (I) defined on g s ( ~ ) . We assume that s - 1 , . . . , r + 1 is an integer. Then, at least for smooth exact solutions u of the variational problem, we may expect that fiw provides a better approximation to u t h a n / t v does. Compare Lemma 1 (resp. 6), and note that W~ locally contains polynomials of degree up to k + 1 which Va does not. Thus, the heuristic assumption that there is a q < 1 such that
Ii~- ~11o _
(5.3.S)
seems to be plausible. (Hopefully, q is close to 0 and not to 1, the choice q - 1 being trivial since Va C W~.) Using the triangle inequality for the energy norm I]" [la and the H~(~)-ellipticity of a(., .), (5.3.8)yields ilu -- uv [i~/,(~) ~ a ( u - fiv, u - fly) ~ a(~tw - uv, ~tw - u v ) ,
(5.3.9)
with constants depending on q, and the ellipticity constants of a(., .). T h e o r e m 5. Under the above assumptions for the construction of Va, I)da
and the variational problem, we have a(uw-~tv,~tw-uv)'~
J E
E
l=jo iEcbl,w
((~(r i,w) - a(itv, r ' dli,w
2
'
52
P. Oswald
where the scaling factors satisfy dz,i,w ~ 2 j ( 2 s - d ) . Together with the heuristic assumption (5.3.8), this two-sided estimate leads to an a posteriori error estimator for the error of the approximate solution fly in the energy norm. The error estimator is based on the residual contributions r,,i,w -- ~(r
- a(fiv, r
and does not require the computation of fiw. The proof uses Lemma 6, the inequalities (5.3.4), (5.3.5), and the hierarchical definition of the bases (5.3.7) in conjunction with the approach to hierarchical error estimation of [4] or [61, Section 3.5]. To use Theorem 5 in practice, one first computes (with sufficiently high accuracy) all residuals rz,i,w from the available current approximation to fly. For a typical elliptic PDE problem, this is a local procedure and can easily be parallelized. Formally, this step yields local error indicators, e.g., we may define r2
t,i,w
(1,i) 9[]nsupp Cz,,,w~0
for all j-cubes [] C f~ and j - j 0 , . . . , J. This choice of an error indicator is not unique. One has to experiment also with the scaling factors dt,i,w. Obviously, J r3
l = j o iEC~j,w
r2 '
[]
where the summation is with respect to all above described []. We do not have theoretical access to the constant c > 0. Its value may significantly influence the quality of error estimation. Note that the above splitting of the global error estimate into local components does not necessarily reflect the true local error of fly. However, the use of such local error indicators to monitor the refinement process is a well-established numerical practice. In a second step, those cubes Q are refined for which e 9 exhibits a large error contribution. Error equidistribution is a common approach, but many other issues might also influence the decision to refine a given cube. After this, one usually cares about keeping the refinement structure sufficiently regular. In our framework, this could lead to additional refinement in order to stay close to the geometric assumptions (G1)', (G2) (which is needed to guarantee an efficient multilevel scheme in the next solution step),^ and (G3)'. This enables us to set up the new cube structures {~j} (resp. {fl~}), and so on.
Multilevel Solvers for Elliptic Problems
53
Acknowledgments. The author thanks his colleagues at the Institute for Algorithms and Scientific Computation at GMD, Sankt Augustin, Germany, and the Department of Mathematics, Texas A&M University, College Station, USA, where large parts of this paper have been prepared, for their support. References [1] Adams, R. A., Sobolev Spaces, Academic Press, Boston, MA, 1978. [2] Auscher, P., Wavelets with boundary conditions on the interval, in: [16], pp. 217-236. [3] Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. User's Guide 6.0, Frontiers Appl. Math., vol. 7, SIAM, Philadelphia, PA, 1990. [4] Bank, R. E. and R. K. Smith, A posteriori error estimates based on hierarchical bases, SIAM J. Numer. Anal. 30 (1993), 921-935. [5] Bank, R. E. and J. Xu, An algorithm for coarsening unstructured meshes, Numer. Math. 73 (1996), 1-36. [6] Beck, R., B. Erdmann and R. Roitzsch, KASKADE User's Guide, Report TR-95-11, ZIB, Berlin, 1995. [7] Bespalov, A. N. and Yu. A. Kuznetsov, RF cavity computations using the domain decomposition and fictitious component methods, Russian J. Numer. Anal. Math. Modelling 2 (1987), 259-278. [8] Bespalov, A. N., Yu. A. Kuznetsov, O. Pironneau, and M.-G. Vallet, Fictitious domains with separable preconditioners versus unstructured adapted meshes, Impact Comput. Sci. Engrg. 4 (1992), 217249. [9] de Boor, C., K. Hhllig, and S. Riemenschneider, Box Splines, Springer, New York, 1993. [10] Bornemann, F. and H. Yserentant, A basic norm equivalence for the theory of multilevel methods, Numer. Math. 64 (1993), 455-476. [11] Bramble, J. H., Multigrid Methods, Pitman Res. Notes Math. Ser., vol. 294, Longman Sci.& Tech., Harlow, 1993. [12] Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Cutup. 55 (1990), 1-22.
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[66] Xu, J., The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing 56 (1996), 215-235. [67] Yserentant, H., Old and new convergence proofs for multigrid methods, in Acta Numerica 93, Cambridge Univ. Press, New York, 1993, 285-326. [68] Zhang, X., Multilevel additive Schwarz methods, Numer. Math. 63 (1992), 521-539. Peter Oswald
Institute for Algorithms and Scientific Computing GMD- German National Research Center for Information Technology D-53754 Sankt Augustin, Germany
[email protected]
W a v e l e t - L i k e M e t h o d s in t h e D e s i g n of Efficient M u l t i l e v e l P r e c o n d i t i o n e r s for E l l i p t i c P D E s
Panayot S. Vassilevski and Junping Wang
A b s t r a c t . This article introduces a stabilization of the classical hierarchical basis (HB) method by modifying the HB functions using some computationally feasible approximate L~-projections onto finite element spaces of coarser levels. It is proved that the corresponding multilevel additive and multiplicative algorithms give spectrally equivalent preconditioners for model elliptic problems. In practical computation, one action of the proposed preconditioner is of optimal order algebraically. Numerical results are included to illustrate the efficiency and stability of the new preconditioning methods.
w
Introduction
In this paper we are concerned with the construction of efficient numerical methods for matrix problems arising from finite element methods for elliptic partial differential equations. In practical computations, the standard nodal basis for the finite element space is often chosen as the computational basis and the resulting matrices are ill-conditioned. Our objective is to seek a substitution for the standard nodal basis so that the stiffness matrix arising from the new basis is well conditioned. A computationally feasible basis should possess the following properties: (a) the basis functions must be computable and have local support; (b) the resulting stiffness matrix is sparse and well conditioned. This paper will introduce a wavelet-like method proposed by the authors in [41] which can be employed to construct a new basis with the above mentioned features for the finite element application to elliptic problems. Multiscale Wolfgang Copyright
Wavelet Dahmen,
Methods
for P D E s
Andrew
g. K u r d i l a ,
59 and Peter Oswald
(~)1997 by A c a d e m i c P r e s s , Inc.
All rights of reproduction in any form reserved. ISBN 0-12-200675-5
(eds.), pp. 59-105.
60
P. Vassilevski and J. Wang
The method has a very close relation with the multiresolution (or multiscale) decompositions exploited especially in the wavelet literature (e.g., [17, 14]). In our approach, the requirements on the L2-orthogonality and the existence of a single generating function r (more precisely, r generates an orthogonal basis by dilation and translation as in {r k. - i ) , i E zd}) commonly imposed in the wavelet literature (see Mallat [26] for details) are relaxed in order to have a computationally feasible basis. The basic idea lies in approximating the wavelets without deteriorating their stability, yielding a stable Riesz basis for the finite element space under consideration. Attempts in the search of a stable Riesz basis with some restrictions, either on the mesh or on the analysis, have been made by Griebel and Oswald [20], Kotyczka and Oswald [24], and Stevenson [35, 36]. For a recent comparative study on the construction of economical Riesz bases for Sobolev spaces we refer to Lorentz and Oswald [25]. Our method is general and provides a satisfactory answer for most elliptic equations. The method is based on modifying the existing (unstable) hierarchical basis by using operators which are approximations of the L 2projections onto coarse finite element spaces. A similar approach was used by J affard [22] in seeking a wavelet basis for finite element spaces. In [22], the construction starts with the standard nodal basis functions which are transformed to an orthogonal multiresolution basis based on an explicit orthogonalization procedure exploiting wavelets in each subspace Wj of ~ . Here Wj is the L2-orthogonal complement of the finite element space Vj_ 1 in the next finer finite element space k). The latter procedure generally leads to basis functions that are not locally supported but have good decay rates and hence allow for locally-supported approximations. The method proposed by Vassilevski and Wang [41] can be regarded as an approximation of the wavelet basis in [22]; the approximate-wavelet basis functions are locally-supported and, therefore, computationally feasible. There are two alternatives in the implementation of the approximatewavelet basis in practical computation. In the first approach, one first computes an explicit form for each new basis function and then assembles the global stiffness matrix corresponding to the new basis. The major drawback for this approach is the difficulty encountered while assembling the global stiffness matrix; the robustness corresponding to the standard nodal basis is no longer retained for the new basis. In the second approach, one makes use of the standard nodal basis as the computational one and constructs a preconditioner for the stiffness matrix by using the new basis functions. This is possible because most of the iterative algorithms such as the CG method require only the action of the stiffness matrix on vectors. Thus, the technique of basis changing can be employed to compute the action of the matrix corresponding to the new basis. The second approach can be viewed as a "black-box fix" of the conditioning of the standard nodal
Wavelet-Like Multilevel Preconditioning
61
basis stiffness matrices. Details can be found in Section 8. The organization of the present paper is as follows. Section 2 contains some preliminaries and the problem formulation in an abstract Hilbert space setting. The importance of having a stable Riesz basis and its relation to the conditioning of discrete matrix problems, especially those from the finite element discretization, is the topic of Section 3. Specific examples of second order elliptic PDEs are presented in Section 4. Section 5 discusses the basic idea in constructing multilevel direct space decompositions. The hierarchical basis of Yserentant [44] is reviewed in Section 6. Section 7 contains a detailed discussion of the wavelet-like method in constructing a stable basis. The implementation issues, including two basic preconditioning schemes (namely the additive and multiplicative algorithms) and the construction of the approximate L2-projections, are discussed in Section 8. Finally, some numerical results are presented in Section 9 for convectiondiffusion problems, to confirm the theory developed in previous sections. w
P r e l i m i n a r i e s a n d basic p r o b l e m s
This preliminary section introduces the basic problem we address in this paper. Let H be a Hilbert space equipped with the inner product (., .) such that H C V C H', (2.1) where H is dense in another Hilbert space V with inner product (., ")0, and H ~ is the dual of H with respect to the pairing of V defined by the inner product (., ')0. Assume that the imbedding from H to V is continuous. More precisely, there exists a constant ~0> 0 satisfying
Ilullo
,ilull
v
u E/-jr.
(2.2)
Here I1" II- v/U, )indicates the norm in the Hilbert space H and i1" II0 = x / ( , .)0 that of V. Let a(., .) be a bounded bilinear form defined on H x H satisfying the following inf-sup condition of Ladyzhenskaya, Babugka, and Brezzi:
sup
Ilwll
>/31lv[i -
V v E H,
(2.3)
where/3 > 0 is a fixed constant. We are interested in approximate solutions of the following problem: Given f E H ~, find u E H satisfying
a(u, r
f(r
V r E H.
(2.4)
62
P. Vassilevski and J. Wang
It is well known that if a(., .) is symmetric and H-coercive (i.e., there exists a positive c~ satisfying a(v,v) >_ c~[[vl]2 for all v E H), then the problem (2.4) is equivalent to the minimization problem for the quadratic functional if(v) - la(v, v) - f(v). To approximate (2.4), let {gk}~=l be a sequence of finite dimensional subspaces of H satisfying the following approximation property: For any v E H one has lim
(2.5)
inf I I v - r
k---* oa CE Hk
The well-known Galerkin method for (2.4) seeks uk E Hk such that
a(uk, Ok) -- f(Ok) V Ok 6 Hk. (2.6) In order to have a well-posed discrete problem (2.6), we assume the following discrete inf-sup condition" There exists a constant /3k > 0 such that sup a(v, w) > ~k[iv[ [
V v E Hk.
(2.7)
It is known that if (2.7) holds true, then the Galerkin problem (2.6) has a
unique solution in the subspace Hk. Interested readers are referred to [18] and [11] for more details. In practical computation, we usually formulate a matrix problem for (2.6) by choosing a suitable basis for the subspace Hk. More precisely, let {r . i -- 1 , 2 , . . . , n k } be a computational basis of Hk. Expand the approximate solution u(k) in terms of this basis, yielding rtk
F_,
(2.s)
i--1
Let u (k) - (Cl, c 2 , . . . , cnk) T be the coordinates of u (k). It is not hard to see that the vector u (k) is given as the solution of the following linear system" A(k)u(k) _ f(k),
(2.9)
where the right-hand side vector f(k) is defined as follows: f(k) _ (bl, b2,..., b,~) T
with bi - f(r
The matrix is given by A (k) - {a(r k), r
(2.10)
One may view A (k) as
a linear operator on Hk defined, for any v - ~ vi
E Hk, by
i=1 nk
A(k)v- E i=1
nk
cir
where c i - E j=l
a(r k)' r
(2.11)
Wavelet-Like Multilevel Preconditioning
63
Of main interest in this paper, we study techniques for solving the linear system (2.9) by iterative methods. It is known that the conditioning of the matrices A (k) is of great importance in practical computation. Let us see how the condition number of A (k) is related to the choice of the basis (I)(k) - {elk)} i=1 '~k and the discrete inf-sup condition (2.7) For any v E Hk, denote in the bold face the coordinates of v with respect to the basis (I)(k). Namely, the vectors v (Vl, v2, 9 vnk )T and v are related as follows" _
..
_
,
rtk
v-
~
vir k).
(2.12)
i--1
Introduce a new inner-product in Hk by using the basis {r
i=1 as follows:
nk
k,0 - ~
viwi - v . w .
(2.13)
i--1
Denote by I]vl]k,0 - x/V v, the norm induced by the new inner-product. Recall that the condition number of the matrix A (k) is defined by ~ ( A ( k ) ) - IiA(k)ll IlA(k)-~ll ,
(2.14)
where [[A(k)l[ indicates the norm of the matrix A (k) with respect to the new norm I1" []k,0 in the vector space Hk. The following result can be checked easily. T h e o r e m 1. Assume that there are positive constants Ok,1 and Qk,2 satisfying ~ok,l[[v[]k,o _< IIA(k)vlik,o <_ ~ok,2[[v[[k,o. (2.15) ~)k,2 Then, ~(A (k)) _< ~-ZT,~" The best estimate for Qk,1 and Qk,2 is given by
~ok,2 -
sup ""llA(k)vIJk,o, ][vllk'~ ,, ,, []A(k)vl]~,o. inf
v6_Hk,
t~k,1 =
(2.16)
vEHk, [[vilk,o=l
Moreover, one has ~(A (k)) _ ~k'2 for the best estimate of ~k, In fact, the inequality (2.15) implies the following []A(k)[[ _< ~ok,2,
I]A(k)-~]l _< 1/~ok,1,
(2.17)
which verifies the validity of the theorem. Therefore, it suffices to establish an estimate like (2.15) in order to gain some knowledge about the condition number of the matrix A (k).
P. Vassilevski and J. Wang
64 w
M a t r i x c o n d i t i o n i n g a n d s t a b l e Riesz bases
Our objective in this section is to figure out the connection between the condition number of A (k) and the selection of the basis {r for Hk ~=1 To this end, we derive the inequality (2.15) by using the mr-sup condition. First, notice that for any v E Hk,
IIA(k)vllk,0 --
sup
~eH~, ll~llk,o=1
(A(k)v, w)k,o.
(3.1)
The definition of A (k) (see (2.11))implies that
(A(klv, w)k,o - a
vie ~'),
wi
It follows that IIA(k)vllk,0
--
= a(v, w).
a(v, w).
sup wEHk, llwllk,o=l
(3.2)
(3.3)
By letting 0 k , 2 - sup sup
,
(3.4)
one obtains the following (3.5)
IIA(k)vl]k,o < ek,el]vllk,o. Next, by letting Ok 1 '
inf
sup
Ilvllk,011
ll
,0
,
(3.6)
one has the following estimate from below:
IIA(k)vl]k,o >_ 0k,ll]Vl]k,0.
(3.7)
To summarize, we have proved the following result: T h e o r e m 2. If Ok,1 and 0k,2 are given by (3.6) and (3.4), respectively, then the estimate (2.15) holds true. Consequently, the condition number of the matrix A(k) is bounded by ~k,2 ~k,1
"
The question now is about the determination of the constants 0k,1 and 0k,2" We would like to select a basis {r for Hk so that the ratio (or 1 the condition number of A (k)) ek,2 is as small as possible. L0k,1 As our first consideration , we assume that {r ,~k is an orthonormal }i=l basis with respect to the original inner product (., .) of H. It is clear that
65
Wavelet-Like Multilevel Preconditioning
the corresponding discrete norm II. IIk,o is the same as the original norm II. II in Hk. Therefore, the constant 0k,2 is bounded from above by the norm of the bilinear form a(., .). Similarly, the constant 0k,1 is bounded from below by the parameter ~k in the inf-sup condition (2.7). Since in practical problems, the norm of a(., .) and the parameter/3k stay uniformly bounded in terms of k, the condition number of A (k) is uniformly bounded. It is, of course, impractical to assume the existence of an orthonormal basis {r which is also computationally feasible. The next best thing to an orthonormal basis is the Riesz basis in any Hilbert space. We recall that in the Hilbert space H, a Riesz basis is a basis {r }~--1 of H satisfying OO
#lllvll 2 < ~ c,2 < #211vii2
v v e H,
(3.8)
i=1
where v - EiC~=l c i r [l" I] indicates the original norm in H, and 0"1 and 0"2 are two absolute constants. Here we have assumed that the Hilbert space is separable. R e m a r k 1: The strong norm I" II in the Hilbert space H was used in the definition of the Riesz stability (3.8). Notice that H is a subspace of V. Thus, it would be feasible to discuss the Riesz property with respect to the norm !1" II0 of V. In practical applications, V often represents L2(Ft) and I1" II0 denotes the corresponding L2-norm. In the wavelet literature, the Riesz property is commonly studied with respect to the L2-norm, but the ultimate goal in the preconditioning analysis is to establish estimates such as (3.8), since (3.8) implies that the discretization matrix A (k) is well conditioned. We now go back to the finite dimensional subspace Hk of H. As in (3.8), we assume that there exist constants 0"1 (k) and 0"~k) such that
~k)llvll2 < k
c,2 < 0"~k)Ilvll 2
v v e Uk.
(3.9)
i=1
Observe that Condition (3.9) can be rewritten as 0"{k)l]vll2 _< I]vl]~,0 _< 0"~k)l]vll2 Since {r
i=1
g v e H~.
(3.10)
is assumed to be a basis of Hk, the equivalence relation
(3.10) obviously holds for some constants 0"~k) and 0"2(k) with 0"2(k) =
sup I]vll2k , 0 ~eH~, I1~11=1
(3 911)
o'~k) -_
2 inf llvl]k,o. ,,eH~, I1,,11=1
(3 12)
and
66
P. Vassilevski and J. Wang
The important thing is that one should have some control over the ratio
Definition 1. The family { ~(k ) - {r
} iS said to be a uniformly sta-
b•e family of Riesz bases for {Hk} if the quotient a~k)/a~ k) of the Riesz bounds in (3.9) is bounded uniformly with respect to k --~ oo, i.e., if there exists a constant M independent of k such that (k). (k) (3.13) a 2 / a 1 <_ M for any k = 1, 2, ....
The rest of this section explains the importance of having a stable Riesz basis. First we estimate the parameter Qk,2 defined in (3.4). Let [la[[ denote the norm of the bilinear form a(.,-). Then, ~k,2
- sup sup
a(v,~) IIvll~,011wll~,0
< sup sup
a(v.~) IIvllll~ll sup sup II~llll~ll ~ . ~ ~ n ~ II~llk.011~llk.0
vEHk w e H k
-~.~
-I[a[I
~.~
sup
I]v[]l]w[[
(3.14)
< I[all/cr~k)
v,,~eH,, Ilvllk,011wllk,0 --
Next, we estimate the parameter Qk,1 as follows: Qk,1
a(v,~) ,,eHk ,,,eHk Ilvll~,011~ll~,0
-- inf
sup
> inf
sup
>
inf
inf
a(v,w)
,i,;ii;,oii ll
sup
a(v,~)
inf
Ilwil
i1 11 ,o Ilvll
inf
(3.15) Ilwll
-,~eH,~,~eH,. Ilvllll~ll veHk IIvIIk,0 ,,,eHk Ilwllk,0 >_ Z,~/,,'~). a(k) It follows that Ok,2/Qk,1 _< ~l~ko~k).
The result can be summarized as
follows: be a computational basis of Hk satisfying(3 10) T h e o r e m 3. Let {r Then the condition number of the matrix A (k) arising from the basis {r is bounded by 1
~
g(A (k)) _< Ilal] cr~k) /~k ~ k ) '
"
(3.16)
Wavelet-Like Multilevel Preconditioning
67
Consequently, the condition number of A (k) is uniformly bounded for stable Riesz bases {r provided that the discrete Jar-sup condition (2.7) holds true with uniformly bounded constants/~k from below by some/3* > O. The conditioning estimate in Theorem 3 contains two important factors:
a(k) ~k and a~-'~" The first one depends on the norm of the given bilinear form and the stability constant/3k from the discrete Jar-sup condition. In practical computations, the space Hk must be constructed so that it ensures the boundedness of f3k from below by some fixed/3* > 0. This is the case for the model problems and their discretization spaces to be considered in Section 4. The second factor is basis-dependent. More precisely, it is a characterization of the difference between the discrete coefficient norm I1" llk,0 and the continuous norm I1" II. Stable Riesz bases are important because the corresponding discretization matrices are well conditioned. Thus, simple iterative methods such as the conjugate gradient (CG) can be successfully applied to solve the matrix problem from the Galerkin discretization with a geometric rate of convergence. As is well known, the convergence factor 1 where x - n (A(k) ) -< ~#k o(k) is bounded by ~/~-v/-E+I, o~k). For nonsymmetric problems, one could apply the CG method to the normal equation or the GMRES method applied to A (k). For symmetric and indefinite problems, the MINRES (minimum residual) algorithm would be a good choice. The convergence rate is no worse than that of the CG-method applied to A (k)2. There is another practical criterion in the choice of the basis {(~Ik)}n~l . It is of great practical importance to represent the matrix entries of A (k) as sparsely as possible. This is trivially achieved (assuming a(., .) is symmetric and positive definite) if the basis is a(., .)-orthogonal. Thus, the matrix A (k) admits diagonal forms and only nk entries need to be stored. Such a situation is too special and rarely happens in practice. In general, we assume that the basis is computationally feasible in the sense that the basis function are computable and the corresponding matrix A (k) is sparse (the number of nontrivial entries in the matrix is of order O(nk)). This is the case in practice for the standard nodal bases of finite element spaces Hk if the bilinear form arises from partial differential equations. However, this choice will not make a stable Riesz basis for most of the PDEs. The following section contains a detailed discussion of this aspect. w
Model problems
Here we consider some model problems of (2.4) in partial differential equations. Boundary value problems for the second order elliptic equations are
P. Vassilevski and J. Wang
68
of major consideration in this discussion. Finite element methods will be applied to approximate the solution defined on an open bounded domain gt in 11%d with d = 2 or 3. 4.1
S e c o n d - o r d e r elliptic e q u a t i o n s
Consider the homogeneous Dirichlet boundary value problem for the following second-order elliptic equation:
L(u) -- -~7 . (a(x)~Tu) + b(x). ~7u + c(x)u = f(x),
x E f2,
(4.1.1)
where a = a(x) is a symmetric and positive definite matrix with bounded and measurable entries, b = b(x) and c = c(x) are given bounded functions, and f = f(x) is a function in H-l(ft). Note that we do not intend to consider problems which are convectiondominated. Let Hl(f~) be the standard Sobolev space equipped with the norm:
I1 11 - (llullo + II ullo
V u E Hl(ft).
(4.1.2)
Here ll" {i0 stands for the L2-norm. Let H01(ft) be the closed subspace of Hi(f2) consisting of functions with vanishing boundary values. The following relation is well known: H~(ft) C L2(ft)C H-I(f2).
(4.1.3)
A weak form for the Dirichlet problem of (4.1.1) seeks u E H~(~) satisfying b(u, v ) - f(v) V v E H~(~), (4.1.4) where
b(u, v) - / f ~ (a(z)Vu . Vv + b ( z ) . Vu v + c(z)uv) dx
(4.1.5)
and f(v) is the action of the linear functional f on v. Assume that the problem (4.1.4) has a unique solution. Then the inf-sup condition (2.3)is satisfied for the bilinear form b(., .) defined on H01(f2) • H01(f2). Let us approximate (4.1.4) by using the Galerkin method with continuous piecewise polynomials. If Sh denotes the finite element space associated with a prescribed triangulation of ft with mesh size h, then the Galerkin approximation is given as the solution of the following problem: Find Uh E Sh satisfying
b(uh, r
f(r
V r E Sh.
(4.1.6)
69
Wavelet-Like Multilevel Preconditioning
It has been shown that the discrete problem (4.1.6) has a unique solution when the mesh size h is sufficiently small. Thus, the discrete inf-sup condition (2.7) is satisfied for this problem. Details can be found from [32, 33]. Choose the standard nodal basis as the computational basis for the finite element space Sh. Let {r }~=1 be the set of nodal basis functions and Ah be the corresponding discrete matrix (also called the global stiffness matrix). The condition number of Ah can be estimated by using Theorem 3. To this end, let us establish Inequality (3.10) for the standard nodal basis. For any v E Sh, let tl
v - ~
with vi - v(xi),
vir
(4.1.7)
i=1
where xi are the interior nodal points of the finite element partition. Relation (3.10) is equivalent to the following: n
~lllvll21~ ~ v~ ~ llvll ,
v v e Sh
(4.1.8)
i=1
for some positive constants &l and #2. It is not hard to see that n
,lvll02
~ h-
(419)
i=1
It follows that &2 - O(h-d). Also, by the standard inverse inequality one sees that #1 is bounded from below by a constant proportional to h 2-d. Thus, from Theorem 3, the condition number of Ah is bounded from above by Ch -2 for some constant C; the lower bound for its condition number is also bounded from below by some Ch -2.
4.2
Stokes equations
Consider the problem which seeks u e [H~(a)] d and p e L 2 ( a ) s u c h that -Au+Vp V.u u
= = =
f, 0, 0,
inf,, in f~, on Oft,
(4.2.1)
where f E [L2(f~)] d is a given vector-valued function and 0f~ denotes the boundary of f~. A weak form of the problem (4.2.1) involves the following bilinear form: A(u, p; v, w) = (Vu, Vv)0 - ( V . v , p)0 - ( V . u , w)0
(4.2.2)
P. Vassilevski and J. Wang
70
defined on W • W with 142 - [H0i(fl)] d • L20(f~). Here L~(~)is the closed subspace of L2(fl) consisting of functions with vanishing mean value. The weak problem seeks (u, p) E )42 satisfying A(u, p; v, w) - (f, v)
V (v, w) E W.
(4.2.3)
The inf-sup condition is satisfied for the bilinear form defined in (4.2.2). Details can be found from [18, 11]. To apply the finite element method to (4.2.3), we employ the HoodTaylor element [21] which satisfies the discrete inf-sup condition (with a mild restriction on the triangulation). The Hood-Taylor element is a combination of continuous piecewise linear functions for the pressure variable p and continuous piecewise quadratic functions for the velocity variable u. Denote by Wh = Xh x Sh the corresponding finite element space, where Xh contains continuous piecewise quadratic functions and Sh contains continuous piecewise linear functions for the pressure variable. The finite element approximation (Uh, Ph ) E Wh satisfies .A(uh, ph ;',,, w) - (f, v)
V (v, w ) E Wh.
(4.2.4)
If the standard nodal basis is selected in formulating a matrix problem for (4.2.4), then the condition number of the global stiffness matrix can be estimated by using Theorem 3. More precisely, let {r be the that of Sh, as discussed in the standard nodal basis of Xh and {r previous section. For any (v, w) E Xh • Sh, let tl
w - E wir
(4.2.5)
i=1
and
m
v - E
vjCj.
(4.2.6)
j=l
Using the inverse inequality and the Poincar~ inequality one can derive the following relations" n
2
(4.2.7)
v~ _< c2h-d]JvlJ 21,
(4.2.8)
i=1
and
m
ci h2-dl]vi] ~ _< ~ j=l
where ci and c2 are two absolute constants. Thus, we have from Theorem 3 that the condition number of the global stiffness matrix is bounded by Ch -2.
Wavelet-Like Multilevel Preconditioning
71
We emphasize that the poor conditioning for the Stokes problem is caused by Relation (4.2.8), where the Hi-norm of v was approximated by a discrete norm stable in L 2 only. The equivalence (4.2.7) indicates that the standard nodal basis is a good choice for the pressure variable in the Stokes problem. Therefore, attention should be focused on stabilizing the velocity component in the Stokes equation. A direct wavelet approach to the Stokes problem has been developed by Dahmen et al. [16]. One could also use block-diagonal preconditioners for the saddle-point discretization matrices A (k) with one block corresponding to Laplace-like preconditioners for the velocity component and a second block corresponding to massmatrix preconditioners for the pressure unknown in the MINRES method. Details of this approach can be found in Rusten and Winther [31] and Silvester and Wathen [34]. 4.3
Mixed methods
Here we consider the mixed method for the second order elliptic equation (4.1.1). For simplicity, assume that b = 0, c _= 0, and the following Dirichlet boundary condition u =-g on 0f~ (4.3.1) is imposed on the solution. Let (
, V . v C L2(f~)},
H(div; f ~ ) - ~v" v G which is equipped with the following norm: []Vl[H(div; a) -Let c~(x)=
(Iv[ 2 + [V. vl2)dx
a-l(x) be the inverse of the coefficient matrix a = a(x) and .A(q, u; v, w) = (a(x)q, v)o - (V- v, P)o - (V -q, w)o
be a bilinear form defined on W x W, where W = H(div; f~) x L2(fl). Then, a mixed weak form for (4.1.1) with Boundary condition (4.3.1)seeks (q, u) E W satisfying A(q, u; v, w) = (g, v . n)or~ - (f, w)o
V (v, w ) E W,
(4.3.2)
where (., ")0~ denotes the inner product in L2(O~). The inf-sup condition (2.3) can be verified for the bilinear form A(.; .). Furthermore, finite element spaces satisfying the discrete inf-sup condition (2.7) are available for this bilinear form. Details can be found in the book by Brezzi and Fortin [11]. If the standard nodal basis for the finite element
P. Vassilevski and J. Wang
72
space of Raviart and Thomas is employed in practical computation, the condition number of the global stiffness matrix is known to be proportional to h -2. The question is how to stabilize the nodal basis. Observe that in this application, one needs to construct a basis for the mixed finite element space (called Wh) so that the discrete norm is equivalent to the following norm: /l(v; w)ll~v -Ilvll~(div; ~) + IIwli~. One sees from above that the standard nodal basis is a good choice for the pressure unknown. Thus, the difficulty lies in stabilizing the flux component with the norm in H(div; ft). We do not yet have a positive answer yet for this stabilization, but the approach of ttelmholtz decomposition for vectors might provide a partial answer for problems of two-space variables. The method decomposes each vector-valued function v as follows: v - curl r -b Vr
(4.3.3)
where r and r are functions in Hl(f~). A discrete version of (4.3.3) can be studied in order to apply it to the mixed method. Results for the standard conforming and non-conforming finite elements should be investigated first in this direction. We point out that this approach may have some difficulty for problems of three-space variables. For multilevel methods relying on the above Helmholtz decomposition, see Vassilevski and Wang [40] and Arnold, Falk, and Winther [1]. In the following sections we will devote ourselves to modifying the nodal bases in the application to second order elliptic problems. Our goal is to construct some stable Riesz bases that are computationally feasible for elliptic equations. More precisely, the basis should be so constructed that the resulting discretization matrices are both well conditioned and sparse. w
M u l t i l e v e l direct d e c o m p o s i t i o n s
To construct a computationally stable basis for the second-order elliptic and the Stokes equations, we take advantage of the fact that the weak problem (2.4) is discretized on a sequence of finite element subspaces. In particular, a sequence of nested subspaces may be possible in practical computations. Our objective in this section is to exploit ways of constructing stable basis by using the information from each approximating subspace. 5.1
T h e basic idea
The basic idea comes from the fact that an L 2 orthonormal basis of wavelets is also H 1-stable in applications to partial differential equations. There-
Wavelet-Like Multilevel Preconditioning
73
fore, wavelet bases are good candidates in formulating matrix problems for (2.6). Since the conventional wavelet bases have complicated structures which limit their application in the numerical methods, we shall focus our attention on approximations of wavelet bases. Below we present a detailed discussion. Assume that we have a sequence of nested subspaces {l~ }~=0 satisfying
VoCV~C...cvkc....
(5.1.1)
Each vector space V) shall be referred to as a coarse subspace of Vk when j < k. In applications, they are finite element spaces consisting of continuous piecewise polynomials over a sequence of finite element partitions for the domain f~. Upon viewing Vj as a subspace of L2(f~), one has (:x:}
U V~ - L2(a),
(5.1.2)
j=0
where the closure was taken in the strong topology induced by the L2-norm. We assume that V0 is a very coarse subspace of L2(ft) whose dimension is a small number. For every j > 1, define Wj to be the L2-orthogonal complement of Vj-1 in I~. We have Vj -
Vj _ 1 (~ W j
(5.1.3)
and
Wi_I_Wj if i C j ,
(5.1.4)
where we have assumed that W0 - V0. It follows that k
(5.1.5)
j=0
where all these subspaces are orthogonal. By virtue of (5.1.2) and (5.1.5), this implies C~
j=l
which is a decomposition of L2(f~) into mutually orthogonal subspaces. A wavelet basis for L2(f~) can be constructed if one is able to find an orthonormal basis for each subspace Wj. Such a basis would be ideal in preconditioning the discrete problem (2.6), if it is computationally feasible. In practice, it is very hard to find an L2-orthonormal wavelet basis which is also computable. Therefore, the orthogonality requirement in the
74
P. Vassilevski and J. Wang
decomposition (5.1.5) shall be relaxed to allow only a direct decomposition of the following form" - Voe
e Vl
v:,
(5.1.6)
where each Vj1 is a complement of ~ _ 1 in Vj such that the corresponding two-level decomposition is direct. But in order to attain the stability of the wavelet basis, it is crucial to have some approximate orthogonality among the subspaces Vj1. 5.2
A general approach
A general method for deriving the hierarchical complement of each ~ _ 1 in V) is based on the existence of some computationally feasible projections 7rj from C, a dense subspace of the Hilbert space H, onto Vj. In particular, we assume that C D UI>,VI and zrjr = r for any r E Y). Thus, one has 7rjTri - 7ri for j >_ i ifV~ C V). With Vj1 - ( I - T r j _ l ) ~ , one has the following two-level direct decomposition: (5.2.1)
Definition 2. (Multilevel Hierarchical Basis) For j - O, 1 , . . . , k, let {r 1 , . . . , nj } be a c o m p u t a t i o n a l l y feasible basis o f l ~ . A s s u m e that {r
ii --
1 , . . . , n j _ l } U{ r ), i - nj_ 1 "~- 1 , . . . , nj } forms a basis of k~ . A multilevel hierarchical basis for Vk is defined as follows: k
(I) k -- U { ( / - - 7 r j _ l ) r j=O
j)
i--nj
1-~" 1
nj}
(5.2.2)
Here we have assumed that 7r_ 1 - 0 and n-1 - O. We now discuss the stability of the multilevel hierarchical basis. The following result sets a guideline for the selection of the operators 7rj.
Theorem 4. A necessary condition for ~k to be a stable Riesz basis o f Vk is that the projection operators 7rr be uniformly bounded on Vk with respect to r and k for any r <_ k.
75
Wavelet-Like Multilevel Preconditioning
Proofi
Let v E Vk be expanded as follows:
v-
k
cl )Sl i--1
{r
nj
E
Z
j=r+l
i=nj-l+l
'~r~=~
where is the multilevel hierarchical basis of V~ and ~s) - ( I 7rs_l)r ~). Introduce the notation c(~) c~r+l) C --
ci where c~j) -- "( C(j) n j _ l + l , . . . ,c ~) ) T f o r j - r + l , . . . , k ,
c(~) -(c?),...,
and
c~?) ~ .
Since, by assumption, we have a stable Riesz basis, then there exist (rl and cr2 independent of k such that 0" 1
Observe that 7r~v 7r~v, we obtain
(5.2.3)
c T C < Ilvl12 < ~ C r c .
~
. Thus, from (5.2.3), with v replaced by
i=1
Ill, vii ~ < ~c(~)'c(~)
< ~
(
c(~)~c(~) + ~
c~)~c~ ~)
j=r+l
)
< ~-~llvll ~. O'1
Here we have used the lower bound of (5.2.3). This shows the boundedness of the projection operator 7r~ for any r. w
T h e h i e r a r c h i c a l basis
In this section we review the classical hierarchical basis decomposition. First, partition the domain f2 into large elements. Let To denote this initial coarse triangulation and V0 be the corresponding finite element space of continuous piecewise linear functions. The fine finite element space V - Vj corresponds to the triangulation Tj which is obtained by J _> 1 successive refinements of the coarse triangulation To. For problems of two space variables, one can use the triangular element and the refinement of one triangle at level k - 1 will generate four congruent triangles of level k by connecting
P. Vassilevski and J. Wang
76
the midpoints of its edges. Similar techniques are available for problems of three space variables with tetrahedra being used as elements. We refer to Ong [29] for more details of this discussion. It is also possible to use bisection refinement for both two- and three-space problems. Details about this technique can be found from Mitchell [28] and Maubach [27]. Let 7~ denote the finite element partition at level i, and 17/ be the corresponding finite element space of continuous piecewise linear functions. Thus, we obtain a nested sequence of conforming finite element spaces
VoCVxC...cvj, which can be used to discretize the second-order elliptic equation. To describe the classical hierarchical basis, let Afi be the node set of nodal degrees of freedom at level i which consists of vertices of triangles (or tetrahedra in 3-d) in 7~. One has the following natural direct decomposition: = X:
with A/'~ being the set of newly-introduced nodal points. For example, in 2d, the nodal set A/'~ contains the vertices of the triangles at level k that are midpoints of the edges of the triangles from level k - 1. We also introduce the mesh size hi - 2-ih0 for the ith level triangulation Ti. Here, ho stands for the maximum diameter of the elements in To. Recall that the standard nodal basis functions {r k), xi E A/k} are defined to satisfy the condition r k) (xj) = $i,j, where 6i,j is the Kronecker symbol, with xj running over all the nodes in A/'k. One can then define a two-level hierarchical basis of Vk by adding to the nodal basis of Vk-1 the nodal basis functions of Vk corresponding to the complementary nodal set A/'~ - A/'k \ A/'k-1. Consider now the nodal interpolation operator Ik " C(f~) ~ Vk defined by Ikv -v(xi)r k). It is clear that Ik is a projection and Ikr -- r if @ E Vk. x~EA/'k With the choice of Try ~_ Ij, one obtains the classical hierarchical basis by using the general decomposition method discussed in Section 5. We comment briefly on the stability of the classical hierarchical basis in applications to the second order elliptic problems. In these applications, the natural norm for the finite element space is the norm in the Sobolev space H 1(f~). Since the interpolation operator Ik is not bounded in the H 1norm (e.g., [13]), Theorem 4 implies that the classical hierarchical basis is not absolutely stable. Thus, the resulting stiffness matrices computed with respect to the hierarchical basis will not be well conditioned. However, as is well known, the condition number for problems of two-space variables is practically acceptable. In fact, the condition number for the second-order elliptic problems can be verified to be proportional to k 2 at level k. This condition number grows more slowly than that from the standard nodal
Wavelet-Like Multilevel Preconditioning
77
basis, which behaves like O(h~2). For problems of three-space variables, the condition number for the classical hierarchical basis is of order O(h~-1) which is better than the one from the standard nodal basis but worse than in the case d - 2. w
A s t a b l e R i e s z basis b y w a v e l e t m e t h o d
In this section we will construct appropriate projections 7rk which are H Istable and provide a computationally feasible Riesz basis for Vj. The bilinear forms of main interest are those arising from the Hilbert space method for second-order elliptic problems discussed in Section 4. The method presented here was proposed by Vassilevski and Wang in [41]. 7.1
O n t h e basis c o n s t r u c t i o n
Define the L2-projection operators Q k ' L 2 ( ~ ) ~ Vk as follows:
(Qkv, r
- (v,r
vtE
vk.
Also, assume that there are computationally feasible approximations Q~ 9 L2(f~) ~ Vk of Qk such that for some small tolerance r > 0 the following estimate holds:
[I(Q~ - Qf~)vll0 _< ~lIQkvll
V v E L2(f~).
(7.1.1)
The projection operators of major interest are defined as follows: J-1 (7.1.2)
7rk - 1-I (ij + Q;(Ij+x - i i)), j=k
with ~'j - I. It is clear that ~'kr - r i f r E Vk s i n c e / j r 1 6 2 / i t - r for j >_ k based on ( I j + l - / j ) r - 0 and Ijr - r This also implies that 7r~ - 7rk. Note that 7 r k - l ( / k - I k - 1 ) r Q ~ _ l ( X k - Ik-1)r and 7rk - 7rk_l - ( I - Q~_l)(Ik - Ik-1)Trk. Then, the components in the definition (5.2.2) for the wavelet-like multilevel hierarchical basis read as follows" k {r ~ i -
1
no} U { ( / - j=l
Q;_1)r j) i -
nj 1 + 1
hi}
(7.1.3)
The above components { ( I - Q~_l)q~l j), i - nj_ 1 + 1,..., nj) can be seen as a modification of the classical hierarchical basis components based on the interpolation operator Ik, since ( I - Q]_I)r j) - ( I - Q~_I)(Ij -
78
P. Vassilevski and J. Wang
Ij_l)r
j)', the modification of the classical hierarchical basis components
{(Ij
/j_l)r j),
--
+ 1 , . . . , n j } comes from the additional term In other words, the modification was made by sub-
i -- n j _ l
Q ~ _ I ( I j - Ij-1)r
tracting from each nodal hierarchical basis function r its approximate L2-projection Qa_lr onto the coarse level j - 1. Such modifications of the hierarchical basis function r for some particular choices of Q~_I will be shown in Figures 1-3 in Section 8. It can be seen that the modified hierarchical basis functions are close relatives of the known Battle-Lemari~ wavelets [17]. Observe that in the limit case of Q~ - Qk, so that r - 0 in (7.1.1), we get 7rkv - Q k l k + l Q k + l l k + 2 . . . Q j _ l l j v
- QkQk+l...Qj_lv
- Qkv.
Therefore, :rk reduces to the exact L2-projection Qk. As is well known, the L2-projection operators are bounded in both H~(ft) and L2(Ft). This gives us hope that the hierarchical multilevel basis corresponding to the above choice of the operators 7rk may yield a stable Riesz basis if r is sufficiently small.
7.2
Preliminary estimates
For an analysis of the multilevel basis (7.1.3), we need some auxiliary estimates already presented in Vassilevski and Wang [41]. The following result on estimating the error ej - (~rj - Q j ) v will play an important role in our analysis. L e m m a 1. There exits an absolute constant C such that k
j=l
k
h72il~j 11o2_< c~ 2 ~ h~-211(Qj j=l
Q~-~)vll~),
u
(7.2.1)
The estimate (7.2.1) relies on the following recursive relation"
~,_~ - (Q,_~ + R,_~)~, + R,_~(Q, - Q,_~),, where R~-I - ( Q , - 1 e8-1
Q~_I)(/,-1 - / , ) .
It can be seen as follows"
-- 7 r s - I V -- Q s - I v
= (I,_~ + Q L ~ ( I ,
-
~,_~))~,v
-
Q,_~v
= (Q8-1 - Qa_l)Is-17r, v + Qa_ l~r,v - Q , _ l V .
(7.2.2)
79
Wavelet-Like Multilevel Preconditioning Thus, one has e,_~
-- (Qs-1 - Qas_l)Is-l(Trsv - Qsv) + Qa_l(Trsv - Q , v ) +(Q,-1
-
Qa_l)Is-les-1 -b Qa_lIse8
+Qs-I(Is-IQsv
- Qsv) - Q a _ l ( I s _ l Q s v - Qsv)
= (Qs-~ - Q~_~)(18-1 - I8)e8 + (Qs-~ - Q~_~)e8 + Q~_le8
+(Qs_I-Qa_~)(18_I-18)Qsv = [Qs-~ + (Qs-~-
Q~_~)(18-1- 18)] e8
+(Qs_I-Qa_I)(Is_I-I~)Q,v. The latter together with the fact that (Is-1 - I s ) Q s - 1 desired recursive relation (7.2.2).
-
0 implies the
Proofi We now verify L e m m a 1. Let C n be a mesh-independent upper b o u n d of the L2-norm for the operator I8 - I8-1 " V8 ~ Vs. Then, IlR,-xvii0 _< CRr[bv[}0
for all v C Vs.
(7.2.3)
We assume that, C R r _< q 0 - Const < 1.
(7.2.4)
Then, 1
1 + qo = Const < 1 2
(7.2.5)
Next, observe that ek - 0 (since v E Vk). Then a recursive use of (7.2.2) leads to
ILe,-lll0
(1 + CRr)l]e, [io + CRrl[(Q, - Q,-x)v[[o k j=s
P. Vassilevski and J. Wang
80
Therefore, with hj - 2-Jho and h0 being the coarsest mesh size,
II~s-lli0
k
<_ C n r h s _ l ~_, (1 + Cnr)J-sh-21[[(Q i - Qj_l)v[Io j-s k
- Cnrh~_x ~ (1 + Cnr)J-~h-i~hjhy~ll(Q j - Qj-1)v[10 j=s k
1 j-s - CR~-h._~ F, (1 + CR~-)J -~ (~) h-/~ll(Qj
Qj_l)Vll0
-
j=s k
<_ CRvhs-1 ~ qJ-Sh21[l(Q j -Qj-1)v[}o j=s
1 <- CRvhs-11vff-:~-q
=s qJ-Sh-j-2 i](Qj - Qj-1)v[12o
9
The latter inequality shows k E h~---21[[es- 1[[0 s=l
k
<- C~r21-~q E
k
E qJ-Sh;2[l(Qj - Q/-1)v[[02
s=l j=s k
_< c~,2(~_q)~
h;21](Q~ - Qj - ~)vll 2o.
which proves the lemma. The above proof also shows the following corollary. Corollary 1. For any ~r E (0, 1], the following estimate holds: k E
k
-20" ll~,-~ll~ _< C ~ ~ h~_~
1 ~ hj2~ ( 1 - q ) 2 j= 1
Qj 1)vii 2 --
0,
provided that v satisfies the estimate C n r _< q2 ~
1,
(7.2.6)
for a mesh-independent constant q E (0, 1) (actually q > 2-a). R e m a r k 2: Corollary 1 indicates that in order to have the L2-stability of the deviations, one has to assume a level dependence on the tolerance r. More precisely, there exists a 7"o > 0 such that if v < r0J -1, then k
E s--1
[}e~-1[[02 -< C[Iv[[~)
for all v e Vk.
(7.2.7)
Wavelet-Like Multilevel Preconditioning
81
L e m m a 2. Let V~ - ( I - Mk_I)Vt:(1), with V(1) - (Ik - Ik-1)Vk, be the modified hierarchical subspace of level k for any given L2-bounded operator Mk-1. Then, there are positive constants Cl and c2 independent of k such that
c:11r for any r
_ (I-
_< IIr
_< c=11r
r=0,1,
(7.2.8)
Mk_l)r 1 E V1, (/)1 E V(1). Here I1" I1: ~ . , . d ~ tot th~
. o t t o i . th~ Sobo1~v ~ p ~
Hg(~) ~.d
II Iio denotes the L2(f2)-norm.
Proof: The following strengthened Cauchy inequality is valuable: There exists a constant 7 E (0, 1), independent of the mesh size or the level index k such that
(Vr 1, V(~) < "y(Vr 1, Vr
89(V(~, V~) 89 ,
(7.2.9)
for all (~1 E Yk(1) and r E Yk-1. In fact, we shall make use of the following version of (7.2.9). For any r
e V(1) and r e V~-I, one has
(:7(rI + r V(r 1 + 8)) >_ (i - 72)(Vr I,Vr
(7.2.10)
A derivation of (7.2.9) and (7.2.10) can be found from Bank and Dupont [5] or Axelsson and Gustafsson [3]. We first establish (7.2.8) for the case r = 1. With r = - M k _ : r 1 we see from (7.2.10) that
(1
-
~=)11r
< i1r
Thus, the inequality on the left-hand side of (7.2.8) is valid with cl = i - 7 2. To derive the part on the right-hand side, we use the standard inverse inequality to obtain
2 where we have used the L2-boundedness of the linear operator Mk-1. Observe now'that since r E V(1), there exists a constant C such that
IIr ~ ~ Ch~llr
(7.2.11)
It follows that Itr _< ciir for some constant C. This completes the proof of (7.2.8) for r - 1. Similar arguments can be applied to verify the case r = O.
P. Vassilevski and J. Wang
82 Lemma 3. F o r a n y r 1 - ( I - M k _ l ) r w i t h r X1 e V(1) -- (Ik - I k - 1 ) V k ,
1E
V 1 a n d ~o1 - ( I - M k _ l ) X
1 E V~,
del~ne
(A~kl)~l, ~pl) _ a(~l, ~pl) _ / a ( x ) ~ l .
~1.
fl Here, the bilinear form a(., .) is equivalent to the H I - i n n e r product. there are positive constants ri such that
vlh~-2[lr
2 _< (A~kl)r r
r2h~-2llr
Then
2
Since a(.,.) is equivalent to the H~-inner product, there are two
Proof:
positive c o n s t a n t s ~l and ~2 such t h a t
~111~)1[[2 < (A~)$I r m
~
--
h11r
l"
Using the n o r m equivalence (7.2.8), e s t i m a t e (7.2.11), and the inverse inequality, we o b t a i n (with possibly different constants 7"1 and r2),
nh~-2[1r
_< (A~)r
r
(7.2.12)
r2h-~2llr
T h e above inequalities conclude the l e m m a . L e m m a 4. Given v, let v( k)' - (Trk - 7rk-1)v. There exists a sutticiently s m a l l constant 7"0 > 0 such that if the approximate projections Q~ satisfy (7.1.1) with r e (0, 7"0) (see (7.2.4) a n d (7.2.5)), then
J
Ilvll
h;- llv
k=0
II02.
(7.2.13)
Proof." Let v E V. S t a r t i n g with v (J) - v, for s - J down to 1, one defines V(S-- 1) __
( I , _ 1 + Q ~ - I (I~ - I~_ 1)v(~) - 7r._ 1 V. T h e n the d e c o m p o s i t i o n v
in t e r m s of entries in Vs1 - (I -- Q~-I) V(1) _ (i - Q~_ 1)(18 - 18_1 ) v reads as
d v - v (~ + Z
v(/)l'
v(')l - v(S) - v ( s - l ) .
(7.2.14)
j=l F r o m the r e p r e s e n t a t i o n v (~)~ - v (~) - v ( s - l ) - (Q~ - Q , - 1 ) + es - es-1, one arrives at the e s t i m a t e
J k=0
<_ C
d
J
2 E h~21](Q~- Qk-~)vll2o + c E h~2I[ek[Io
k=0
J
_< C(1 + r 2) ~
_< Cllvll
.
k-O
k=0
h ~ 2 [ l ( O k - Qk_l)V[I 2o
Wavelet-Like Multilevel Preconditioning
83
Here we have used the norm equivalence result of Oswald [30] and Lemma 1. Thus, it suffices to establish the upper bound" J
Ilvll~ ~ c ~ h~-211v(k)~1102.
(7.2.15)
k=0
It is possible to give a direct proof for (7.2.15) by using the strengthened Cauchy-Schwarz inequality (see Vassilevski and Wang [41]). Here we would like to adopt an alternative approach by using the following characterization of the H~-norm for finite element functions: J
Ilvll~ ~
inf
,
(7.2.16)
Eh;=llvkll02"
v= ~'~ vk, v k E V k
k=0
k---O
A proof of the above equivalence can be found from [30]. Thus, for the J
particular decomposition v -
~ v (k)l, v (k)~ C Vk, one immediately has k=0 J
ttvllx2 ~ c ~ h~-211v(k)~llo2, k=O
which completes the proof of the lemma. 7.3
Stability analysis
Here we study the Riesz property of the wavelet-like multilevel hierarchical basis defined in (7.1.3). For any v E V, let J
v-
~
co,ir ~ + ~
xi EA/'o
y~
ck,i(I-Q~_l)r
k)
(7.3.1)
k=lx,EX~x)
be its representation with respect to the given wavelet basis. The corresponding coefficient norm of v is given by / 1/2 JJJvrir-
h0
E x, E.Afo
+
9' h d-2 k-i
,
(7.3.2)
x, E./V'(X)
where d - 2 or 3 according to the number of space variables. Our main result in this section is the following norm equivalence"
P. Vassilevski and J. Wang
84
T h e o r e m 5. There exists a small (but fixed) r0 > 0 such that if the approximate projections Q~ satisfy (7.1.1) with 7" E (0, vo), then there are positive constants Cl and c2 satisfying
c~lllvlll 2 _< Ilvll~ _< c2111vlll2
v v E V.
(7.3.3)
In other words, the modified hierarchical basis is a stable Riesz basis for the second order elliptic and Stokes problems. The equivalence relation (7.3.3) shall be abbreviated as IIIvlll2 ~_ Ilvll~, Proofi
We first rewrite (7.3.1) as follows: J
- ~ v(~) ',
(7.3.4)
k=0
where with Q~_ - 0 V(]g)I ----
~/.
c k , i ( I - Q~_ 1)elk) E V~ .
(7.3.5)
xiEAfk(1)
Furthermore, by letting r
E
ck,,r k) e V(1), we see that v(k)' =
xiEAf(1)
( I - Q~_I)r (k). Thus, by using (7.2.8)in Lemma 2 (with r Mk-1 -- Q~-I) we obtain
IIr Since r
~ II~(k)'ll~o .
(7.3.6)
E V(1), then
IIr
~h~
~
c~,,.
x, EA/'( ~)
Combining the above with (7.3.6) yields J
IIIvlll~ ~ ~ h;~ll~(k)'tlo~. k=O
This, together with Lemma 4, completes the proof of the theorem. R e m a r k 3: For any fixed a E (0, 1], define i/2
mvmo -
h ~o- ~~ F~ c~,, + ~ h ~- ~~ F_, c~ ,, xiEAfo
k=l
0 and
x,,E
85
Wavelet-Like Multilevel Preconditioning
with d -
2 or 3. Then, for sufficiently small r, one has
IIIvlll,, ~ Ilvll~,
(7.3.7)
for all finite element functions v E Vj. Here I1" I1~ denotes the H~'(a)norm defined by interpolating H0i(f~) with L2(f~). The constants in the norm equivalence depend on cr as indicated in Corollary 1. For ~r = 0, the equivalence (7.3.7) holds true provided that the tolerance satisfies r < r0J-1 for some small 70. w
I m p l e m e n t a t i o n s for a m o d e l p r o b l e m
In this section we discuss techniques which implement the wavelet-like multilevel hierarchical basis for approximate solutions of PDEs. For simplicity, we consider the self-adjoint second-order elliptic equation discussed in Section 4. The Stokes equation can be covered in a similar manner. The bilinear form under consideration is defined as follows: a(~, r
- f a(x)~z~. X7r
V ~p, r E H0i(Ft).
(S.1)
s
Let d(~, r
f - L V ~ . V r be the Dirichlet form defined
on
Hl(f~)
x
H01(f~).
Since the two bilinear forms a(., .) and d(., .) are equivalent, we have from (7.3.3) that
cl[llvlll 2 _< a(v, v) < c~lllvlll ~ for some positive constants
C1
(8.2)
and c2. Moreover, the following result holds"
L e m m a 5. Let c > 0 be a parameter and ac(~, r
9V r + / ~
r
V ~, r E H~(Ft).
(8.3)
I f the approximation in (7.1.1) is sufficiently accurate such that 7" < voJ -1 for some constant to, then there exist 7"1 and r2 independent of e and the mesh size hk such that J
J
< a,(v, v)< for any finite element function v E Vj. Here ak - eh~ 2 + 1 and ck,i are the coemcients of v in the expansion (7.3.1).
P. Vassilevski and J. Wang
86
The spectral bounds (8.4) can be used for the bilinear form arising from discretizing time-dependent Stokes problems. The appearance of e is due to the time stepping parameter At. Similar bilinear forms can be obtained for the pressure unknown by eliminating the vector unknown u in the steadystate Stokes equation. For more details, we refer to Bramble and Pasciak [7]. We remark that the bilinear form a~(., .) is equivalent to the Dirichlet form d(., .) if the parameter e is bounded away from zero by a fixed positive constant Co (i.e., e >_ Co). In this case, the equivalence (8.4) holds true under the condition of Theorem 5. In practical computations, one has two alternatives in solving the discrete problem (2.6) with the wavelet-like basis presented in the previous sections. The first makes use of the explicit form of the basis functions (I - Q]-1)r j) to assemble the corresponding stiffness matrix. The second uses the stiffness matrix assembled from using the standard nodal basis and then performs a change of basis in the process of iterations. The first approach has a difficulty in that the assembly of the global stiffness matrix is no longer local (element-wise). In general, one is recommended to adopt the second approach because the corresponding stiffness matrix is much easier to assemble. In this case, the wavelet-like hierarchical basis actually provides a preconditioning technique for solving the discrete problem (2.6). The rest of this section will describe some preconditioning procedures for the model problem. A more detailed discussion can be found in [41] and [42]. 8.1
Preconditioners
Here we outline two preconditioners for the elliptic operator A (k) : Vk --* Vk arising from the bilinear form a(.,.). The preconditioners will be constructed by using the following wavelet-like multilevel hierarchical decomposition of Vk:
= Voe vl e v) e . . . e
v:,
where
Vj 1 -- ( I - Q ; _ I ) ( I j - I j _ I ) V . The preconditioners (to be defined below) shall be called AWM-HB (Approximate-Wavelet Modified Hierarchical Basis) preconditioners. The following operators are needed in the construction of the AWM-HB preconditioners: 9 In each coordinate space V~, there exists a discretization operator
A~kl)" Vk1 --* Vk1 as the restriction of A (k) onto the subspace V~ defined by
(A~kl)r ~pl) _ a(r
~1)
~, ~1,
~1 E Vk1.
(8.1.1)
87
Wavelet-Like Multilevel Preconditioning
9 Similarly, we define A ~ )" -
Vk-1 ~
V1
and A ~ )" V~ V~eVk_I
-
---* V k - 1
by
eVk 1.
(8.1.2)
Thus, the operator A (k) naturally admits the following partition" A(k)--[
A~kl) A ~ )
A~kl) A (k-l)
] } Vkl } Vk-1.
(813) " "
Let B~k) 1 be given approximations (symmetric positive definite operators) to A~ ) such that for some positive constant bl the following holds" (A~kl)~pl, ~ 1 ) ~ (B~)~I, ~ 1 ) < (1 + bl)(A~kl)9~1, 9~1), ~/~1 E Vk1. (8.1.4) Let A - A(J) be the discretization operator for which a preconditioner is necessary in practical computation. The following explains how the preconditioners can be constructed based on the block structure of A(~) in (8.1.3). Definition 3. (Multiplicative A W M - H B preconditioners) Let B - B (J) be the multiplicatiye A W M - H B preconditioner of A. It is defined as follows: 9 Set B (~ - A (~ 9 For k -
1 , . . . , J , set
B(k) -
B~ A21)
0 B (k-l)
I 0
Bll
I
.
Definition 4. (Additive A W M - H B preconditioners) Let D - D (J) be the additive A W M - H B preconditioner of A. defined as follows: 9 Set D (~ - A (~ 9 For k -
1 , . . . , J , set 0
0 ]
D (k-l)
"
It is
P. Vassilevski and J. Wang
88 8.2
M a i n r e s u l t s for t h e A W M - H B
preconditioners
A spectral equivalence between A and its preconditioners B and D has been established in [41]. The result can be stated as follows. If the tolerance in (7.1.1) is sufficiently small such that r _< 7"0 for some small r0, then there are two absolute constants cl and c2 satisfying
cl(Sv, v) <_ (Av, v) <_c2(Sv, v)
g v C Vj,
(8.2.1)
where S - B (J) or D (a). The estimate (8.2.1) is based on the following results: (A) There exists a constant (7"N ) 0 such that J
IQovl~ + ~ 22'11(Q,- Q,-~)vll~ ~_ ~NII~II~
V v e V.
(8.2.2)
s--1
(B) There exist constants ~ri > 0 and (5 C (0, 1) (in fact, if h i - 7hi-l,1 then ~ - ~22) such that the following strengthened Cauchy-Schwarz inequality holds for any i < j"
a(~i, ~j)2 _ o.ir
ii,jl12o
V~iCV~, ~j C ~ .
(8.2.3)
Here Aj - O ( h - f 2) is the largest eigenvalue of the operator A(J). The inequalities in (A) and (B) have been respectively verified by Oswald [30] and Yserentant [44, 45]. One important feature of the partition (8.1.3) is that the block A~ ) is well conditioned; this can be seen from Lemma 3. In particular, the block A~ ) is spectrally equivalent to its diagonal part. Thus, the J acobi preconditioner would be a good choice for B ~ ) (see (8.1.4))in approximating A~ ). R e m a r k 4: If one does not assume the strengthened equality (B), then the estimate (8.2.1) for S = B still cl = O(1) and c2 = O(log 2 25). In the case S = D, not required in the equivalence (8.2.1). See Griebel related details. 8.3
Cauchy-Schwarz inholds with constants the condition (B) is and Oswald [19] for
On t h e a p p r o x i m a t e L 2 - p r o j e c t i o n
Denote by Q~-I the approximate L2-projections onto the subspace Vk-1. We begin with describing algorithms for computing the action of Q~-l. For any v E V(1) , let v - [ V0l ] }N'k\A/'k-1 }A/k- 1
beitscoefficientvector
with respect to the standard nodal basis of Vk; the second block-component
89
Wavelet-Like Multilevel Preconditioning
of v is zero since v vanishes on Ark-1. The operator Q~_I can be designed by approximately solving the following equation: (Qk-l V, w) -- (v, w),
V w E
(8.3.1)
Vk-1.
1--.. 0.9---. 0.80.7--. 0.60.5-. 0.40.30.20.12
20 0
0
F i g u r e 1. Plot of a HB function (no modification).
Let Ik_l
--"
(with the abbreviation
}./V'k-1
12-
and /kk-1 -- Ik_rl be the natural coarse-to-fine, and respectively, fine-tocoarse transformation matrices. For example, if the nodal basis coefficient vector of a function v2 E Vk-1 in terms of the nodal basis of Vk-1 is v2, then its coefficient vector with respect to the nodal basis of Vk (note that V2
C Vk-1 C Vk) will be
k
Ik-lV2
--
[
t.
J12v2 V2
}JV'k \ .hi'k-1 }-~k-1
"
Denote by Gk -- {(r r EAf~ the mass (or Gram) matrix at the k-th level. Then (8.3.1) admits the following matrix-vector form:
w2rGk-lV2-
(/k_ 1
w2)TGkv,
V W 2.
Here v2 and w2 are the nodal coefficient vectors of Qk-1 v and w E Va-1 at the ( k - 1)th level respectively. Therefore, one needs to solve the following mass matrix problem" Gk-lV2 - - Ik-lGkv. (8.3.2)
P. Vassilevski and J. Wang
90
In other words, the exact L2-projection Qk-lv is actually given by
G-~l_lI~-lakv. Hence
(6;!11k
IIQ -,vll0 -
l Ck v) T a k _ l ( -1
V)
k
(8.3.3)
-
= IIGS 89I,k- 1G~ vii 2.
1-,
0 . 8 "--
0.6-, 0.4--
0 . 2 ""
0 "-.
-0.2 20
15 ~
10
~ 0
0
10
15
20
Figure 2. Plot of a wavelet-modified HB function; m = 2.
To have a computationally feasible basis, we replace G~-ll by some approximations (~-11 whose action can be computed by simple iterative methods applied to (8.3.2). Such iterative methods lead to the following polynomial approximations of G~-11"
Gk!1 - [ I - ?rm (ak_1)]ak11, where 7rm is a polynomial of degree m >_ 1. The polynomial ~rm also satisfies 7rm(0) - 1 and 0 _< 7r,~(t) < 1 for t E [a, fl], where the latter interval contains the spectrum of the mass matrix Gk-1. Since Gk-1 is
91
Wavelet-Like Multilevel Preconditioning
well conditioned, one can choose the interval [a,/3] independent of k. Thus, the polynomial degree m can be chosen to be mesh-independent so that a given prescribed accuracy r > 0 in (7.1.1) is guaranteed. More precisely, given a tolerance r > 0, one can choose m = re(r) satisfying
i]G~_I (G-kll - O-k:l) Ik-lGkv]l = [[G~_lgrm(Gk-1)Gklllk-lGkv[I
ilQ~_,v-e~-,,ilo
-
< -
=
max
7rm(tlllG-~)lI~-lGkvll
m~x
~',,,(t)ilek-,vllo.
te[o,,~']
tED,#]
Here we have used identity (8.3.2) and the properties of ~rm. The last estimate implies the validity of (7.1.1) with r > max 7rm (t). - tED,~] A simple choice of ~rm(t) is the truncated series m-1
(1 - 7r.~(t))t -1 - p . ~ - l ( t )
- fl-1 E (1 - fit) k,
(8.3.4)
k=0
which yields G~-_I1 - p m - l ( G k - 1 ) . from the following expansion:
We remark that (8.3.4) was obtained
OO
1 - tf1-1E(1
-tfl-1)
k
t E [a fl] ~
9
k=O
With the above choice of the polynomial ~m(t), we have ~m(t) -
1 - tp~_l(t)
- tZ - ~ ~
(1 - Z - ~ t ) k - (1 - Z - ' t ) ~ .
k>m
It follows that max
~m(t)-
1-
In general by careful selection of ~m, we have max 7rm(t) < Cq m for some ' te[~,Z] constants C > 0 and q C (0, 1), both independent of k. Since the restriction on r was that r be sufficiently small, one must have m - O(log r - l ) .
(8.3.5)
The requirement (8.3.5) obviously imposes a very mild restriction on m. In practice, one expects to use reasonably small m (e.g., m - 1, 2). This
P. Vassilevski and J. Wang
92
observation is confirmed by the numerical experiments performed in Vassilevski and Wang [42]. We show in Figure 1 a typical plot of a nodal basis function of V(1), and in Figure 2, we show its approximate-wavelet modification for m - 2. The conjugate gradient method was employed to provide polynomial approximations for the solution of the mass-matrix problem (8.3.2). 8.4
M a t r i x f o r m u l a t i o n s of t h e A W M - H B p r e c o n d i t i o n e r s
We now turn to the description of the multiplicative and additive AWM-HB methods in a matrix-vector form. Let us first derive matrix representations for the operators A~ ), A ~ ), and A(~) introduced in (8.1.1)and (8.1.2). In what follows of this section, capital letters without overhats will denote matrices corresponding to the standard nodal basis of the underlined finite element space. For example, A (k) denotes the standard nodal basis stiffness matrix with entries ra~(k) ~9i , l(k),~ q~j )~,,xj~fk. For any v E Vk and its nodal coefficient vector v, we decompose v as follows:
where w2 E Vk- 1 is uniquely determined as w2 - Ik- 1v + Q~_ 1(Ik - Ik- 1)v. Our goal is to find a vector representation for components of v. Since the above decomposition is direct, it is clear that there are vectors 91 and v2 satisfying
v-(I-I:-lG-~l-lI:-lGk)[ 91 ]0
}Afk-1}A/' \A/'k-1 k + i:_192.
[,1]
(8.4.1)
The vectors 91 and 92 represent the components of our wavelet-modified two-level HB coefficient vector ~ -
"~2
of v.
Now, consider the following problem A v - d,
(8.4.2)
which is in the standard nodal basis matrix-vector form. We transform it into the approximate wavelet modified two-level HB by testing (8.4.2)
(I-I~_IG~llI~-IGk)""[ @1
[ and I~ 1@2 for 0 J arbitrary @1 and @2. By doing so, we get the following two-by-two block system for the approximate wavelet modified two-level HB components of (denoted by Vl and ~r with the two components
k
,1
Wavelet-Like Multilevel Preconditioning
93
where ^
A~I)-[I [I
0]
(
) ( ~" )[I] i - a ~ 2 _ ~ a ; ~ l I 2 -1 A(~) i-i2_~a-i!1~2-~a~ o
O] .(I-
GkI~_IG-~I_II~ -1)
A(k)I~_l,
A~kl) - I~-IA (k) I - I~_~a;~_lI~-lak
o
^ k 1 _ A(k-1) A~k2)I k - 1A(k )Ik_ Note that having computed Vl and v2, the solution v of (8.4.2) can be recovered by using the formula (8.4.1), i.e., V -- g l V l nu Y2"r
where r,
-
I-ILIO;_~II~-*c~
Y~
= I~_1.
,-
~=
o
'
We have, v-Y~,
, Y-[Y1, Y,], Y1-Y1 (~), y~-y~(~)
The transformed right-hand side vectors of (8.4.3) read similarly as follows: dl d2
-
[I O] (I-GkI~_l(l-~llI~ / ~ - l d - yTd.
-1)
d-
yTd,
Therefore, the multiplicative AWM-HB preconditioner B (k) from Definition 3, starting with B(~ = A (~ takes the following block-matrix form:
~ ][ 0 1
844,
The preconditioner B (k) is related to/~(k) in the same way as A (k) to ~(k). More precisely, one has
~(k)_
[y~,y2]rB(k)[yl,y2] ' B(k)-' _ [yl,y2]~(~)-'[yl,y2]r"
We will show below that the inverse actions of B (k) can be computed only via the actions of A (k), Y1, Y2, and yT, yT in addition to the inverse actions o f / ~ ).
'
94
P. Vassilevski and J. Wang
We point out that (8.4.4) has precisely the same form as the algebraic multilevel method studied by Vassilevski [37] (see also Axelsson and Vassilevski [4] and Vassilevski [38]). Observe that, in (8.4.4) , Bii ^(k) is an appropriately scaled approximation of A^~]). We have shown that A~])is well conditioned (see Lemma 3). Thus, it is possible to utilize some simple polynomial approximation B~k1) for A(i]) in the implementation. However, in order to take into account any possible jumps in the coefficient of the differential operator, it would be preferable to compute the diagonal part of .4~]). This is computationally feasible since the basis functions of V~ - ( I - Q~_i)V (i) have reasonably narrow support if m is not too large, which should be the case in practice. Nevertheless, one can employ in actual implementation the CG method to compute reliable approximate actions of ~ ] ) - 1 . A l g o r i t h m 1 (Computing inverse actions of B (k)) The inverse actions of B (k) are computed by solving the system B(k)w = d, with the change of basis w w
-Yi~
dl d2
-Y~d, = Y~d,
^
Y@. Namely, by selling
+Y2~2- [Y~, Y2] ~2
w -- B(k)-ld is computed via the solution of B(k)~r -- ~l as follows: Forward recurrence: 1. compute zi - ~il ~(k)-I ^di, 2. change the basis; z.e., compute z - Yizi, " ^ ~;(k ).. A(k)z), 3. compute d2 "- d 2 - ~2i zi - y T ( d ~. compute ~ r 2 - B(k-i)-l~t2, 5. change the basis, i.e., compute v -
Y2@2.
Backward recurrence: 1. update the fine-grid residual, i.e., compute
di^ "- di^ _ .~(k)~,122 - y T ( d - A(k)Y2@2) - y T ( d - A(k)v),
95
Wavelet-Like Multilevel Preconditioning 2. compute ~r 1 -3. get the solution by w
-
Y l ' ~ r 1 + Y2'~r2 -
Y1'~"1
+ V.
End. Note that the above algorithm requires only the actions of the standard stiffness matrix A (k), the actions of the transformation matrices Y1 and }I2 and their transpositions yT and yT, the inverse action of/3~), and some suitable approximations to the well conditioned matrices .4~). The actions of y - 1 are not required in the algorithm. We now formulate the solution procedure for one preconditioning step using the multiplicative AWM-HB preconditioner B - B(J). A l g o r i t h m 2 (Multiplicative A WM-HB preconditioning) Given the problem Bv- d Initiate d(J) - d. F o r w a r d r e c u r r e n c e . For k -
d down to 1 perform:
1. Compute ~k)_
Ix
O]
(I- GkI~_IG;llI~-1) d (k),
2. Solve 1 __ d~ ^ k), B^(k)~r ll
3. Transform basis
W--(I-- Ik-lGkllik-l~k) [ ~W1
}d~fk_l}'/~fk\'Afk-1
~. Coarse-grid defect restriction d(k-1)
__
= 5. Set k -
k-1.
i~-ld(k) -- ]-i21 7(k)^Wl / ~ - l ( d ( k ) - A(k)w),
If k > O go to (1), else,
6. Solve on the coarsest level A(O)x(O)_ d(O).
P. Vassilevski and J. Wang
96 Backward
recurrence.
I. Interpolate result: Set k := k + 1 and compute x(k)_ i~_lx(k-1),
2. Update fine-grid residual:
._
a~)
Z(~)v(k-~)
-- ~l~k)_[I O](I-Gke2_lS"~llI~-l)A(k)x (k) 3. Solve B1 l^(k)~ 1 _ ~ k ) ,
~. Change the basis 0
5. Finally, set x (k) = x (k) + w.
6. Set k := k + 1. If k < J go to Step (1), else s e t v = x (J).
End. Similarly, one preconditioning solution step for the additive A W M - H B preconditioner D - D (J) takes the following form: 3 (Additive A WM-HB Preconditioning) Given the problem Dv=d
Algorithm
Initiate d (J) -- d. Forward
recurrence.
For k = J down to 1 perform:
1. Compute
~k)_ IX 0] (I--GkX~_lVkll Ik-1) d (k),
97
Wavelet-Like Multilevel Preconditioning 2. Solve
B1 l^(k),~r1 __ ~k)
3. Transform basis
X(k)
--(I--Ikk_lS;!1I~-lGk)[ Wl] 0
\Nk-1 }&-I
~. Coarse-grid defect restriction
d ( k - 1 ) _ i~-ld(k) 5. Set k - k - 1 .
I f k > O go to (1), else
6. Solve on the c o a r s e s t level
A(O)x(O) _ d(O). Backward recurrence. 1. Interpolate result: Set k "- k + 1 and compute
w-
I~_1 x(k-1),
2. Update at level k
x (k) - x (k) + w, 3. Set k "- k + 1. I l k < J go to Step (1), else set
v - x (J). End.
For both the additive and multiplicative preconditioners, it is readily seen that the above implementations require only actions of the stiffness matrices A (k), the mass matrices G (k), and the transformation matrices I~_ 1 and I kk-1 . The approximate inverse actions of A^ ~ ) can be computed via some inner iterative algorithms. Similarly, the action of G~-_I1 can be computed as approximate solutions of the corresponding mass-matrix problem using m steps of some simple iterative methods. Therefore, at each discretization level k, one performs a number of arithmetic operations proportional to the degrees of freedom at that level, denoted by nk. In the case of local mesh refinement, the corresponding operations involve only the stiffness and mass matrices computed for the subdomains where local
P. Vassilevski and J. Wang
98
refinement was made. Hence, even in the case of locally refined meshes, the cost of the AWM-HB methods is proportional to nj. The proportionality constant depends linearly on m = O(log r - l ) , but is independent of J (or h). Some numerical results for the AWM-HB preconditioners can be found from Vassilevski and Wang [42]. A performance comparison with the BPX method [10] and Stevenson's method [36] on more difficult elliptic problems in three dimensions, and in other applications such as interface domain decomposition preconditioning, is yet to be seen. w
Numerical experiments
In this section we present some numerical results to illustrate the efficiency of the method discussed in Section 7. Consider the boundary value problem of seeking u satisfying s
Vu u
-f - g
inFt, on c9~.
(9.1)
Here, / E L2(~), g E H 89 and b - [bib2] are given single-valued or vector-valued functions. We assume that all given functions are sufficiently smooth on their domains. For simplicity, we take ~ to be a square domain and g - 0 on c9~. If u is a solution of (9.1), then it solves the following problem: .A4cu - - S V . (b s
+ s
- - 6 V . (b f) + f
in ~,
(9.2)
subject to the boundary condition u - 0 on 0~. Here 5 > 0 is a parameter. The purpose of considering the problem (9.2) is to get the so-called streamline derivative o~, _ b. Uu in a variational formula for u More precisely, by testing (9.2) against any r E H ] ( ~ ) one obtains
b (u, r
-
r e ) + (r b. b. r e )
+ 6(_b.
b. r e ) (9.3)
-- (f6, r for all r E H~(fl). Here ]'6 - f 9.1
5U. (b f).
Galerkin discretization
The Galerkin method for the approximation of u is based on the variational problem (9.3). Let V - Vh be a C~ finite element space of piecewise polynomials corresponding to a quasiuniform triangulation Th of
Wavelet-Like Multilevel Preconditioning
99
f~. The Galerkin approximation is a function Uh E Vh such that
b~(~h, r
- ~(V~h, r e ) + (r -e5 E
V~) + 6(b. V~, b. r e )
f Auh(b.Vr
TETh T
(9.1.1)
= (f6, r for all r E Vh. For continuous piecewise linear functions, one has AUh -- 0 on each element. It follows that the discrete problem seeks Uh C Vh such that
~(v~h, r e ) + (r b. v~h) + 6(5. v~h, b. r e ) - (f~, r
(9.1.2)
for all r E Vh. The convection term is assumed to satisfy V.b_< 0
in ft.
(9.1.3)
For a convergence analysis of the streamline diffusion finite element approximation Uh, we refer to [23] and [2]. 9.2
Numerical tests
We choose the same test examples as in [2]. Namely, _b- [ ( 1 - x cos a) cos a(1 - y sin a) sin a], for various angles c~. Note that V. _b- - 1 . The right hand side f is chosen so that u - x(1 - x)y(1 - y) is the exact solution. Thus, the right-hand side function f is e-dependent. The stopping criterion is that the relative error of the residual be less than 10 - s in the discrete L2-norm. The objective is to test the number of iterations in the solution procedure by using the wavelet-like hierarchical basis. The matrix form of the discretized problem (9.1.2) reads as follows" Au-f. (9.2.1) Here A is a nonsymmetric matrix. For small e, it is very difficult to find a good preconditioner for A. It was seen in [2] that a block-ILU factorization method turns out to be very robust with respect to arbitrary positive e, though very little is known theoretically about this good performance. For any finite element function v, we use the bold face v to denote the vector with respect to the nodal basis and 9 the vector representation with respect to the modified hierarchical basis. In the implementation, the modified hierarchical basis is employed to provide a preconditioner for the global stiffness matrix A as follows. Let Y be the transformation from -~ to v such that v - Yg. The preconditioner is given by P - ( y y T ) - I with y T being the conjugate of Y. We discuss the computation of Y and y T .
100
P. Vassilevski and J. Wang
Let
and y(k)_ g-l"
(9.2.3)
Here, G (k) stands for the mass matrix at kth discretization level, I2_ 1 is the natural coarse-to-fine interpolation matrix from ( k - 1)th grid to the kth one a n d / ~ - i _ (/~-l)T" Also, (~(8)-1 is an approximate inverse of the mass matrix G(') 9 For example, a good choice for G (8)-1 w would be an approximate solution of G(~)x = w by polynomial iterative methods. In practice, the J acobi and conjugate gradient methods are good candidates. The numerical results in this section are based on the Jacobi iterative method with two iterations. A l g o r i t h m 4 For any given ~ - (9~J) ,''', Y 9 is computed as follows:
~1), ~(0))T,
the action v -
9 Set v(~ - 9(0), 9 Fork-1
toJ
do
9 v - v(J),
Algorithm 5 yTd
For any given d is computed as follows:
(d~ s) , . . - , d~1), d(~ T, the action a
-
. Set d(J) - d (~ 9 Fork-J
down t o l
do
~(k-i) 9 a-(a(J )
a(~ 9
Once the preconditioner P - ( y y T ) - I is known, one can solve the discrete problem (9.2.1) by using the generalized conjugate gradient method employed in [2]. We comment that this simple preconditioner may not work well for convection-dominated diffusion problems. This fact can be seen from the numerical results illustrated in Tables 1-4.
Wavelet-Like Multilevel Preconditioning
101
T a b l e 1. Iteration counts for h -1 = 64, a = 75 ~
iter
c-1 6 - 0.001 30
e-O.1 6 - 0.01
28
c-
c - 10 -2 6-0.1 42
10 -3 5 - 1.0 59
...........
c - - 10 -4 5--10
74
T a b l e 2. Iteration counts for h -1 --64, a = 105 ~
iter
c-1 5 - 0.001 31
c-O1 6 - 0.01 27
c-lO
2
c-lO -a 6 - 1.0
6 - 0.1 70
e - - 10 . 4 6--10
........
*
.
....
T a b l e 3. Iteration counts for h -1 = 128, c~ = 75 ~
iter
e-1 5 - 0.0025
c -0.1 5 - 0.025
32
29
....
c - 10 -2 5 -- 0.25 48
T a b l e 4. Iteration counts for h -1 = 128, a = 105 ~
iter
c-1 6 - 0.0025 32
c-O.1 5 - 0.025 29
c-
10 -2
6 - 0.25 71
In Tables 1-4, " , " is used to indicate a non-convergence in 100 iterations. It is clear t h a t the use of the wavelet-like hierarchical basis gives an efficient p r e c o n d i t i o n e r if the p r o b l e m is not c o n v e c t i o n - d o m i n a t e d . T h e present i m p l e m e n t a t i o n of the modified hierarchical basis is c o m p a rable with the additive p r e c o n d i t i o n i n g m e t h o d discussed in [42]. T h e inform a t i o n is also c o n t a i n e d in A l g o r i t h m 3 of this p a p e r with B"(k) l l _ Ch.~2ik , where Ik s t a n d s for the identity m a t r i x . Acknowledgments. T h e research of Vassilevski is s u p p o r t e d in p a r t by the U.S. NSF g r a n t INT-95-06184 and also in p a r t by B u l g a r i a n M i n i s t r y for E d u c a t i o n g r a n t M M - 4 1 5 , 1994. T h e research of W a n g is s u p p o r t e d in p a r t by the NSF g r a n t INT-93-09286. T h e a u t h o r s are also grateful to Dr.
P. Vassilevski and J. Wang
102
Oswald for his careful comments and remarks on the paper. References [1] Arnold, D. N., R. S. Falk, and R. Winther, Preconditioning in H(div) and applications, Math. Comp., to appear. [2] Axelsson, O., V. Eijkhout, B. Polman, and P. S. Vassilevski, Incomplete block-matrix factorization iterative methods for convectiondiffusion problems, BIT 29 (1989), 867-889.
[3]
Axelsson, O. and I. Gustafsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximations, Math. Comp. 40 (1983), 219-242.
[4]
Axelsson, O. and P. S. Vassilevski, Algebraic multilevel preconditioning methods, II, SIAM J. Numer. Anal. 27 (1990), 1569-1590.
[5]
Bank, R. E. and T. Dupont, Analysis of a two-level scheme for solving finite element equations, TechnicM Report CNA-159, Center for Numerical Analysis, The University of Texas at Austin, 1980.
[6] Bramble, J. H., Multigrid Methods, Pitman Res. Notes Math. Ser., vol. 294, Longman, Harlow, U.K., 1995 (2nd Ed.). [7] Bramble, J. H. and J. E. Pasciak, Iterative techniques for time dependent Stokes problems, ISC Report, Texas A&M University, 1994, preprint.
[8]
Bramble, J. H., J. E. Pasciak, J. Wang and J. Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), 1-21.
[9]
Bramble, J. H., J. E. Pasciak, J. Wang, and J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 5 7 (1991), 23-45.
[10]
Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22.
[11] Brezzi, F. and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. [12] Carnicer, J. M., W. Dahmen, and J. M. Pefia, Local decompositions of refinable spaces and wavelets, Appl. Comput. Harmon. Anal. 3 (1996), 127-153.
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[13] Ciarlet, P., The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1977. [14] Chui, C. K., An Introduction to Wavelets, Academic Press, Boston, 1992. [15] Dahmen, W. and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315- 344. [16] Dahmen, W., A. Kunoth, and K. Urban, A wavelet Galerkin method for the Stokes equations, Computing 56 (1996), 259-301. [17] Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [18] Girault, V. and P.-A. Raviart, Finite Element Methods for NavierStokes Equations, Springer-Verlag, Berlin, 1986. [19] Griebel, M. and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), 163-180. [20] Griebel M. and P. Oswald, Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems, Adv. Comput. Math. 4 (1994), 171-206. [21] Hood, P. and C. Taylor, A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids 1 (1973), 73-100. [22] Jaffard, S., Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 29 (1992), 965-986. [23] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, Cambridge, 1994. [24] Kotyczka, U. and P. Oswald, Piecewise linear prewavelets of small support, in Approximation Theory VIII, vol. 2, C. K. Chui and L. L. Schumaker (eds.), World Scientific, Singapore, 1995, pp. 235-242. [25] Lorentz, R. and P. Oswald, Constructing economical Riesz bases for Sobolev spaces, presented at the 9th Domain Decomposition Conference held in Bergen, Norway, June 3-8, 1996. [26] Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2(IR), Trans. Amer. Math. Soc. 315 (1989), 69-88. [27] Maubach, J. M., Local bisection refinement for n-simplicial grids generated by reflection, SIAM J. Sci. Comput. 16 (1995), 210-227.
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[28] Mitchell, W. F., Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Star. Comput. 13 (1992), 146-167. [29] Ong, M.-E. G., Hierarchical basis preconditioners in three dimensions, SIAM J. Sci. Comput., to appear. [30] Oswald, P., Multilevel Finite Element Approximation: Theory and Applications, Teubner Skr. Numer., Teubner, Stuttgart, 1994. [31] Rusten, T. and R. Winther, A preconditioned iterative method for saddle-point problems, SIAM J. Matrix Anal. Appl. 13 (1992), 887904. [32] Schatz, A. H., An observation concerning Ritz-Galerkin methods with indefinite bilinear forms, Math. Comp. 28 (1974), 959-962. [33] Schatz, A. H. and J. Wang, Some new error estimates for RitzGalerkin methods with minimal regularity assumptions, Math. Comp. 65 (1996), 19-27. [34] Silvester, D. S. and A. J. Wathen, Fast iterative solution of stabilized Stokes systems, part II: using general block preconditioners, SIAM J. Numer. Anal. 31 (1994), 1-16. [35] Stevenson, R., Robustness of the additive and multiplicative frequency decomposition multilevel method, Computing 54 (1995), 331-346. [36] Stevenson, R., A robust hierarchical basis preconditioner on general meshes, Technical Report # 9533, Department of Mathematics, University of Nijmegen, Nijmegen, The Netherlands, 1995. [37] Vassilevski, P. S., Nearly optimal iterative methods for solving finite element elliptic equations based on the multilevel splitting of the matrix, Technical Report # 1989-09, Institute for Scientific Computation, University of Wyoming, Laramie, WY, USA, 1989. [38] Vassilevski, P. S., Hybrid V-cycle algebraic multilevel preconditioners, Math. Comp. 58 (1992), 489-512. [39] Vassilevski, P. S., On two ways of stabilizing the hierarchical basis multilevel method, SIAM Rev., to appear. [40] Vassilevski, P. S. and J. Wang, Multilevel iterative methods for mixed finite element discretizations of elliptic problems, Numer. Math. 63 (1992), 235-248.
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[41] Vassilevski, P. S. and J. Wang, Stabilizing the hierarchical basis by approximate wavelets, I: Theory, Numer. Linear Algebra Appl. (1996), to appear. [42] Vassilevski, P. S. and J. Wang, Stabilizing the hierarchical basis by approximate wavelets, II: Implementation and numerical experiments, SIAM J. Sci. Comput. (1996), submitted. [43] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Reg. 34 (1992), 581-613. [44] Yserentant, H., On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. [45] Yserentant, H., Old and new convergence proofs for multigrid algorithms, in Acta Numerica, Cambridge Univ. Press, New York, 1993, pp. 285--326.
Panayot S. Vassilevski Center of Informatics and Computing Technology Bulgarian Academy of Sciences "Acad. G. Bontchev" street, Block 25 A 1113 Sofia, Bulgaria p an ayot @iscbg.ac ad. bg Junping Wang Department of Mathematics University of Wyoming Laramie, Wyoming 82071 [email protected]
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II. Fast Wavelet Algorithms: Compression and Adaptivity
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An Adaptive Collocation Method B a s e d on I n t e r p o l a t i n g W a v e l e t s
Silvia Bertoluzza
A b s t r a c t . A wavelet collocation method for the adaptive solution of second order elliptic partial differential equations in dimension d is presented. The method is based on the use of the Deslaurier-Dubuc interpolating functions. The method is tested on an advection dominated advection-diffusion problem, and on a Laplace problem posed on a nonrectangular domain.
w
Introduction
The solutions of many differential equations are very smooth in a large part of the domain, but is not as smooth globally. In order to approximate them in an optimal way one needs to use high order methods, capable of taking advantage of the local smoothness of the solution by using a coarse grid in the greatest part of the domain, but capable, on the other hand of coping with singularities. Among the methods that display this potential one can count the methods based on wavelet bases and multiscale analysis. On one hand, such bases allow one to design methods of arbitrarily high order, and on the other hand they display a local behavior, allowing one to take into account the presence of singularities by locally refining the approximation space. Furthermore, such bases actually provide a singularity detection tool that has been successfully employed in designing adaptive schemes for the numerical solution of several classes of differential equations [9, 17, 27,28]. Multiscale Wavelet M e t h o d s for P D E s Wolfgang D a h m e n , A n d r e w J. Kurdila, and P e t e r Oswald (eds.), pp. 109-135. C o p y r i g h t O 1997 by A c a d e m i c Press, Inc. All rights of r e p r o d u c t i o n in any form reserved. ISBN 0-12-200675-5
109
110
S. Bertoluzza
The fundamental idea underlying such bases is that, in a multiscale analysis, the space ~ corresponding to a uniform discretization with step size h - 2-J can be obtained as the span of two different bases, the nodal basis of scaling functions {r and the hierarchical basis of wavelets {r m = j 0 , . . . , j - 1}. Such a basis is obtained by decomposing t~ as - Om=jo Win, where the detail or complement spaces ~ Vm being Wm are defined as Wm = ( P m + ~ - Pm)Vm+~, Prn : L2(IR) the L 2 orthogonal projection, or in the biorthogonal case an L 2 bounded projection operator. The change of basis between the two can be performed by applying the Fast Wavelet Transform (FWT), which takes O(2 j) operations. It is important to underline the known fact that when expressing a function in Vj in terms of the nodal basis, generally, no matter how smooth it is, all the coefficients will be needed in order to get a good approximation, while, when expressing the same function in the wavelet basis, in order to get an approximation of the same order, one would usually need only a subset of the coefficients (essentially the ones corresponding to those basis functions whose center is close to singularities). In other words, the nodal basis naturally corresponds to taking a uniform discretization and the wavelet basis to a nonuniform one. In the solution of PDEs one would like to work with the second basis, which is the one allowing high order approximation in smoothness regions and at the same time grid refinement near singularities. Unfortunately, working with such a nonuniform basis presents some difficulties. Let us consider for example the Galerkin approach. Suppose that we have an equation Au = f, and that we want to find the solution of the form Uh -- ~(rn,k)ehh UmkCrn~ (where Ah is a small subset of all the admissible indices (re, k)). In implementing a Galerkin scheme, the need arises in evaluating integrals of the form f A C j k r or of the form f A u h r We recall that computing such integrals by applying a classical quadrature rule (see [15])is not convenient. In fact, in order not to lose the accuracy properties of the scheme (which we assume to be of order M), one would need to choose a quadrature rule which is exact for polynomials of order 2 M - 2. In order to obtain an order M error estimate one needs however to assume that the approximation space is included in H M. On the other hand, it is well known that wavelets are usually much less regular than accurate ( i.e., Crn,~ ~ HM). This implies the need of using a lower order quadrature rule, with a much finer grid of quadrature nodes, which raises the cost of the whole procedure. This difficulty can be overcome by applying a result by W. Dahmen and C. Micchelli [21]. The values of the integrals f ACjkCjn can be computed by taking advantage of the autosimilarity properties of the functions r
Adaptive Collocation Method
111
and then a change of basis can be performed in order to retrieve the desired values. This procedure, however, requires one to express u (or r in terms of the nodal basis ej k. We are therefore obliged to perform this computation on a uniform fine grid. This difficulty is even stronger when we deal with nonlinear operators .T'(u). In fact, when linear expressions are concerned, one can assemble the stiffness matrix once and for all, while when dealing with nonlinearities, (especially the ones which are not of multilinear type, e.g., e ~ , for example), one has to go back to the uniform grid each time one computes f.~(U)r One possible way to overcome such a problem is to use a collocation approach, by which one totally avoids integration. In particular, we propose here a collocation method based on the multiscale decomposition relative to the Deslaurier-Dubuc interpolating functions [23, 24]. This is a class of compactly supported scaling functions, which will be denoted by 0, that satisfy a property of interpolation, i.e., 0(0) = 1,
O(n) = O, for n r O,
(1.1)
rather than the usual property of orthonormality (or biorthogonality). Instead of defining the complement spaces as (Pj+I - Pj)t~+l, in such a construction, which is described in some detail in Section 2, the complement spaces are defined as Wj -- (Lj+I - Lj)~/~+I,
(1.2)
where Lj is a Lagrange interpolation operator. Clearly such an operator is not bounded in L 2, i.e., this multiscale decomposition is somewhat out of the usual L 2 framework. Nevertheless, it turns out that most of the properties of the usual wavelet decomposition still hold, provided that the decomposed function is smooth enough. In particular, a property of smoothness characterization through wavelet coemcients still holds. It is possible to write an equivalent norm for H ~(IRd) in terms of the wavelet coefficients for all s such that H 8 C C u, i.e., in dimension d, s > d/2 (see Theorem 2.1 in the following). This property is perhaps the most relevant one from the point of view of applications to the numerical solution of PDEs, for preconditioning and adaptivity. Another interesting feature of this basis is the particular structure of the fast wavelet transform (which in this case takes the name of Interpolating Wavelet Transform). The algorithmical structure is the same as in the L -~ bounded case, but due to the particular properties of the scaling and wavelet functions, it exhibits some extra features. First of all, due to the interpolation property (1.1), the coefficients of the developement of a given functon f in Vj with respect to the scaling
S. Bertoluzza
112
functions basis are its values at the Lagrange interpolation nodes,
f E Vj,
f-
~f(k/2J)ojk,
(1.3)
k
i.e., the wavelet transform and its inverse allow one to go back and forth from the coefficient space to the physical space. Moreover, with this basis it is particularly simple to work with nonuniform discretizations. In fact, it is possible to assign to each wavelet function a corresponding dyadic point and vice versa. Suppose now that one needs to work with a nonuniform subset of the wavelet basis Bh = {r (m, n) E Ah}. A function f in span{Bh} is uniquely determined by its values at those points corresponding to the reduced set of basis functions. Given the values of f at such points, the coefficients of the expansion of f in the basis Bh are computed by applying a reduced version of the interpolating wavelet transform which works only on such points. Its matrix form is obtained from the matrix form of the full interpolating wavelet transform by simply taking a square submatrix. Moreover, such a submatrix can be assembled very easily, independently of the full interpolating wavelet transform matrix. The aim of this paper is to describe how such a basis can be used for the numerical solution of PDEs by means of a collocation method. Consider, for example the very simple problem: -u"+u=f
inlR.
(1.4)
The solution method proposed consists in first selecting a grid Gh of dyadic points and the corresponding subspace Uh C V), and then looking for Uh E Uh such that for all points p in Gh
- u ' ( p ) + u(p) = f(p).
(1.5)
We remark that in the case Uh = ~ , the method falls into the framework studied in [19] for the wider class of Petrov-Galerkin schemes, and has been extensively studied and tested in [8]. The method is stable, convergent, and good preconditioning techniques are available. In the case where Uh is much smaller than Vj, as we already stressed, the use of a collocation scheme allows us to work at the nonuniform grid level, and there is no need of performing any computation on a uniform grid. On the other hand, such a choice imposes the constraint of dealing with problems whose solution is a least C 2. In other words, the method we propose here is directed to the solution of problems that have such a minimal smoothness, but still have some kind of localized singularity (for instance in an higher order derivative). In particular, the method that we
Adaptive Collocation Method
113
propose aims at getting a high order approximation (possibly in high order norms) for such kinds of problems. There is then the problem of selecting a grid which is well suited to the solution of a given problem. This is usually done through the application of an adaptive scheme. Adaptive refinement based on different types of a posteriori error estimations has been studied recently in several papers [25, 31]. In particular, wavelet based adaptive schemes have been studied in different papers and in different approaches [17, 26, 27], which have in common the fact of taking advantage, in one way or another, of norm equivalence properties of the form (2.17). In the framework of Galerkin discretizations it is known that a hierarchical decomposition of the approximate solution provides an a posteriori error estimator [3, 4, 33]. Although such a theory does not apply to collocation schemes, one can nevertheless try to apply the same principle to such a case. In particular, in the framework of collocation with Deslaurier-Dubuc interpolating multiscale decomposition, the ideas of [27] have been tested on Burgers eqaution in [4]. Here we propose to use such techniques in the framework of the solutions of linear elliptic PDEs in dimension d. The paper is organized as follows. In Section 2 we describe the construction of the Deslaurier-Dubuc interpolating scaling function and the corresponding multiresolution decomposition. We consider both functions on IRd and on (0, 1) d. In Section 3 we review some of the definitions and properties related to the collocation method on a uniform grid. In Section 4 we describe the collocation method on nonuniform grids, while in Section 5 we introduce the adaptive strategy and the resulting adaptive collocation method, which in Section 6 we test on several examples in one and two dimensions. In particular we consider a problem whose solution presents boundary and internal layers, and a Laplace problem on a nonrectangular domain. This is treated by mapping the domain onto a square and solving the resulting problem on the square. The results of all tests are very promising and show that collocation with Deslaurier-Dubuc interpolating wavelets is a valid alternative to Galerkin approximation, especially in the case of adaptive discretization. w
Deslaurier-Dubuc interpolating wavelets
This section is devoted to the multiresolution framework in which the method will be stated. In the following, we will denote by t1" [Is,v,~ the WS'V(f~) norm, and by [[. [[8,~ the H~(D) norm. Let eL be the Daubechies compactly supported scaling function of order L [22]. Recall that eL satisfies, among other properties, (i) supp eL = [0, 2L + 1]. (ii) eL E W R/2,~ for some R > 0 (R is proportional to L).
S. Bertoluzza
114
(iii) eL is refinable with refinement equation of the form (in the Fourier domain' CL(~) - mo((/2)r (2.1) where m0 is a trigonometric polynomial of degree 2L + 1. (iv) eL is orthogonal to its integer translates. (v) Polynomials up to order L can be written as a linear combination of the integer translates of eL. A new scaling function 0 is defined as the autocorrelation of eL. Definition 1. The Deslaurier-Dubuc fundamental function 0 of order N - 2L + 1, is defined by
o(=) - s r162
1.5
Fig l a - Haar scaling function ......................................
0.5 0
-0.5
x)dy.
(2.2)
Fig. l b - D e s l a u r i e r - D u b u c - N = I 1.5
0.5
0
0.5
1
1.5
0
-1
0
1
Figure 1. The Haar function r and its autocorrelation. As a consequence of (i-v) the function 0 satisfies the following properties" 1) supp0-[-N,N]
and
0 E W R'~176
(2.3)
2) The function 0 is clearly refinable. In fact, it satisfies 0 - ICLI2, and hence (2.1) gives 0~(~)- Imo(~/2)12b'(~/2). (2.4) 3) As a direct consequence of the orthogonality of the translates of eL, the function 0 satisfies the following interpolation property:
o(n) -/,~ CL(V)r (Y- n)dy- ~,~o.
(2.5)
4) As a consequence of (v), polynomials up to order N can be written as linear combinations of the integer translates of 0.
Adaptive Collocation Method
115
R e m a r k . The function 0 was originally introduced by G. Deslaurier and S. Dubuc [23] as the limit function of an interpolatory subdivision scheme. Its relation with the minimal phase Daubechies orthonormal scaling functions was pointed out by G. Beylkin and N. Saito [11]. Based on such a function, we can build multiscale decompositions on IR and, by tensor product, on IRd, which we will denote respectively {V)(lR)} and {Vj(IRd)}. In o r d e r to work on (0, 1) d o n e can apply the technique of [15]. The basis functions interacting with the boundaries can be suitably modified in order to get a multiscale analysis {Vj ((0, 1)d)}. In all these cases there will be a Riesz basis for Vj (f~) which will satisfy an interpolation assumption, relatively to the grid-points ,~ - x~ - k/2J E Gj(f~), with Gj(f~) - z d / 2 j N-~. It will be convenient in the following to index the basis functions by the corresponding interpolation grid-point, that is we will use the notation .
lO (f~) - span{O~,, ,X C Gj(t2) - 2 - J z n t2}.
(2.6)
The above mentioned interpolation property will read: for all points )~, # C Gj(a)
1, C~
o,
r #.
(2.7)
It is therefore natural to introduce an interpolation operator Lj ~ Vj (~) defined by
Lj(f)-
~ f(A)oJ),. ~eaj(a)
(2.8)
In such a framework the complement spaces are introduced according to the interpolation operator
W j ( a ) - (Lj+I - Lj)Vj+I(~).
(2.9)
Such spaces admit Riesz bases of the form {Ca,
A C Jj(gt)},
with J j ( a ) -
aj+l(a)\aj(~)
(2.10)
which can be defined, for instance, by simply taking (see Figure 2)
- 0{
c aj+l(a)\aj(a).
(2.11)
R e m a r k 1. In dimension d _> 2 other choices for defining the basis functions CA for the complement spaces are possible. For example, one could define the basis starting with the one dimensional basis of the form (2.11), and use the usual construction of tensor product wavelets (the one that leads
S. Bertoluzza
116 Function theta and psi for N=5 !
!
1
............................
I
!
theta: solid line
i
i
i
!
/%
psi: dashed line 0.5
-5
-4
I
I
-3
-2
l
-1
I
0
.....
|,
I
1
2
I
3
I
4
5
Figure 2. Interpolating Scaling and Wavelet functions. in two dimensions to the three types of wavelet functions r162r162and r 1 6 2 see [28]). Also in such case there is a well-defined correspondence between basis functions for Wj and points of the grid Gj+I (12)\Gj(12). We can use such a basis in the following. For simplicity we will make the choice (2.11), which allows us to avoid some technicalities. Any continuous function f defined on ft for 12 - lRd and for 12 = (0, 1) d can be mapped into its interpolating wavelet transform (IWT), i.e., the sequence of its coefficients IWT(f)-
{{fjo,~, A e Gjo(ft)}, {e~,
)~ e A(12)}},
(2.12)
where A(f~) = Uj>joJj(f~ ) indicates the set of admissible dyadic points. Any function f E C~ may be reconstructed from its transform by means of f - - E fJo,AOi~ + E e),r (2,13) AEGj 0
Proposition
AEA(f~)
I. The following estimates hold: Jackson Inequality. Let f E H'(ft), with d/2 < s <_ N + 1, and let 0
B e r n s t e i n Inequality.
Let f E Yj(~) and let 0 ~_ r < t ~ R. Then llfl]t,~ ~_ C2J(t-r)ilfl]r,~.
(2.15)
Proof: Inequality (2.15) follows from a standard scaling argument. By a standard argument, a sufficient condition for (2.14) is that the operator Lj is bounded from HS(ft) to L2(f~), and that it acts locally as the identity on polynomials of order Y. Since []Ljf[[ 2 ~ 62 -jd ~ k [f(x~)[ 2, the boundedness of Lj is equivalent to the inequality
2 -jd
~ ]f(x~)] 2 ~ C]]f]]s. kEIj(D)
(2.16)
Adaptive Collocation Method 0
.
117
Fig 3 a - M a l n x M_j, j = 6
.
.
.
.
0
.
.
Fig. 3 b - M a t r i x M _ j . .
in dimension d=2
,
50
N 100
250
20
40
8o
nz = 1483
~
Figure 3.
1oo
i~o
0
5o
100
150
nz = 8449
woo
25o
The matrices realizing the IWT.
Now, in the case N - 1, the operator Lj is bounded (we are in the classical case of Ql-finite elements [15]) and hence inequality (2.16) holds. B As a consequence of inequalities (2.14) and (2.15) the following theorem can be proved using the results of [18]. T h e o r e m 1. Let I W T ( f ) denote the sequence of wavdet coefficients of f E C~ as in (2.13). Then
IIIWT(f)II~,2- 2-J~
}11~ +
2(2s-1)j \j_j
~
[e~l2
,
(2.17)
~EJj
gives an equivalent norm for the Besov space B~'2(f~), where d/2 < s < R. We remark that the basis functions are normalized in L ~ rather than in L 2 . When the I W T is computed on a function belonging to ~(f~) it can be evaluated with a fast algorithm, which is equivalent to multiplying the vector of coefficients of f with respect to the basis of ~ by a matrix My. Due to the particular structure of the I W T and of the Deslaurier-Dubuc basis functions, the matrix My can be row permuted to a lower triangular form (see Figures 33-3b). This turns out to be an advantage over the computational point of view in the sense that it allows one to compute the I W T on nonuniform discretization spaces, which will be defined in the following, by multiplying by a suitable squared submatrix of Mj. w
W a v e l e t collocation on u n i f o r m grids
In [8], the function 0 described in the previous section has been used as a trial function in the framework of a collocation scheme for the numerical solution of partial differential equations.
118
S. Bertoluzza
Consider the following model problem. Let ,4 be a second order differential operator of the form
A -
01~1 ~ c , ( x ) - f i ~ + .~(x, u, Vu). 1~1_<2
(3.1)
q..] ~ t ,
Assume that for each of the problems which will be considered in the following, the coefficients ca and the nonlinear operator $" are always chosen in such a way that the operator A is well posed and that the corresponding problem admits a unique solution on either H01(~) or on H I ( ~ ) , depending on whether boundary conditions are imposed or not. Assume also that the coefficients ca are continuous. Consider first a problem defined on IRd: P r o b l e m P1. that
Given f E HS(IRd), (s > d/2), find u E Hs+2(IR d) such
. A u - f,
in IRd.
(3.2)
The following discretization scheme for the numerical solution of problem P1 was proposed: Problem Plj.
Find u j E ~ ( I R d) such that V )~ E Gj(IR d)
Au j (A) - f(A).
(3.3)
R e m a r k . The smoothness assumptions on f imply that f E C~ hence equation (3.3) makes sense. In the case of 9c - 0 (a linear equation), such a scheme is stable and convergent. Applying the results of [19] it is possible to prove the following proposition" P r o p o s i t i o n 2. Assume that in (3.1) we have .7z - 0, and that the coefficients ca are of class C t-2. The following error estimate holds: let u be the solution of P1 and uj the solution of P l j . Suppose d/2 <_ s <_ R, s < t <_ N + 1 and f E H t-2. Then the error satisfies Iluj - ull~ _< c2-JCt-~)lult.
(3.4)
In particular we point out the existence of efficient preconditioning techniques for such a scheme, based either on the use of the orthonormal transform associated to the function eL which appears in Definition 1 [8], or on the interpolating transform (IWT) described in Section 2 (see [4, 29]). An analogous scheme can be used for the solution of Dirichlet boundary value problems on [0, 1]d. Consider for example the following problem.
119
Adaptive Collocation Method
Problem P2. Given f E C~ C2([0, 1]d) satisfying Au-f u - g
1]d) and g E C~
1]d), find u E
in(0,1) d on 0[0, 1] d.
(3.5)
The discretization of problem P.2 by the collocation method as proposed in [8] reads Problem P2j.
Find u j E ~([0, 1]d) such that
A u J ( , ~ ) - f(,~),
V,~ E Gj([0, 1]d) such that ,~ E (0, 1) d,
u j (,~) = g(,~),
V,~ E Gj([0, 1]d) such that ~ E 0[0, 1]d
(3.6) (3.7)
Although no a priori estimates are available for such a scheme, the results of numerical tests show that a good order of approximation is preserved in this case (see [8]). w
C o l l o c a t i o n on n o n u n i f o r m grids
The aim of this section is to discuss how the method described in the previous section (which is based on a uniform discretization) applies to unevenly distributed subspaces of V), corresponding to nonuniform grids. For finite grids of dyadic points Ch of the form Gh = Gjo W Ah,
Ah C A(ft),
(4.1)
(in order to have a minimum degree of approximation, the grid Gh will be assumed to contain the coarse grid Gjo), we will denote VAh = Vjo | span{r
E Ah},
(4.2)
the space spanned by the basis functions corresponding to the selected points. The criteria that will lead to the choice of such a grid in the case of a given problem will be discussed in Section 5. It will also be convenient in the following to introduce the corresponding interpolation operator LAb : C~ ) VAh. Given f E C~ LAh I E VAh is the unique function satisfying Lhhf(,~) = f(A)
for all points A E Gh = Gjo U Ah.
(4.3)
The natural way of generalizing the method of Section 3 is to look for a solution in VAh by collocation at the points of the nonuniform grid Gh. We point out that other choices are possible (see [6]). The discretization of problem P1 with collocation on the nonuniform grid Gh results in
S. Berloluzza
120
Problem P l h . Given Gh -- Gjo U Ah, Ah C A(IRd), find Uh E Vhh such that for all points A in Gh we have
.Auh(A) = f(A).
(4.4)
In the case of a problem with boundary conditions, the grid Gh is split as the union of the set of interior nodes and the set of boundary nodes:
Gh - G~ U G~
(4.5)
with -
n
-
n
(4.6)
Problem P2 can then be discretized as follows: Problem P2h. that
Given Gh -- Gjo m Ah, Ah C A((0, 1)d), find u E YAh such
AUh(A) -- f(A) Uh (A) -- g(A)
for all points A E G~, for all points A G Gbh.
(4.7)
As already pointed out in the introduction, from the point of view of implementation, in the framework of wavelet methods, the choice of a collocation approach has numerous advantages over other approaches, as does a straightforward implementation of a wavelet G alerkin method. The main advantage is that no evaluation of integrals is needed, and therefore the assembling of the stiffness matrix and the evaluation of nonlinear terms do not require any con~putation on the uniform fine grid. Let us consider in more detail how such a scheme can be implemented as opposed to wavelet Galerkin schemes. Typically, four different kinds of operations will be involved (apart from the resolution of the resulting linear system).
Pre-processing. After fixing a finest level jmax, compute the values of the derivatives of 0 at dyadic points t~n
,k
_ 0(8) (2k_~]
'
m -- O,...,jmax -- jo, k - 0 , . . . , N 2 m, s - 0 , 1 , 2 .
(4.8)
This is done according to [30]. Due to the symmetry of the function 0, just the values at positive points are needed. Such a task requires C2 jm"x-j~ operations, independently of the dimension d of the domain of definition of the problem to be solved. Moreover, such values are computed once and for all independently of the problem considered, and then stored (the storage needed is proportional to 2jm"x).
Adaptive Collocation Method
121
Assembling the stiffness matrix. The entries of the collocation matrix relative to the linear part of the operator take the form
, - Z
141<2
Ox~ (,~),
(4.9)
(with possibly Ojo,,~ in the place of r or x j~ in the place of ,~), i.e., computing each entry of the matrix involves the evaluation of derivatives of a basis function at a dyadic grid point, which is performed in a constant number of operations, once the fundamental quantities (4.8) are known. On the opposite, in a Galerkin approach integrals have to be computed. Since, as observed in the introduction, quadrature formulas of the right order are available for computing integrals of the form f a(x)(Ol~loJ~/Ox~)Oj but not for computing f a(x)(Olc~lr162 directly, the assembling of the stiffness matrix has to be performed on the uniform finest grid, and it costs 0(2 ajm~ This consideration also holds if one chooses not to assemble the stiffness matrix but to compute its action on given vectors directly. Moreover in such a case the passage through the uniform grid has to be repeated for each matrix-vector multiplication, and therefore such a choice is not convenient.
Evaluating nonlinear terms. The advantage of using a collocation approach in the context of nonuniform wavelet discretization is particularly evident when dealing with nonlinear expressions, especially those which are not of multilinear type. In fact, in a straightforward implementation of the Galerkin approach, in order to evaluate f Jr(x, tth, ~7Uh)ff),k dx for uh E VAh, one would need to perform the following steps: - Evaluation of Uh(XJkm"x) and VUh(X{ ""x) (0(2 ajm"x) operations). Evaluation of 5c at the points x~'~''~ (0(2 gym's) operations). - Evaluation of f .T(Uh)r176 through a quadrature formula (0(2 djr"a*) operations). - Application of the Fast Wavelet Transform. -
All the four steps need O(2 djm"x ) operations. The simplest way of reducing the complexity of such a stage consists in using, if the regularity of the problem allows it, the collocation schemes (4.4) and (4.7), in which one only needs to evaluate the nonlinear term at the points of the grid Gh. For each point A E Gh this implies computing u(,~) and V'u(,~) (a computation which can be performed in O(jmax- jo) operations), and then evaluating r
Reassembling the matrix. In the implementation of adaptive schemes, there is the need of reassembling the matrix after each refinement step. Practically one needs to assemble the matrix for a problem posed on a grid
S. Berloluzza
122
Gh2 while already knowing the matrix relative to the same problem posed on a different grid Ghl. Working with a hierarchical basis implies, independently of the approach, Galerkin with exact integration or collocation, that if Ghl N Gh2 is nonempty, a part of the matrix is already assembled. T h a t is, we will have to compute only the entries in the rows and columns corresponding to new grid-points. In the proposed collocation approach all the information needed for such an updating is contained in the vectors ~ t~ ~k" Such a vector needs to be recomputed only if the level jma~ increases and, even then, computing t~n,k given t~, k would take only C2 (m'-m) operations. We also want to remark that, in the nonconstant coefficient case, the matrix entries for the Galerkin approach are not exact but they are approximatively computed through some quadrature procedure. Changing the m a x i m u m level jma~ implies the need of changing, at least locally, the precision with which the matrix entries are computed. This implies that some matrix entries relative to old basis function also have to be updated. R e m a r k . Once that the undersampling operation corresponding to the choice of the subspace of the form (4.2) is performed, different choices of the basis for Wj (see Remark 1) give rise to different discretization spaces corresponding to the same nonuniform grid Gh, and therefore, to different collocation methods on the nonuniform grid. The method discussed here is the one relative to the basis of the form (2.11), although numerical tests were also performed for the classical wavelet tensor product basis described in Remark 1. w
The adaptive scheme
Consider now the problem of selecting a discretization space which is well adapted to describe the solution of the continuous problem. One of the ways to accomplish such a task is through the use of an adaptive grid refinement technique. This consists of solving a sequence of problems on a sequence of grids, each chosen by an analysis of the numerical solution at the previous step. The use of the information provided by a multilevel decomposition of an approximate solution as error indicator has already been studied in the framework of Galerkin schemes based on finite elements [2, 3, 33], where the authors proved that it provides an a posteriori error estimation. The use of wavelet transforms for accomplishing such a task has been proposed in several papers ([4, 9, 10, 17, 26, 27]), also in the framework of Galerkin schemes. W h a t we propose here is to apply the same ideas in the framework of wavelet collocation. For the sake of simplicity, let us focus, throughout this section, on problem (P2). Assume that we aim at well approximating u in the W 8,~ norm, for some s >_ 0.
Adaptive Collocation Method
123
Let t = m a x { s - 2, 0}. A first grid can be selected by looking at the behavior of the data of the equation. Assume for simplicity that the coefficients and the boundary data are smooth. We select the first grid G~ -- Gjo tOA0 by requiring that the right-hand side f is well approximated on such grid, in the sense that
Ilf - Laoflit,o~ <_ e.
(5.1)
If no a priori knowledge of the structure of the right-hand side is available, a way of selecting a grid A0 according to such requirement is the following. Assume that f C Hr(f~), r > t. Then, by estimate (2.13) for J big enough, we will have I l f - La fllt,~ ~ e/2. For each j, j0 _< j _< J -
(5.2)
1, we can then retain the points ,~ E
Jj(~) whose corresponding coefficients in the expansion of f is bigger than e'/(2Jtj2), and discard the remaining points, obtaining in such a way a grid A0. It is easy to prove that the following estimate holds.
ILj f(x) - LAof(X)l <_ Ce'.
(5.a)
By choosing e' _< e/(2C) we get the desired grid. The complexity of such a procedure is proportional to the size of the full fine grid Gj. Such a complexity can be reduced by taking advantage of an a priori knowledge of the structure of f, enabling us to substitute Lj by the interpolation on a nonuniform grid in tile a priori estimate (5.2). This could be obtained by knowing the position of tile singularities or that f exhibits a higher regularity in part of the domain. A particularly interesting case is when the considered problem derives from the time discretization of an evolution equation, in which case the right-hand side is constructed using the solution at the previous time step, whose expansion in a nonuniform basis is already available. In any case, we want to remark that such a starting step would also be necessary in a Galerkin approach (see [17]). Once the first grid G~ has been selected, compute a first approximate solution u0 by applying a collocation method in the corresponding space U0 : VAo, according to the formulation (P2h). Following the idea of [27], the grid is then refined with a procedure where the leading element is the IWT. A very simple interpretation of the coefficients of the IWT is possible, which is particularly significant from the point of view of approximation. In fact, it is quite easy to check that the wavelet coefficients eA of a function have the expression e~ = f ( A ) - Ljf(A),
)~ E Jj(f~).
(5.4)
S. Bertoluzza
124
As far as higher derivatives are concerned, it is also trivial to check that e)~ > e/2 is, for A E Jj(f]) implies
[If -- Lj fils,~,suppr
> ce.
(5.5)
In other words, for ~ E Jj(f2) the coefficient e~ measures the lack of approximation of f by Lj f in the neighborhood of the corresponding point. The size of the coefficients of the IWT is also an indicator of the local relevance of high frequency components in u, since the function r ~ E Jj(f2), is oscillating at frequencies around 2J. Both facts imply the need of locally refining the grid. On the other hand, a very small coefficient in the IWT tells us that the corresponding point is not relevant, that is the solution is well approximated already on a coarser grid. The point can then be removed without deteriorating the quality of approximation. We then fix two tolerances 5~ < 5a " one is used to decide whether a coefficient is "small" and the other whether it is "relevant." We remove the points )~ E Jj(~) corresponding to the wavelet coefficients smaller (in absolute value) than 5~/(2J'j 2) and add some neighboring points at the higher level in the neighborhood of the points ,~ E Jj (f2) corresponding to wavelet coefficients bigger that 5a/2 `j. On the new grid G1 that we obtain, we solve the problem by means of a collocation scheme and then we repeat the refining procedure. At the n-th step, given a grid a n and the relative approximate solution un with coefficients u~, n }, - {{ Cjo
e h.}},
(5.6)
we compute the next grid G "+1 by removing the useless points and refining where the approximation is bad. For each point ,~ - ( k l , . . . , kd)/2 j define a set Ux of neighboring points UA - - ~ N {X~ +1 , ] r
(2kl "~" rll, 9
2kd + rid), r l i - --1, 0, 1},
(5.7)
and let A . + I - {A'
> 5,/(2J*J2)}
u L/x, {~: 1~1>6./2J~}
G n+t - Gjo u An+l.
(5.8) (5.9)
The (n + 1)-th approximation space is defined as U ~ + I - VA,+,.
(5.10)
The (n + 1)-th approximate solution is computed by solving the collocation problem: find u '~+1 E U n+l verifying the boundary condition u~+l(A) g(A) at boundary points A E G n+l'b and the equation at interior points .Attn+l(,~) -- f(/~),
,'~ E a n+l'i.
(5.11)
125
Adaptive Collocation Method
We want to remark that one of the strengths of the proposed refining procedure is its extreme simplicity. In particular, it does not need extra computation of any quantities, the coefficients of the approximate solution themselves being the parameters driving the decision on whether to refine or not. Moreover, since the discretization is not based on any triangulation, adding or removing each point has no influence on the other points of the grid, since there is no conformity requirement to fulfill, which causes some difficulty when refining and derefining a finite element triangulation. In this paper we will not address the problem of solving the sequence of linear systems resulting from the adaptive procedure. We want to remark that in order to be able to apply the proposed strategy to real life problems, there is the need of proposing an efficient solver for the resulting linear system. The author is actually studying the problem of designing a suitable iterative solver, by exploring the possiblility of applying multilevel preconditioning techniques [12, 29] and algorithms of the cascadic multigrid type [13] to the collocation scheme proposed here. The aim of this paper is, however, to study the behavior of the collocation scheme and of the refining strategy from the point of view of approximation and of reliability of the grid selection procedure. w
Numerical tests
The method has been tested with the aim of assessing the behavior in terms of approximation, the sensitivity to the choice of the parameters co, ca, and er, and the reliability of the refinement procedure. For the time being the linear systems arising from the discretization on different grids of the equation considered have been solved by a direct method. Two types of tests have been performed. Quantitative tests in one dimension and qualitative tests in two dimensions. 6.1. O n e - d i m e n s i o n a l test p r o b l e m s We tested the proposed scheme on the following two boundary value problems in one dimension. Problem T1.
Find u such that
{
-ul'-f u(0)-a,
in(0, 1), u(1)-b,
for f, a, and b chosen in such a way that the true solutions of P1 are respectively Test 1. Ul
--
Iix a -X 2
- ~ x1 +
for x < .5, 2-!4 f o r x > . 5 .
(6.2)
S. Bertoluzza
126
Test 2. u2 -
{ sin(Trx) 6
~i=oCiX'
9
for x < .5, -
for x > .5,
(6.3)
where the coefficients ci are chosen in such a way that the function u is C 6 (but not C7).
Test 3.
(6.4) P r o b l e m T2.
Find u such that
"-cu"+ u'- 1 u(1)-
in (0, 1), 0,
(6.5/
for c - .01 and c - .001. The method has been tested on both problems for different values of the tolerances Ca and or. All the tests are performed with the DeslaurierDubuc interpolating wavelet of order 5. Depending on the different smoothness properties of the three different solutions, one can observe different behavior of the method. For Tests 1, 2, and 3, the error is considered as a function of the number of degrees of freedom. The function u l has a discontinuity in the third derivative at x - .5. Far from such point the solution is C ~ . Using a uniform discretization is not covenient, since the solution is not regular enough and the high order m e t h o d does not show up. However, by applying the proposed adaptive strategy the method takes advantage of the high regularity of the solution far from the point x - .5, while coping with the singularity by a local refinement. The results of the numerical test (Figure 4a) indicate that for such a problem the adaptive strategy (o 9 cr - 10 -4 , * 9 c~ - 10 -7, + 9 Cr - - 1 0 - 1 0 ; Ca -- 5Or), is clearly winning over the uniform discretization (displayed as a d,=,~ted line). For comparison we also display in a dashed line the results of using piecewise linear finite elements on a uniform discretization ill which the point x - .5 is a node (so that the singularity is treated in an optimal way), and with a dashdot line the results obtained by spectral methods. In Table 1, we display the convergence history for such a test. The number of points discarded (resp. added) at each grid refinement is displayed along with the L ~ error. The solution u2 of test problem 2 has a discontinuity in the seventh derivative, that is, it is everywhere as regular as the method demands in order to have optimal convergence. As a result, the best approximation is obtained by using a uniform discretization. Also the adaptive strategy, both without and with analysis of the right-hand-side (Figure 4b, co - c~ - *" c~ - 10 -7, o: c~ - 10 -13, +" co - c~ - 10 -7) eventually converges to the solution on a uniform mesh. We nevertheless remark that an analysis
Adaptive Collocation Method
127 Problem 1 . - T e s t 1.
10 -2
9
\ 10 -3
\
\
\
'
.
.
.
.
.
.
.
i
\ \
9
1 0 -4
\
tO
\
L~ 10 -5
".
\
\ \
10 -6
"''''....
I
"''''.....
I !
1 0 -7
"'... "''''''''r
10 .8 10 ~
i
|
,
i
= , I
,
10 2 10 3 Number of Degrees of Freedom F i g u r e 4a.
10 4
Results of Test 1.
Problem 1 . - Test 2.
10 .2
.
.
.
.
.
.
i
.
.
.
.
''
'
'~
'1
II
10 -4
+ il(
+
10 -6
O. .9
s
4-
lO -8 O. "'-
0.. . . . . . . . . .
1 0 -10
10 -12 _
1 0 -14
. . . . . . . . . . . . . . . .
101
I
|,,
i
,
i
,,
~
,
,
I
14 10 2 10 3 Number of Degrees of Freedom Figure
4b.
R e s u l t s of T e s t 2.
10 4
S. Bertoluzza
128
of the right-hand side allows us to start from the beginning with a better grid. Also the case comparison with piecewise linear finite elements (dashed line) and with spectral methods (solid line) is displayed. In particular the spectral method can exploit the regularity of the solution, and therefore it gives the best results. Table 1
Test 1- Convergence history of the adaptive procedure.
Tolerances
Nadd
Nrem
N. d.o.f.
L ~ Error
0
16
33
3.616E-5
12
20
41
9.617E-6
18
22
45
5.968
5~ = 5 . E - 7
0
16
33
3.616E-5
5~= 1 . E - 7
12
20
41
9.617E-6
18
16
39
9.892E-6
5a
"--
1.E- 7
5~= 1 . E - 7
1 . E - 10
0
16
33
3.616E-5
5~ = 1 . E - 10
6
26
53
9.607E-6
22
30
61
2.472E-6
24
36
73
6.269E-7
34
38
77
1.581E-7
40
36
73
3.965E-7
5 . E - 10
0
16
33
3.616E-5
5~= 1 . E - 10
6
22
49
9.609E-6
18
30
61
2.472E-6
24
28
65
6.278E-7
26
26
65
1.580E-7
5a =
5a =
It is well known that if a hierachical decomposition of an approximate solution to a PDE is to provide an error indicator, the discretization grid can not be chosen arbitrarily, but has to be coupled to the (unknown) solution in a suitable way. This is expressed by means of the saturation
Adaptive Collocation Method
129
assumption (see [2]). This also holds for the method proposed. Consider, for example, Test 3 (see Table 2). If the first grid Go is too coarse, the adaptive scheme converges in one iteration to a wrong solution. We want to stress that the use of the more refined error indicator of [17] allows one to avoid the need of making such an assumption, whose validity is naturally enforced by the chosen adaptive strategy, which, on the other hand, starts with a choice of a starting discretization providing a good approximation of the right-hand side. The resulting adaptive strategy is proved to be convergent. The numerical tests show that our adaptive collocation method exhibits a similar behavior. If the grid Go is chosen by an analysis of the right-hand side, as proposed in Section 5, the results of the numerical tests show that one gets a good solution as a result of the adaptive procedure. In other words, choosing Go according to Section 5 seems to guarantee also in this case that the grid Go is chosen properly and it is a good starting point for the scheme proposed. For fixed cr = 10 .8 and ca = 510 -s, we tested the scheme with different values of co. In Table 2, we report the number n of refinements performed, the number of points of the initial grid Go and of the final grid G,~-I, and the L 2 error on the final grid. Table 2
Test 3 - Results of the procedure after convergence.
N. points of Go
N. points of G . _ 1
L2 Error
1
17
17
0.707
10.
4
113
505
7.6E-9
.1
4
177
505
7.6E-9
1.E-3
2
245
505
7.6E-9
1.E-5
2
497
505
7.6E-9
1.E-7
2
497
505
7.6E-9
~50
n
No check
,
,
.........
As far as problem T2 is concerned, the structure itself of the equation is different. Here the problems do not derive from a lack of regularity of the data but on a degeneracy of the ellipticity of the operator considered. The operator -ud2/dx 2 + d/dx is elliptic, but its coercivity constant (u) is much smaller than the continuity constant (which is of order O(1)). As a consequence its solutions may develop boundary layers, which are
S. Bertoluzza
130
not detectable by an a priori analysis. The results of the numerical tests are displayed in Figure 5. The L 2 error versus the number of degrees of freedom during the successive refinements is plotted in log-log scale, for different values of the tolerances ea and cr ( o: cr - 10-4; *" er - 10-6; +" e~ - 10-s; Ca - 5Or ). The method gives good results by properly refining the grid in the vicinity of the boundary layer. Test n.4 - Viscosity = .01 1 0 -~
...........
" ".'.~11.
or '
1 0 -2
10 -3
.. 1 0 -'i
\
ill
\
e'l lO_S
-
9 X
.
\
. 9
..
10 -6 .
1 0 -7
10 -8
,
,
,
,
,
,
103
Number of Degrees of Freedom
Figure 5. 6.2
,
10 2
10 ~
Two-dimensional
Adveetion-diffusion
Results of Test 4.
tests problem
Among the problems in which the use of adaptive schemes is necessary, are advection-diffusion problems when the advection term is dominating. Consider for example the following model problem. P r o b l e m PA-D.
Find u such that
-uAu+fl. Vu-f
u - g
i n ( 0 , 1 ) 2, on 0(0, 1) 2.
(6.6)
It is well known that when u < < [/3] the solution of such a problem can develop boundary and internal layers, which need to be resolved. A straightforward application of a uniform dicretization is either too expensive or inaccurate, since Gibb's phenomena (see [34]) may arise if, at least in proximity of the layer, the discretization is not fine enough. Therefore an adaptive procedure is needed. Such problem is the two-dimensional problem corresponding to Test 1. The aim of this test is to check that in the twodimensional case the adaptive method proposed behaves qualitatively in
Adaptive Collocation Method Test 5. ,
131 Test 5. -
Visc=,(X)8 - jmax=6 - d.o.f.=lg71
§ § :::::::::::::::::::::::::::::::::::::t4§
~+nm,_g~
+
+ +
::;laRt;:g~;W,:: ;::+ +
o ~+
1.2,
+ + + + + + + + + + + + + + + + § § + + + § ++++~.++++++++
+ §
o.7
1,
*+ +§+ + + + + + + + + 1 - + + +. +. +.+.+ + + ++~+ . . . . . . . .
+++. +++ ++++§ ++++ § + + + + +++++++++ + .H-+++ § + ++++§ 0.6 +++ §
+
+
+,+
0.5 H, + + + + +
0.4
Vise=.008 - tmax---6 - d.o.f.=Ig71
9
+
+
+ + + + +
+
+
+
+
+
+
+
+
0.8, 0.6,
+
+
+ + + + +
+
+
t.
+ + + + §
+++++++++ ++-~+++ +
+ + , + + + +
+
+
+
0.4,
++++++
~t+:: +§§:,, :; :: ~: +::,* . . . . . . + ;,+:::++:; :1," * +
0.
o . ~
-0.2
.
1
0.4 0
0.1
0.2
0.3
0.4
0.5
F i g u r e 6.
0.6
0.7
0.8
0.g
0.2
,,,<."~,....~
1
02
0.4
"
oo
Test 5. - Selected mesh and computed solution.
the same way as in the one-dimensional case. In particular we want to test how well the grid refinement stategy behaves in the case of an oblique layer. In fact, since the method used is based on a tensor product structure, it might a priori display some preferred directions. The method proposed has been tested on the following:
Test 5. Consider Equation (6.6) with the following data ( 5 - 008, 9
g(x,y)-~O
/3-(-1,-2),
. 1
x<.5, x >
.5.
(6.7)
The problem on each grid has been solved by a direct method. In the first steps of the adaptive procedure we used an artificial viscosity in order to stabilize the scheme. Such a viscosity (which we set equal to 1/2 jm~ eventually becomes zero in the last steps of the procedure. The results of such a test are shown in Figure 6. One can remark that most of the nodes are concentrated in the proximity of the boundary layer, and near the j u m p at the boundary. Several nodes are also concentrated around the internal layer. The result is qualitatively correct, and Gibbs phenomena do not show up. Laplace problem on a nonrectangular
bounded
domain
Consider the following problem. Let f~ be a bounded open set of IR2. P r o b l e m PL.
Find u such that -Au-f
u-g
inf', on cOf~.
Assume that f2 can be mapped onto a square by means of a conformal m a p p i n g of class C" with a C 2 inverse, which are denoted by {~-~(x,y) i-i(x,Y)
{x-x(~,(:) Y-Y(g,i)
'
x, y
E
f~
g,C
C
(0,1) 2'
(6.8)
132
S. Bertoluzza
- {(x, y), x = x(~, ~), y - y(5, ~), (5, ~) C (0, 1)2}.
(6.9)
Then, the solution u of problem PL has the form u = ~(~(x, y), ~(x, y)) where ~ is the solution of
_((r
+ ( ~ ) ~ ) ~ _ ((r + (r162 -2(~.r + ~yCy)~r162 +
(6.10)
-(~** + Cyy)~ - (~** + Cyy)~r = f(.(~, ~), ~(~, ~)), ~(~, ~) = g(x(~, ~), y(~, ~)),
(~, ~) on 0(0, 1) 2.
(6.11)
The method has been tested on problem PL on the domains Test 6. f~-{(x,y),
x-{(-2~ 2+2~+1),
y-~,
({, ~) C (0,1)2}.
(6.12)
Test 7. ~2 = { ( x , y ) ,
x=~(-~+2),
y=~,
(~,~)E(0,1)2}.
(6.13)
The results of such tests are displayed respectively in Figure 7. Note that the grid has been refined near the corners of the domain, where the continuous solution is less regular. w
Conclusions
We proposed an adaptive collocation method based on the use of the Deslaurier-Dubuc interpolating bases. The use of a collocation approach and the particular structure of the chosen basis allows us to avoid computations in the uniform fine grid, which are usually needed in the framework of wavelet Galerkin schemes. A very simple refining strategy, based on the natural hierarchical decomposition relative to the basis chosen, is proposed in analogy with the Galerkin adaptive schemes based on hierarchical bases. Although it lacks a rigorous theoretical justification, mainly due to the fact that we are dealing with a collocation scheme, the proposed refining strategy turns out to be quite effective, as shown by the results of the numerical tests. In particular, numerical results show that the proposed method gives good results when the continuous solution (which has to satisfy the minimal smoothness assumption u E C 2, needed in order to define the method) is very smooth in a large part of the domain, but has some localized singularity in some high order derivative.
Adaptive Collocation Method
Figure 7.
133
Results of Test 6 (top) and 7 (bottom).
A c k n o w l e d g m e n t s . The author would like to thank Prof. Wolfgang Dahmen for the useful discussions on this topic. Work supported in part by C.N.R., Progetto Strategico "Applicazioni della matematica per la tecnologia e la Societs References [1] Babuska, I., A posteriori error estimates for the finite element methods, Internat. J. Numer. Methods Engrg. 12, (1978), 1597-1615. [2] Bank, R. E. and R. K. Smith, A posteriori error estimates based on a hierarchical bases, SIAM J. Numer. Anal. 30 (1993), 921-935. [3] Bank, R. E. and A. Weiser, Some a posteriori error estimates for elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg. 61 (1987), 283-301. [4] Bertoluzza, S., Adaptive wavelet collocation method for the solution
134
S. Bertoluzza
of Burgers equation, Transport Theory and Statist. Phys. 25 (1996), to appear. [5] Bertoluzza, S., A Posteriori error estimates for the wavelet Galerkin method, Appl. Math. Lett. 8 (1995), 1-6. [6] Bertoluzza, S. and A. Cohen, work in progress [7] Bertoluzza, S. and G. Naldi, Some remarks on wavelet interpolation, Mat. Apl. Comput. 13 (1994), 1-32. [8] Bertoluzza, S. and G. Naldi, A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harmon. Anal. 3 (1996), 1-9. [9] Bertoluzza, S. and G. Naldi, An adaptive wavelet collocation method, IAN-CNR, 1994, preprint. [10] Bertoluzza, S. and P. Pietra, Adaptive wavelet collocation for nonlinear BVP's, in Proc. of ICAOS '96, Lecture Notes in Control and Inform. Sci., Springer, London, 1996, pp. 168-174. [11] Beylkin, G. and N. Saito, Multiresolution representation using the auto-correlation functions of compactly supported wavelets, in Progress in Wavelet Analysis and Applications, Y. Meyer and S. Rouques (eds.), Edition Fronti@res, Paris, 1993, pp. 721-726. [12] Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. of Cutup. 55 (1990), 1-22. [13] Bornemann, F. A. and P. Deuflhard, The cascadic multigrid method for elliptic problems, Numer. Math., to appear. [14] Chen, M. and R. Temam, Incremental unknowns for solving partial differential equations, Numer. Math 59 (1991), 255-271. [15] Ciarlet, P. G., The Finite Element Method for E11iptic Problems, North-Holland, Amsterdam, 1978. [16] Cohen, A., I. Daubechies I., and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1993), 5481. [17] Dahlke, S., W. Dahmen, R. Hochmuth, and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, IGPMReport Nr. 124, RWTH Aachen, January 1996. [18] Dahmen, W., Stability of multiscale transformations, J. Fourier Anal. Appl. 2 (1996), 341-364. [19] Dahmen, W., S. Pr6Bdorf, and R. Schneider, Wavelet approximation methods for pseudodifferential operators: I Stability and convergence, Math. Z. 215 (1994), 583-620. [20] Dahmen, W. and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315-344. [21] Dahmen, W. and C. Michelli, Using the refinement equation for evaluating integrals of wavelets, SIAM Jour. Numer. Anal. 30 (1992), 507537.
Adaptive Collocation Method
[22] [23] [24]
[25] [26] [27]
[28] [29] [3o]
[31] [32] [33] [34]
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Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. Deslaurier G. and S. Dubuc, Symmetric iterative interpolation processes, Constr. Appr. 5 (1989), 49-68. Donoho, D., Interpolating wavelet transform, Department of Statistics, Stanford University, 1992, preprint. Eriksson, K. and C. Johnson, Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems, Math. Comp. 60 (1991), 167-188. Gottlieb, D. and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1977. Liandrat, 3., V. Perrier, and P. Tchamitchian, Numerical solution of non-linear partial differential equations using the wavelet approach, in Wavelets and Their Applications, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai (eds.), Jones and Bartlett, Cambridge, MA, 1992, pp. 227-238. Maday Y., V. Perrier and J. C. Ravel, Adaptivit~ dynamique sur bases d'ondelettes pour l'approximation d'~quations aux deriv~es partielles, C. R. Acad. Sci. Paris Sdr. I Math. 312 (1991), 405-410. Meyer, Y., Ondelettes et Opdrateurs, vol. 1-2, Hermann, Paris, 1990. Schneider, R., Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten LSsung grofler vollbesetzter Gleichungssysteme, Habilitation Thesis, Technical University Darmstadt, 1995. Strang, G., Wavelets and dilation equations: a brief introduction, SIAM Rev. 31 (1989), 614-627. Verfiirth, R., A posteriori error estimation and adaptive mesh refinement techniques, J. Comput. Appl. Math. 50 (1994), 67-83. Yserentant, H., On the multi-level splitting of finite element spaces, Numer. Math. 49 (1986), 379-412. Zienkiewicz, O. C., D. W. Kelly, J. Gago, and I. Babuska, Hierarchical finite element approaches, error estimates and adaptive refinements, in The Mathematics of Finite Elements and Applications, Academic Press, New York, 1982, pp. 313-346.
Silvia Bertoluzza I.A.N.-C.N.R. v. Abbiategrasso 209 27100 Pavia Italy aivlis@dr agon .ian. pv. cnr.it
This Page Intentionally Left Blank
An Adaptive
Pseudo-Wavelet
Nonlinear
Approach
Partial Differential
for
Solving
Equations
G r e g o r y Beylkin and J a m e s M. Keiser
A b s t r a c t . We numerically solve nonlinear partial differential equations of the form ut = s + A/'f(u), where s and A/" are linear differential operators and f(u) is a nonlinear function. Equations of this form arise in the mathematical description of a number of phenomena including, for example, signal processing schemes based on solving partial differential equations or integral equations, fluid dynamical problems, and general combustion problems. A generic feature of the solutions of these problems is that they can possess smooth, nonoscillatory and/or shock-like behavior. In our approach we project the solution u(x, t) and the operators s and A/" into a wavelet basis. The vanishing moments of the basis functions permit a sparse representation of both the solution and operators, which has led us to develop fast, adaptive algorithms for applying operators to functions, e.g., f~u, and computing functions, e.g., f(u) = u 2, in the wavelet basis. These algorithms use the fact that wavelet expansions may be viewed as a localized Fourier analysis with multiresolution structure that is automatically adaptive to both smooth and shock-like behavior of the solution. In smooth regions few wavelet coefficients are needed, and in singular regions large variations in the solution require more wavelet coefficients. Our new approach allows us to combine many of the desirable features of finite-difference, (pseudo) spectral and front-tracking or adaptive grid methods into a collection of efficient, generic algorithms. It is for this reason that we term our algorithms as adaptive pseudo-wavelet algorithms. We have applied our approach to a number of example problems and present numerical results.
Multiscale Wolfgang Copyright
Wavelet Dahmen,
Methods
for PDEs
Andrew
J. K u r d i l a ,
(~)1997 b y A c a d e m i c
All rights of reproduction ISBN 0-12-200675-5
137 and Peter
Press, Inc.
in a n y f o r m r e s e r v e d .
Oswald
(eds.), pp.
137-197.
138
G. B e y l k i n and J. K e i s e r
{}1
Introduction
This paper describes a wavelet-based methodology for solving a class of nonlinear partial differential equations (PDEs) that have smooth, nonoscillatory solutions and can exhibit shock-like behavior. Generally speaking, the approach takes advantage of the efficient representation of functions and operators in wavelet bases, and updates the solution by implementing two recently developed adaptive algorithms that operate on these representations. Specifically, the algorithms involve the adaptive application of operators to functions ('special' matrix-vector multiplication) and the adaptive evaluation of nonlinear functions of the solution of the PDE, in particular, the pointwise product. These algorithms use the fact that wavelet expansions may be viewed as a localized Fourier analysis with multiresolution structure that automatically or adaptively distinguishes between smooth and shock-like behavior. The algorithms are adaptive since they update the solution using its representation in a wavelet basis, which concentrates significant coefficients near singular behaviour. Additionally, and as we will show, the algorithm for evaluating nonlinear functions is analogous to the approach used to update the solution of a PDE via pseudo-spectral type algorithms. These two features of the algorithms allow us to combine the desirable features of finite-difference approaches, spectral methods and front-tracking or adaptive grid approaches into a collection of efficient, generic algorithms. We refer to the overall methodology for updating the solution of a nonlinear PDE via these algorithms as an adaptive p s e u d o wavelet method.
In this paper we are concerned with computing numerical solutions of (1.1)
ut = ~.u + A / ' f ( u ) ,
with the initial condition u ( x , O) = u o ( x ) ,
0 _< x <_ 1
(1.2)
0 _< t <_ T.
(1.3)
and the periodic boundary condition u(0, t) = u(1, t),
We explicitly separate the evolution equation (1.1) into a linear part, s and a nonlinear part, All(u), where the operators s and hf are differential operators that do not depend on time t. The function /(u) is typically nonlinear, e.g., f ( u ) = u p. Examples of equation (1.1) in one-dimensional space include reactiondiffusion equations, e.g., ut - vux~ + u p ,
p>l,
v>0,
(1.4)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
139
equations describing the buildup and propagation of shocks, e.g., Burgers Equation ut + u u x = v u x ~ , v > O, (1.5) [15], and equations having special soliton solutions, e.g., the Korteweg-de Vries equation ut + a u u x + / 3 u ~ = O, (1.6) where a and 3 are constant [1, 24]. Finally, a simple example of equation (1.1) is the classical diffusion (or heat) equation ut = v u x x ,
v > 0.
(1.7)
Although we do not address multi-dimensional problems in this paper, we note that the Navier-Stokes equations may also be written in the form (1.1). Consider u t + ~ 1 [U" VU -~- V(U" U)] -- V V 2 U - Vp,
(1.8)
div u - 0
(1.9)
where and p denotes the pressure. Applying the divergence operator to both sides of (1.8) and using (1.9), we obtain
A p = f(u),
(1.10)
where/(u) - - 8 9 [u. V u + V(u- u)] is a nonlinear function of u. Equation (1.1) is formally obtained by setting Lu-
vV2u,
(1.11)
and N u - -31 [u. Vu + V ( u - u ) ] - V ( A - I f ( u ) ) .
(1.12)
The term A - i f ( u ) is an integral operator which introduces a long range interaction and has a sparse representation in wavelet bases. A one-dimensional model that may be thought of as a prototype for the Navier-Stokes equation is ut = 7-l(u)u, (1.13) where ~(.) is the Hilbert transform (see [18]). The presence of the Hilbert transform in (1.13) introduces a long range interaction which models that found in the Navier-Stokes equations. Even though in this paper we develop algorithms for one-dimensional problems, we take special care that they generalize properly to several dimensions so that we can address these problems in the future.
140
G. Beylkin and J. Keiser
Several numerical techniques have been developed to compute numerical approximations to the solutions of equations such as (1.1). These techniques include finite-difference, pseudo-spectral and adaptive grid methods (see e.g. [19, 24]). An important step in solving equation (1.1) by any of these methods is the choice of time discretization. Standard explicit schemes (which are easiest to implement) may require prohibitively small time steps, usually because of diffusion terms in the evolution equation. On the other hand, implicit schemes allow for larger time steps but require solving a system of equations at each time step and, for this reason, are somewhat more difficult to implement in an efficient manner. In our approach [11] we have used new time discretization schemes for solving nonlinear evolution equations of the form (1.1), where s represents the linear and A/'(f(u)) the nonlinear terms of the equation, respectively. A distinctive feature of these new schemes is the exact evaluation of the contribution of the linear part. Namely, if the nonlinear part is zero, then the scheme reduces to the evaluation of the exponential function of the operator (or matrix)/: representing the linear part. We show in [12] that such schemes have very good stability properties and, in fact, describe explicit schemes with stability regions similar to those of typical implicit schemes used in, e.g., fluid dynamics applications. In this paper we simply use one such scheme. The main difficulty in computing solutions of equations like (1.1) is the resolution of shock-like structures. Straightforward refinement of a finitedifference scheme easily becomes computationally excessive. Specialized front-tracking or adaptive grid methods require some criteria to perform local grid refinement. Usually in such schemes these criteria are chosen in an ad hoc fashion (especially in multiple dimensions) and are generally based on the amplitudes or local gradients in the solution. Pseudo-spectral methods, as described in, e.g., [24], usually split the evolution equation into linear and nonlinear parts and update the solution by adding the linear contribution, calculated in the Fourier space, and the nonlinear contribution, calculated in the physical space. Pseudo-spectral schemes have the advantages that they are spectrally accurate, relatively straightforward to implement and easy to understand analytically. However, pseudo-spectral schemes have a disadvantage in that the linear and nonlinear contributions must be added in the same domain, either the physical space or the Fourier space. For equations which exhibit shocklike solutions such transformations between the domains are costly. The Fourier transform of such solutions possesses frequency contributions across the entire spectrum as the shock becomes more pronounced. The wavelet approach, described next, is comparable to spectral methods in their accuracy, whereas the automatic placement of significant wavelet coefficients in regions of large gradients parallels general adaptive grid approaches.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
141
Let the wavelet transform of the solution of (1.1) consist of Ns significant coefficients concentrated near any shock-like structures which may be present in the solution. We describe two adaptive algorithms that update the solution using O(Ns) operations, using only the significant wavelet coefficients. In other words, the resulting algorithmic complexity of our approach is proportional to the number of significant coefficients in the wavelet expansions of functions and operators. The algorithms we describe have the desirable features of specialized adaptive grid or front-tracking algorithms and pseudo-spectral methods. We also recall that in the wavelet system of coordinates, differential operators may be preconditioned by a diagonal matrix, see e.g. [7, 28, 20]. For a related approach used in finite elements, see e.g. [14]. In addition, a large class of operators, namely Calder6n-Zygmund and pseudo-differential operators, are sparse in wavelet bases. Therefore, efficient numerical algorithms can be designed using the wavelet representation of these operators. These observations make a good case for developing new numerical algorithms for computing in wavelet bases. The theoretical analysis of the functions and operators appearing in (1.1) by wavelet methods is well understood, [21, 16, 30, 36]. Additionally, there have been a number of investigations into the use of wavelet expansions for numerically computing solutions of differential equations, see e.g. [34, 29, 25]. In our approach we emphasize the adaptive aspects of computing the solution. Any wavelet-expansion approach to solving differential equations is essentially a projection method. In a projection method the goal is to use the fewest number of expansion coefficients to represent the solution since this leads to efficient numerical computations. We note that the number of coefficients required to represent a function expanded in a Fourier series (or similar expansions based on the eigenfunctions of a differential operator) depends on the most singular behavior of the function. Since we are interested in solutions of partial differential equations that have regions of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shock-like structures, using a basis of the eigenfunctions of differential operators would require a large number of terms due to the singular regions. Alternately, a localized representation of the solution, typified by front-tracking or adaptive grid methods, may be employed in order to distinguish between smooth and shock-like behavior. In our approach the number of operations is proportional to the number of significant coefficients in the wavelet expansions of functions and operators and, thus, is similar to that of adaptive grid methods. The basic mechanism of refinement in wavelet-based algorithms is very simple. Due to the vanishing moments of wavelets, see e.g. [22], we know that (for a given accuracy) the wavelet transform of a function 'automat-
142
G. Beylkin and J. Keiser
ically' places significant coefficients in a neighborhood of large gradients present in the function. We simply remove coefficients below a given accuracy threshold. This combination of basis expansion and adaptive thresholding is the foundation for our adaptive pseudo-wavelet approach. In order to take advantage of this 'adaptive transform' and compute solutions of (1.1) in wavelet bases using O(Ns) operations, we have developed two algorithms: the adaptive application of operators to functions, and the adaptive pointwise product of functions. These algorithms are necessary ingredients of any fast, adaptive numerical scheme for computing solutions of partial differential equations. The algorithm for adaptively multiplying operators and functions is based on a 'vanishing-moment property' associated with the B-blocks of the so-called Non-Standard Form representation of a class of operators (which includes differential operators and Hilbert transforms). The algorithm for adaptively computing ](u), e.g., the pointwise product, is analogous to the method for evaluating nonlinear contributions in pseudo-spectral schemes. The spectral expansion of u is projected onto a 'physical' subspace, the function f(u) is evaluated, and the result is projected into the spectral domain. In our algorithm, contributions to ](u) are adaptively computed in 'pieces' on individual subspaces. Each of our adaptive algorithms uses O(Ns) operations, where N8 is the number of significant coefficients of the wavelet representation of the solution of (1.1). The adaptivity of our algorithms and the analogy with pseudo-spectral methods prompt us to refer to our overall approach as an
adaptive pseudo-wavelet method. The outline of this paper is as follows. In Section 2 we use the semigroup approach to replace the nonlinear differential equation (1.1) by an integral equation and describe a procedure for approximating the integral to any order of accuracy. We provide a brief review of wavelet "tools" relevant to our discussion in Section 3. In Section 4 we are concerned with the construction of and calculations with the operators appearing in the quadrature formulas derived in Section 2. Specifically, we describe a method for constructing the wavelet representation, derive the vanishing-moment property, and describe a fast, adaptive algorithm for applying these operators to functions expanded in a wavelet basis. In Section 5 we introduce a new adaptive algorithm for computing the pointwise product of functions expanded in a wavelet basis, and discuss the calculation of general nonlinear functions. In Sections 4 and 5 we give simple numerical examples illustrating the algorithms. In Section 6 we illustrate the use of these algorithms by providing the results of a number of numerical experiments. Finally, in Section 7 we draw a number of conclusions based on our results and indicate directions of further investigation.
143
Pseudo-Wavelet Algorithms for Nonlinear P D E s
w
The semigroup approach and quadratures
We use the semigroup approach to write the partial differential equation (1.1) as a nonlinear integral equation in time. We then approximate the integrals to arbitrary orders of accuracy by quadratures with operatorvalued coefficients. These operators have wavelet representations with a number of desirable properties described in Sections 4.1 and 4.2. The semigroup approach is a well-known analytical tool that is used to express partial differential equations in terms of nonlinear integral equations and to obtain estimates associated with the behavior of their solutions (see e.g. [37]). The solution of the initial value problem (1.1) is given by u(x, t) - e(t-t~
+
f;
(2.1)
e(t-')z~A/'f(u(x, 7))dT,
where the differential operator Af is assumed to be independent of t and the function f ( u ) is nonlinear. For example, in the case of Burgers equation, the operator Af - o and f ( u ) - ~1U 2 , s o that A l l ( u ) - uux appears as products of u and its derivative. Equation (2.1) is useful for proving the existence and uniqueness of solutions of (1.1) and computing estimates of their magnitude, verifying dependence on initial and boundary data, as well as performing asymptotic analysis of the solution, see e.g. [37]. In this paper we use equation (2.1) as a starting point for an efficient numerical algorithm for solving (1.1). A significant difficulty in designing numerical algorithms based directly on (2.1) is that the matrices representing these operators are dense in the ordinary representation. As far as we know, it is for this reason that the semigroup approach has had limited use in numerical calculations. We show in Sections 4.1 and 4.2 that in the wavelet system of coordinates these operators are sparse (for a fixed but arbitrary accuracy) and have properties that allow us to develop fast, adaptive numerical algorithms. Discrete evolution schemes for (2.1) were used in [11] and further investigated in [12]. The starting point for our discrete evolution scheme is (2.1) where we consider the function u(x, t) at the discrete moments of time tn - to + n A t , where At is the time step. Let us denote un - u(x, tn) and Nn H ( f ( u ( x , tn))). Discretizing (2.1)yields UnZr. 1 -- e q l A t U n + l _ l
-3[- A t
(
.1
7Nn+l + Z
~mNn-m
)
,
(2.2)
m--0
where M + 1 is the number of time levels involved in the discretization, and 1 _< M. The expression in parenthesis in (2.2) may be viewed as the quadrature approximation of the integral in (2.1). To simplify notation, we suppress the dependence of the coefficients 7 and/~m o n l.
144
G. Beylkin and J. Keiser
The discrete scheme in (2.2) is explicit if 7 - 0, otherwise it is implicit. For a given M, the order of accuracy is M for an explicit scheme and M + 1 for an implicit scheme due to one more degree of freedom, 7. This family of schemes is investigated in [12] and is referred to as exact linear part (ELP) schemes. Applying this procedure to Burgers equation (1.5), we approximate I(t) =
(2.3)
e(t--r)s to
and list the results for m = 1 and m = 2. For m = 1, equation (2.3) can be approximated by I(t) - ~10s or
I(t) -
(u(to)ux(to) + u ( t l ) u x ( t l ) ) + O((At) 2),
~I0f.,i(u(to)u~(tl)
+ u(tl)ux(to)) + O((At) 2),
where 01:,m -
(e marl" -
I)s -I,
(2.4) (2.5)
(2.6)
I is the identity operator, u(ti) - ui and v(ti) - vi. Note that (2.4) is equivalent to the standard trapezoidal rule. For m - 2 our procedure yields an analogue of Simpson's rule 2
ci,iu(ti)u~(ti) + O((At)3),
I(t) - E
(2.7)
i=0
where c0,0
-
gx0s
- 1s
Ci,1
--
~0s
C2,2
--
610E,2_~_1~.
(2.8)
(2.9) (2.10)
For the derivation of higher order schemes (m > 2) and the stability analysis of these schemes we refer to [12], since our goals in this paper are limited to explaining how to make effective use of such schemes in adaptive algorithms. {}3
Preliminaries
a n d c o n v e n t i o n s of w a v e l e t a n a l y s i s
In this section we review the relevant material associated with wavelet basis expansions of functions and operators. In Section 3.1 we set a system of notation associated with multiresolution analysis. In Section 3.2 we describe the representation of functions expanded in wavelet bases, and
Pseudo-Wavelet Algorithms for Nonlinear PDEs
145
in Section 3.3 we describe the representation of operators in the standard and nonstandard forms. In Section 3.4 we discuss the construction of the nonstandard form of differential operators, following [5]. Much of this material has previously appeared in a number of publications, and we refer the reader to e.g. [22, 16, 36] for more details. M u l t i r e s o l u t i o n analysis and wavelet bases
3.1
We consider a multiresolution analysis (MRA) of L2(N) as 9"" C V2 C V l C V0 C V - 1 C V - 2 C ' ' ' ,
(3.1.1)
see e.g. [21, 22], such that 1. ~ j e z Vj = {0} and Ujez l/) is dense in L2(R), 2. For any f E Le(R) and any j E Z, f(x) E Vj if and only if f(2x) E Yj-l~
3. For any f E L2(R) and any k E Z, f(x) E V0 if and only if f ( x - k ) V0, and
E
4. There exists a scaling function ~o E Vo such that {~o(x- k)}ke z is a Riesz basis of Vo. In our work, we only use orthonormal bases and will require the basis of condition 4 to be an orthonormal rather than just a Riesz basis. 4'. There exists a scaling function ~o E Vo such that {~o(x- k)}ke z is an orthonormal basis of V0. As usual, we define an associated sequence of subspaces W j as the orthogonal complements of Vj in Vj_I, Vj-1 = Vj ~
Wj.
(3.1.2)
Repeated use of (3.1.2) shows that subspace Vj can be written as the direct sum
vj -
wj,.
(3.1.3)
j'>j
We denote by ~o(.) the scaling function and r the wavelet. The family of - 2-J/2~(2-Jx- k)}ke z forms an orthonormal basis of functions {r - 2-J/2r z forms an orthonormal Vj and the family {r basis of W j .
146
G. Beylkin and J. Keiser
An immediate consequence of conditions 1, 2, 3, and 4 ~ is that the function ~a may be expressed as a linear combination of the basis functions of V - - l , L! -1
~(x) -- ~
E
hkqo(2x- k).
(3.1.4)
g k ~ ( 2 x - k).
(3.1.5)
k=0
Similarly, we have L1-1
r
= vf2 E k--0
L!
The coefficients g - {hk}L=11 and G - {gk}k=l are the quadrature mirror filters (QMFs) of length Lf. In general, the sums (3.1.4) and (3.1.5) do not have to be finite and, by choosing Lf < oc, we are selecting compactly supported wavelets, see e.g. [22]. The function r has M vanishing moments, i.e., 0,
~r
0 _< m <_ M - 1.
(3.1.6)
oo
The vanishing moments property simply means that the basis functions Cj,k(X) are chosen to be orthogonal to low degree polynomials. We note that additional conditions may be imposed on the basis functions ~ and r In the development of the algorithm for adaptively computing nonlinear functions, described in Section 5, we will use a scaling function that has M shifted vanishing moments (see [8, 22]), ~ ~ ( x ) ( x - a ) m d x - O,
1 < m < M,
(3.1.7)
oo
where a -
F
~(x)dx.
(3.1.8)
oo
Such basis functions have been called 'coiflets', and are described in [8, 22]. The quadrature mirror filters H and G, which are defined by the wavelet basis, are related by gk -- ( - - 1 ) k h n l - k - 1 ,
k = 0 , . . . , L f - 1.
(3.1.9)
The number Lf of the filter coefficients is related to the number of vanishing moments M, and Lf = 2M for the wavelets constructed in [21]. If additional conditions are imposed (see [8] for an example where L f = 3M), then the relation might be different, but L] is always even. In fact, if one does not insist that a be an integer in (3.1.8) then the filter length may satisfy Lf = 3 M - 2, [10].
Pseudo-Wavelet Algorithms for Nonlinear PDEs l=Lf The filter G - {gl lJl=o
--1
147
has M vanishing moments, i.e.,
LI -i
~
lmgl --0,
m-
0,1,2,...,M-
1.
(3.1.10)
/--0
We observe that once the filter H has been chosen, it completely determines the functions ~o and r and therefore, the multiresolution analysis. Moreover, in properly constructed algorithms, the values of the functions qo and r are usually never computed. Due to the recursive definition of the wavelet bases, via the two-scale difference equations (3.1.4) and (3.1.5), all of the manipulations are performed with the quadrature mirror filters H and G, even if these computations involve quantities associated with q0 and
r
We will not go into the full discussion of the necessary and sufficient conditions for the quadrature mirror filters H and G to generate a wavelet basis and refer to [22] for the details. The coefficients hk and gk of the quadrature mirror filters H and G are computed by solving a set of algebraic equations (see e.g. [22]). The first and simplest example of a multiresolution analysis satisfying conditions 1, 2, 3, and 4' is the chain of subspaces generated by the Haar basis [26]. The scaling function in this case is the characteristic function of the interval (0, 1). The Haar function is defined as 1,
h(x) -
for 0 < x < 1 / 2
-1, for 1 / 2 _ < x < 1 O, elsewhere,
(3.1.11)
and the family of functions hj,k(x) -- 2 - J / 2 h ( 2 - J x - k), j, k E Z, forms the Haar basis. For the Haar function M = 1, (3.1.6) is easily verified, and the Haar function is indeed trivially orthogonal to constants. For numerical purposes we define a 'finest' scale, j = 0, and a 'coarsest' scale, j = J, such that the infinite chain (3.1.1) is restricted to Vj
C V j-1
C "'" C V0,
(3.1.12)
where the subspace V0 is finite dimensional. In numerical experiments, specifying the QMFs H and G defines the properties of the wavelet basis. We will also consider a periodized version of the multiresolution analysis that is obtained if we consider periodic functions. Such functions have projections on Vo which are periodic of period N - 2 n, where N is the dimension of V0. With a slight abuse of notation we will denote these periodized subspaces also by Vj and W j . We can then view the space V0 as consisting of 2 n 'samples' or lattice points and each space Vj and W j as consisting of 2 n - j lattice points, for j - 1, 2 , . . . , J, where J _< n.
G. Beylkin and J. Keiser
148
3.2
Representation of functions in wavelet bases
The projection of a function f(x) onto subspace Vj is given by
(Pjf)(x)- ~
s~j,k(X),
(3.2.1)
kE Z
where Pj denotes the projection operator onto subspace V j. The set of coefficients {sJk}ke Z, which we refer to as 'averages', is computed via the inner product
9
s3k =
f (x)~j,k (x)dx.
(3.2.2)
OO
Alternatively, it follows from (3.1.3) and (3.2.1) that we can also write (Pjf)(x) as a sum of projections of f(x) onto subspaces Wj, ,j' > j,
( P j f ) ( x ) - ~ ~ dJk'r j'>j kE Z
(3.2.3)
where the set of coefficients {dJk}ke Z, which we refer to as 'differences', is computed via the inner product
d3k --
f (x)r
(3.2.4)
(x)dx.
O0
The projection of a function on subspace W j is denoted (Qjf)(x), where Qj = Pj-1 - Pj. Since we are considering a 'periodized' MRA, on each subspace Vj and W j the coefficients of the projections satisfy 83k -- S jk+2,~_j,
d~ : dJk+2~,_j,
(3.2.5)
for each j = 1, 2 , . . . , J and k E F2.-J - Z/2 n-j Z, i.e., ]F2.-J is the finite field of 2n-j integers, e.g. the set {0, 1 , . . . , 2n-j - 1}. In our numerical algorithms, the expansion into the wavelet basis of (Pof)(x) is given by a sum of successive projections on subspaces W j , j - 1, 2 , . . . , J, and a final 'coarse' scale projection on V j , J
Z
Z
j--1 kE]F2n_ j
Sk Cpg,k (X).
(3.2.6)
kEF2n-j
Given the set of coefficients {S~}keY2n, i.e., the coefficients of the projection of f(x) on V0, we use (3.1.4) and (3.1.5) to replace (3.2.2) and (3.2.4) by
149
Pseudo-Wavelet Algorithms for Nonlinear PDEs
the following recursive definitions for s~ and d~, L I --i 8~9 --
E
j--I hlSl-t-2k+l'
(3.2.7)
y-1 glSl+2k+l ,
(3.2.8)
/=1
L1-1
4 -
~ l--1
where j - 1 , 2 , . . . , J and k E F2n-j. Given the coefficients s o - P o f E V0 consisting of N - 2n 'samples,' the decomposition of f into the wavelet basis is an order N procedure, i.e., computing the coefficients d~ and s~ recursively using (3.2.7) and (3.2.8) is an order N algorithm. Computing the J-scale decomposition of f via (3.2.7) and (3.2.8) by the pyramid scheme is illustrated in Figure 1. Figure 2 9
{d~}
{d~}
{d 3}
...
{d J}
Figure 1. Projection of the coefficients {s ~ into the multiresolution analysis via the pyramid scheme. illustrates a typical wavelet representation of a function with N - 2 n, n - 13 and J - 7. We have generated this figure using 'coiflets', see e.g. [21], with M - 6 vanishing moments and an accuracy (cutoff) of e - 10 -6. We note that a similar result is obtained for other choices of a wavelet basis. The top figure is a graph of the projection of the function f on subspace V0, which we note is a space of dimension 213. Each of the next J - 7 graphs represents the projection of f on subspaces W j , for j - 1, 2 , . . . 7. Each W j is a space of dimension 213-j, i.e. each consists of 213-j coefficients. Even though the width of the graphs is the same, we note that the number of degrees of freedom in W j is twice the number of degrees of freedom in W j + l . Since these graphs show coefficients d~ which are above the threshold of accuracy, c, we note that the spaces W l , W2, W3, and W4 consist of no significant wavelet coefficients. This illustrates the 'compression' property of the wavelet transform: regions where the function (or its projection ( P o f ) - fo) has large gradients are transformed to significant wavelet coefficients. The final (bottom) graph represents the significant coefficients of the projection of f on V j . This set of coefficients, { S J } k ~ , is typically dense and in this example there are 61 significant coefficients, for the threshold of accuracy 10 -6 .
G. Beylkin and J. Keiser
150
-I
----11 7m
70~
-II
F i g u r e 2. Graphical representation of a 'sampled' function on Vo and its projections onto W j for j - 1, 2 , . . . 7 and VT. Entries above the threshold of accuracy, c - 10 -6, are shown. We refer to the text for a full description of this Figure.
Pseudo- Wavelet Algorithms for Nonlinear PDEs 3.3
151
R e p r e s e n t a t i o n of operators in wavelet bases
In order to represent an operator T : L 2 (JR) --+ L 2 (JR) in the wavelet system of coordinates, we consider two natural ways to define two-dimensional wavelet bases. First, we consider a two-dimensional wavelet basis which is arrived at by computing the tensor product of two one-dimensional wavelet basis functions, e.g.,
Cj,j',k,k' (Z, y) -- Cj,k
(Y),
(3.3.1)
where j , j ' , k , k ' E Z. This choice of basis leads to the standard form (S-form) of an operator, [5, 8]. The projection of the operator T into the multiresolution analysis is represented in the S-form by the set of operators 9!
{ A j , {B~
T-
9!
.!
}j'_>j+l, {r} }j'>_j+l }jE Z,
(3.3.2)
.!
where the operators Aj, B~ , and F} are projections of the operator T into the multiresolution analysis as follows
Aj B 3., 3. F~
=
.!
QjTQj QjTQj, Qj, TQj
9 W j --~ w j , 9 w j , --+ w j 9 w j -~ w j , ,
(3.3.3)
for j - 1 , 2 , . . . , n and j ' - j + 1 , . . . , n . If n is the finite number of scales, as in (3.1.12), then (3.3.2) is restricted to the set of operators
To _ {Aj ' {BJ "~j'=n Jj':j+l
~
{F~" }j,=n J' =j+l
'
B2 +1 ~ F~ +1 ~ Tn}j=l , . . .
,n
'
(3.3.4)
where To is the projection of T on V0. Here the operator T~ is the coarse scale projection of the operator T on V~,
T~
-
PnTP~ . V~ ~ V~.
(3.3.5)
The subspaces Vj and W j appearing in (3.3.3) and (3.3.5) can be periodized in the same fashion as described in Section 3.2. The operators Aj, B~, F~, and Tn appearing in (3.3.2) and (3.3.4) are .!
.!
represented by matrices aJ, ~J,J', 7 j,j' and s ~ with entries defined by
J
O~k, k'
=
f fCj,k(x)K(x,y)r
kk ~3,J , k,k n 8k,M
=
f fCj,k(x)K(x,y)r
= =
f fCj,k(x)g(x,y)r f fqon,k(x)g(x,y)qon,k,(y)dxdy,
~J,J',
(3.3.6)
G. Beylkin and J. Keiser
152
where K(x, y) is the kernel of the operator T. The operators in (3.3.4) are organized as blocks of a matrix as shown in Figure 3. In [8] it is observed that if the operator T is a Calder6n-Zygmund or pseudo-differential operator, then for a fixed accuracy all the operators in (3.3.2) are banded. In the case of a finite number of scales, the operator Tn and possibly some other operators on coarse scales can be dense. As a result the S-form has several 'finger' bands, illustrated in Figure 4. These 'finger' bands correspond to interactions between different scales. For a large class of operators, e.g. pseudo-differential, the interaction between different scales (characterized by the size of the coefficients in the bands) decays as the distance ] j - J~l between the scales increases. Therefore, if the scales j and j~ are well separated, then for a given accuracy the operators B~ and F~ can be neglected. For compactly supported wavelets, the distance I J - J~] is quite significant; in a typical example for differential operators IJ -J~l = 6. This is not necessarily the case for other families of wavelets. For example, Meyer's wavelets [30] are characterized by 9!
;(r
-
.!
(27r)-~12ei~12 s i n ( ~ v ( ~ l ~ I - 1)),
< I, 1 <
(27r)-112ei~12 cos(
< -
,(
lfl- 1)),
0,
471"
< -8: T,
-
(3.3.7)
otherwise,
where v is a C c~ function satisfying 0,
[ 1,
(3.3.8)
x _> 1,
and v ( z ) + v(1 - z) = 1.
(3.3.9)
In this case the interaction between scales for differential operators is restricted to nearest neighbors where ]j-j~] _ 1. On the other hand, Meyer's wavelets are not compactly supported in the time domain which means the finger bands will be much wider than in the case of compactly supported wavelets. The control of the interaction between scales is more efficient in the nonstandard representation of operators, which we will discuss later. Another property of the S-form which has an impact on numerical applications is due to the fact that the wavelet decomposition is not shift invariant. Even if the operator T is a convolution, the B~ and F~ blocks of the S-form are not convolutions. Thus, the S-form of a convolution operator is not an efficient representation, especially in multiple dimensions. An alternative to forming two-dimensional wavelet basis functions using the tensor product (which led us to the S-form representation of operators) 9!
.!
Pseudo-Wavelet Algorithms for Nonlinear PDEs
A I
2 1
B1
4 S B 1B 1
A 2
3 B2
4 S B21 B 2
As
Bs Bs
B
3
2
F 1
r'2 4
4
F, S F1
153
4
F2
Fs
s F 2,
s
r~
4
$
s
A4 B 4 5
F 4 T4
Figure 3. Organization of the standard form of a matrix.
is to consider basis functions which are combinations of the wavelet, r and the scaling function, ~(.). We note that such an approach to forming basis elements in higher dimensions is specific to wavelet bases (tensor products as considered above can be used with any basis, e.g., the Fourier basis). We will consider representations of operators in the nonstandard form (N S-form), following [8] and [5]. Recall that the wavelet representation of an operator in the N S-form is arrived at using bases formed by combinations of wavelet and scaling functions, for example, in L2(N 2)
Cj,k (x) r
(y),
Cj,k (x) ~j,k, (y),
~j,k (~) Cj,k, (y),
(3.3.10)
where j, k, k' E Z. The NS-form of an operator T is obtained by expanding T in the 'telescopic' series
T - ~ (QjTQj + QjTPj + PjTQj), jEz
(3.3.11)
where Pj and Qj are projectors on subspaces Vj and W j , respectively. We observe that in (3.3.11) the scales are decoupled. The expansion of T into the NS-form is, thus, represented by the set of operators
T = { & , B ~ , r j } j e z,
(3.3.12)
G. Beylkin and J. Keiser
154
L
Figure 4. Schematic illustration of the finger structure of the standard form.
where the operators Aj, Bj, and F j act on subspaces V j and W j,
Aj Bj Fj
--
QjTQj QjTPj PjTQj
9 Wj--+Wj, "
Vj---}Wj,
(3.3.13)
9 Wj-+Vj,
see e.g. [8]. I f J _ n is the finite number of scales, as in (3.1.12), then (3.3.11) is truncated to J
To - ~ ( Q j T Q j
+ QjTPj + PjTQj) + P j T P j ,
(3.3.14)
j=l
and the set of operators (3.3.12) is restricted to
T o - { { A j , B j , F j } ~ - J1 , T j } ,
(3.3.15)
where To is the projection of the operator on Vo and T j is a coarse scale projection of the operator T,
Tj - P j T P j " V j --+ V j ,
(3.3.16)
using (in L2(]R2)) the basis functions
J,k
J,k, (y),
(3.3.17)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
155
~ii}i~i~iii!iii~iiil!iiiiii!i!!ii!~
1 IW F i g u r e 5. Organization of the non-standard form of a matrix. Aj, Bj, and Fj, j = 1, 2, 3, and 7"3 are the only non-zero blocks.
for k, k ~ E Z. Figure 5 shows the NS-form of a matrix for J - 3. The price of uncoupling the scale interactions in (3.3.11) is the need for an additional projection into the wavelet basis of the product of the N S - f o r m and a vector. The term "nonstandard form" comes from the fact that the vector to which the NS-form is applied is not a representation of the original vector in any basis. Referring to Figure 6, we see that the NSform is applied to both averages and differences of the wavelet expansion of a function. In this case we can view the multiplication of the N S-form and a vector as an embedding of matrix-vector multiplication into a space of dimension M2 n - J ( 2 g + l - 1), (3.3.18) where n is the number of scales in the wavelet expansion and J _< n is the depth of the expansion. The result of multiplying the N S - f o r m and a vector must then be projected back into the original space of dimension N - 2 n. We note t h a t N < M < 2 N a n d , forJ-n, wehaveM-2N-1. It follows from (3.3.11) that after applying the NS-form to a vector we arrive at the representation J
Z Z
j = l kEIF2,.,_ j
J
g3kqOj,k(X). j = l kElF2n_ i
(3.3.19)
G. Beylkin and J. Keiser
156
\ d'
d1
\ \ S
l
2
d
S2
d3 S3
Figure 6. Illustration of the application of the non-standard form to a vector.
The representation (3.3.19) consists of both averages and differences on all scales which can either be projected into the wavelet basis or reconstructed to space V0. In order to project (3.3.19) into the wavelet basis we form the representation, J
sk~g,k(x ), j--1 kE]F2n_ j
(3.3.20)
kEF2n_ J
using the decomposition algorithm described by (3.2.7) and (3.2.8) as follows. Given the coefficients {~J }j=l J and (dJ}g=l, we decompose {~1} into {~2} and {~2} and form the sums {s 2} - {~2 + ~2} and (d 2} - {(~2 § (~2}. Then on each scale j = 2, 3 , . . . , J - 1, we decompose {s j} = {~J + ~J} into {~j+l} and {~j+l} and form the sums {s j+l} - {~j+l + ~j+l} and {dJ+l} _ {~j+l + dj+l}. The sets {s J} and {dJ}J_l are the coefficients of the wavelet expansion of (Tofo)(x), i.e., the coefficients appearing in (3.3.20). This procedure is illustrated in Figure 7. An alternative to projecting the representation (3.3.19) into the wavelet basis is to reconstruct (3.3.19) to space V0, i.e., form the representation
(3.2.1)
(Pof)(x) = E S~176
(3.3.21)
kE Z
using the reconstruction algorithm described in Section 3 as follows. Given
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Pseudo-Wavelet Algorithms for Nonlinear PDEs
{,~0}
~
{~1 _[._,~1} __ {81 }
._.),
"'"
-'+
{ ~1 + ~1} _ { d 1}
{SJ "Jr"8J}-- {8 J} {dg + i1g ) - {d J }
Figure 7. Projection of the product of the NS-form and a function into a wavelet basis.
{~0)
t-"
{S 1} -- {~1 + ~1)
"'"
t--" {8 J - l } --" {~J-1 .~_ ~J-1 }
{d l} _ {dx ..[_~1)
...
{d J - l ) _ {~J-1 ..[..6~J-1 }
/....
{8 J} {d J}
Figure 8. Reconstruction of the product of the NS-form and a function to space Vo. the coefficients {~J } ] 1 and {dJ }j=l, g we reconstruct {dJ} and {sg} into {g J - l } and form the sum {s J-1 } = {~g-1 + gg-i }. Then on each scale j - J - 1, J - 2 , . . . , 1 we reconstruct {~J} and {dJ} into {gj-1} and form the sum {s j - l } = {~j-1 + gj-1}. The final reconstruction (of {d 1} and {s 1}) forms the coefficients {s ~ appearing in (3.3.21). This procedure is illustrated in Figure 8. 3.4
The nonstandard
f o r m of d i f f e r e n t i a l o p e r a t o r s
Following [5], in this section we recall the wavelet representation of differential operators 0 p in the N S-form. The rows of the N S-form of differential operators may be viewed as finite-difference approximations on the subspace V0 of order 2M - 1, where M is the number of vanishing moments of the wavelet r The N S-form of the operator 0 p consists of matrices A j, B j, F j, for j - 0, 1 , . . . , J and a 'coarse scale' approximation T g. We denote the J ]~2i,1, and 0,i,t, 3 for j - 0, 1 , " - , J, and elements of these matrices by hi,l, s i,l J " Since the operator 0 p is homogeneous of degree p, it is sufficient to compute the coefficients on scale j = 0 and use
4
-
-
2- Jz ~
(3.4.1)
We note that if we were to use any other finite-difference representation as coefficients on V0, the coefficients on Vj would not be related by scaling
158
G. Beylkin and J. Keiser
and would require individual calculations for each j. Using the two-scale difference equations (3.1.4) and (3.1.5), we are led to 9 ~-~LI-I ~-~LI-1 j-1 a~ - 2 z..~k=0 z..,k,=O gkgk' S2i+k_ k, , t3Jl ~
-
~-.~LI-I ~-~LI-1
j-1
~-~LI-1
j-i
(3.4.2)
2 z_.,k=O ~k'=O gkhk, S2i+k_k, , LI-1
2 z.,k=O ~-~k'=O hkgk' s2i+k_k,.
-
Therefore, the representation of 0 p is completely determined by s~ in (3.3.6), or in other words, by the representation of 0 p projected on the subspace V0. To compute the coefficients s o corresponding to the projection of 0 p on V0, it is sufficient to solve the system of linear algebraic equations
8 0 -- 2 p
801 "b ~
a2k-1(801_2k+l
"-[- 821+2k_1)
,
(3.4.3)
k=l
for - L f + 2 _ 1 _< Lf - 2 and L! - 2
E l=-Lf+2
1p s o - (-1)Pp! ,
(3.4.4)
where a 2 k - 1 are the autocorrelation coefficients of H defined by L I-l-n
an=2
~
hi hi+n,
n=l,...,Lf-1.
(3.4.5)
i--0
We note that the autocorrelation coefficients an with even indices are zero, a2k = O,
k - 1 , . . . , L I / 2 - 1,
(3.4.6)
and a0 - x/~. The resulting coefficients s~ corresponding to the projection of the operator 0 p on V0 may be viewed as finite-difference approximations of order 2 M - 1. Further details are found in [5]. We are interested in developing adaptive algorithms, i.e., algorithms such that the number of operations performed is proportional to the number of significant coefficients in the wavelet expansion of solutions of partial differential equations. The S-form has 'built-in' adaptivity, i.e., applying the S-form of an operator to the wavelet expansion of a function, (3.2.3), is a matter of multiplying a sparse vector by a sparse matrix. On the other hand, as we have mentioned before, the S-form is not a very efficient
Pseudo-Wavelet Algorithms for Nonlinear PDEs
159
representation (see, e.g., our discussion of convolution operators in Section 3.3). In the following sections we address the issue of adaptively multiplying the NS-form and a vector. Since the NS-form of a convolution operator remains a convolution, the AJ, B j, and FJ blocks may be thought of as being represented by short filters. For example, the NS-form of a differential operator in any dimension requires O(C) coefficients as it would for any finite difference scheme. We can exploit the efficient representation afforded us by the N S-form and use the vanishing-moment property of the B j and F j blocks of the N S-form of differential operators and the Hilbert transform to develop an adaptive algorithm. In Section 4.1 we describe two methods for constructing the N S-form representation of operator functions. In Section 4.2 we establish the vanishing-moment property which we later use to develop an adaptive algorithm for multiplying operators and functions expanded in a wavelet basis. Finally, in Section 4.3 we present an algorithm for adaptively multiplying the NS-form representation of an operator and a function expanded in the wavelet system of coordinates. w
N o n s t a n d a r d form representation of operator functions
In this section we are concerned with the construction of and calculations with the nonstandard form (NS-form) of operator functions (see, e.g. (2.2)). We show how to compute the NS-form of the operator functions and establish the vanishing-moment property of the wavelet representation of these operators. Finally, we describe a fast, adaptive algorithm for applying operators to functions in the wavelet system of coordinates. 4.1
T h e nonstandard form of operator functions
In this section we construct the NS-forms of functions of the differential operator 0x. We introduce two approaches for approximating the NS-forms of operator functions: (i) compute the projection of the operator function on V0, Po f (Oz)Po, (4.1.1) or, (ii) compute the function of the projection of the operator,
f (PoO~Po ).
(4.1.2)
The difference between these two approaches depends on how well i~(~)l 2 acts as a cutoff function, where ~(x) is the scaling function associated with a wavelet basis. It might be convenient to use either (4.1.1) or (4.1.2) in applications. The operator functions we are interested in are those appearing in solutions of the partial differential Equation (1.1). For example, using (2.1)
160
G. Beylkin and J. Keiser
with (2.5), solutions of Burgers equation can be approximated to order (At) 2 by
u(x, t + ~xt) - eA'Lu(x, t)
--1-OE, 1 [U(X, t)OxU(X, t + At) + U(X, t + At)Oxu(x, 2
t)]
(4.1.3)
where s = vO 2 and Os is given by (2.6). Therefore, we are interested in constructing the NS-forms of the operator functions e ~ts
and
(e~'~ -- I)~-~
OE,1 =
(4.1.4)
(4.1.5)
,
for example. In the following we assume that the function f is analytic. In computing solutions of (1.1) (via, e.g., (4.1.3)) we can precompute the N S-forms of the operator functions and apply them as necessary. We note that if the operator function f is homogeneous of degree m (e.g., m = 1 and 2 for the first and second derivative operators), then the coefficients appearing in the NS-form are simply related, see e.g. (3.4.1). On the other hand, if the operator function f is not homogeneous then J , ~ ,k', and k,~ via (3.3.6) and compute the coefficients ak,k, we compute Sk,
7g,k, via equations (3.4.2) for each scale j - 1, 2 , . . . , J _< n. We note that if f is a convolution operator then the formulas for sk_ 0 k, are considerably simplified (see [5]). We first describe computing the N S-form of an operator function by projecting the operator function into the wavelet basis via (4.1.1). To compute the coefficients J = 2-J 8k,M
/_~oo oo ~ ( 2 - J x -
let us consider f(Ox)~(2-Jx-
k)f(Ox)~(2-Jx-
If?
k') - x / ~
k')dx,
oo f(-i~2-J)~(~)e-iek'ei2-~Xed~,
(4.1.6)
(4.1.7)
where ~(~) is the Fourier transform of ~(x),
~(~)- ~
1 / _ ~~176
~ ~o(x)e~dx.
(4.1.8)
Substituting (4.1.7) into (4.1.6) and noting that 8k,kl j " , we arrive = S~_k, at
s~ -
f (-i(2-~)l~(()12e~d(.
(4.1.9)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
161
We evaluate (4.1.9) by setting
~' -
f(-i2-J(~ + 2~k))l~(~ + 2~k)l~,
f0 27r ~
(4.1.10)
kEZ or
s~ where
g(~) - ~
g(~)ei~Id(,
f(-/2-~(~ + 2~k))i~(~ + 2~k)l ~.
(4.1.11)
(4.1.12)
kEZ
We now observe that for a given accuracy e the function 195(~)12 acts as a cutoff function in the Fourier domain, i.e., 195(~)12 < e for I~! > 77 for some > 0. Therefore, equation (4.1.10) is approximated to within e by K
(7(~) -
Z
f ( - i 2 - J ( ~ + 27rk))lq5(~ + 27rk)12'
(4.1.13)
k=-K
for some K. Using (4.1.13) (in place of g(()) in (4.1.11) we obtain an approximation to the coefficients s~, N-1
~ - -~ Z ~ ( ~ ) ~ ' .
(4.1.14)
n--0
The coefficients s~ are computed by applying the F F T to the sequence {g(~n)} computed via (4.1.13). In order to compute the NS-form of an operator function via (4.1.2), we use the DFT to diagonalize the differential operator 0x, and apply the spectral theorem to compute the operator functions. Starting with the wavelet representation of 0x on Vo (see Section 3.4 or [5]) of the discretization of 0x, we write the eigenvalues explicitly as L
k, + s-l e-27rz~.k, ), Ak -- So + ~ j ( sl e21r, ~-
(4.1.15)
/=1
where the wavelet coefficients of the derivative, sl - s ~ are defined by (3.3.6). Since f ( A ) - ~ ' f ( A ) ~ --1, (4.1.16) where A is a diagonal matrix and ~- is the Fourier transform (see [37]), we compute f (Ak) and apply the inverse Fourier transform to the sequence
l(~k),
N
~0 _ Z I ( ~ ) ~ ' ~ (~-~)~'-~ , k=l
(4.1. it)
162
G. Beylkin and J. Keiser
to arrive at the projection of the operator functions ](0~) on the subspace V0, i.e., the wavelet coefficients sT. The remaining elements of the N S form are then recursively computed using equations (3.4.2). 4.2
V a n i s h i n g m o m e n t s of t h e B-blocks
We now establish the vanishing-moment property of the B-blocks of the N S-form representation of functions of a differential operator described in Section 4.1 and the Hilbert transform. We note that a similar result also holds for the B-blocks of some classes of pseudo-differential operators, see e.g. [31]. Additionally, we note that these results do not require compactly supported wavelets, and we prove the results for the general case. In Section 4.3 we use the vanishing-moment property to design an adaptive algorithm for multiplying the N S-form of an operator and the wavelet expansion of a function. Proposition 1. If the wavelet basis has M vanishing moments, then the B-blocks of the N S - f o r m of the analytic operator function f (Ox), described in Section 4.1, satisfy q-c~
1 ~l - O,
(4.2.1)
l-~--oo
for m - O, 1, 2, . . . , M - 1 and j - 1 , 2 , . . . J.
Proof:
Using definition (3.3.6), we obtain lm~l = l----oo
F
r
k)f(Ox)Pm(x)dx.
(4.2.2)
oo
We have used the fact that -~- CX3
Z
Im~(x - l ) - Pm(x),
(4.2.3)
l'--oo
where Pm (x) is a polynomial of degree m, for 0 _ m ___M - 1, see [30]. Since the function f(.) is an analytic function of 0x, we can expand f in terms of its Taylor series. Therefore, the series for f(Ox)Pm(x) is finite and yields a polynomial of degree less than or equal to m, f (Ox)Pm (x) - Pm' (X),
where m' _ m. Due to the M > m vanishing moments of r integrals (4.2.2) are zero, and (4.2.1) is verified.
(4.2.4) the
Pseudo-Wavelet Algorithms for Nonlinear PDEs
163
Proposition 2. Under the conditions of Proposition 1, the B-blocks of the N S - f o r m of the Hilbert transform
1 /? -f(s) - - ds,
(7-lf)(x)- -p.v. 7r
oo
8 --
(4.2.5)
X
(where p.v. indicates the principle value), satisfy" nt-oo
E
lm/~i' - O,
(4.2.6)
l=-c~
for0 <: rn _< M Proof: by
1 and j - 1,2,... J.
The/31 elements of the N S-form of the Hilbert transform are given
/3, ,
~b(x -l)(7-lqo)(x)dx, (:X) I_ ~~176
(4.2.7)
and proceeding as in Proposition 1, we find
Z
--
l=--~
=
Z
l=-- cx:) +oo
-
E
(x:) -kc<)
lm f _
l=--oo
=
-
(7-&b)(x)qo(x + 1)dx oO
(74r
(4.2.8)
O0
where, once again, we have used (4.2.3). To show that the integrals in (4.2.8) are zero, we establish that (7/r has at least M vanishing moments. Let us consider the generalized function
/? (~r
-i-mO~(~)(~).
(4.2.9)
(2O
In the Fourier domain the Hilbert transform of the function g defined by A
(7-/g)(~)- - i sign(~)~(~),
(4.2.10)
may be viewed as a generalized function, whose derivatives act on test functions f E C~(IR) as dm ( ,-7-~-~ aq..~ ( - i sign(~)t~(~)), f)
_ _ i E m j--1 ( Im)
f ( j - 1 ) ( 0 ) ~ ( m - j ) (0)
+i f _ ~ sign(~) ~(m)(~)f(~)d~.
(4.2.11)
G. Beylkin and J. Keiser
164
In order to show that (7-/r has M vanishing moments, we recall that in the Fourier domain vanishing moments are characterized by dm d~ m r - O, for m - 0, 1 , . . . , M - 1, (4.2.12) where r is the Fourier transform of r Setting ~0(~) = r in (4.2.11), the sum on the right-hand side of (4.2.11) is zero. We also observe that the integrand on the right-hand side of (4.2.11), i.e. sign(~)r (m) (~)](~), is continuous at ~ = 0, once again because r has M vanishing moments. We can then define functions w(m)(~) for m - 0, 1 , . . . , M - 1, as u
(~) _
0, i
> 0, ~-0, < 0,
(4.2.13)
such that )/~;(m)(~) coincides with the m-th derivative of the generalized function (4.2.10) on the test functions f E C~(IR). Since )/~(m)(~) are continuous functions for m = O, 1 , . . . , M - 1, we obtain
/~
(Ttr
_ )/~;(m)(~)
(4.2.14)
(x)
instead of (4.2.9). Since )/~;(m)(~)1~=o - O, the integrals (4.2.8) are zero and (4.2.6) is established. 4.3
Adaptive
calculations
with the nonstandard
form
In [8] it was shown that Calder6n-Zygmund and pseudo-differential operators can be applied to functions in O ( - N log e) operations, where N = 2 n is the dimension of the finest subspace V0 and e is the desired accuracy. In this section we describe an algorithm for applying operators to functions with sublinear complexity, O(CNs), where N8 is the number of significant coefficients in the wavelet representation of the function. We are interested in applying operators to functions that are solutions of partial differential equations having regions of smooth, nonoscillatory behavior interrupted by a number of well-defined localized shocks or shocklike structures. The wavelet expansion of such functions (see e.g. (3.2.6)) then consists of differences {d j ) that are sparse, and of averages {s j } that are dense. Adaptively applying the NS-form representation of an operator to a function expanded in a wavelet basis requires fast evaluation of
=
Bk+tS~k+l, l
--
Z Fk+Idk+l, j J
(4.3.1)
l
(4 3.2)
Pseudo-Wavelet Algorithms for Nonlinear PDEs
iii!ii:2!iiii!iiiiiiiii!i~!iii~ ~:::........................::::: ii!:ii::!i~!iii!ili::i!i:!:i!i!i!ili
i!i~i:i,!:!~i~:~:~:!i'~;:i! i!i~iii!:!:i!i!i!~i~i!i~
165 I I 1
~!!i!ii:iiii:~i~:!ii~iiii!i!ili:i! ::::::::::::::::::::::::::::::
ii!i!i~i~i:!i!i:i!;~!i~iil
i!iii~iii!i:!iiiiiiiiii:~!!!i~!ii; iii!ii!ii!i;ii!i!iii!ii~:~i;!ii~i
ii!i!i!!i~i!~i!i!i)i} i!!i!iii~' iiii~!ii:i!i !iiiiii!~ii!!ii~iliiii~:i:~i~i
:i!:ii!i::!;!:i:i!i:!!:!;ii!iiiii:~: ii;iii!!~:~!:~!ii:~ii~:ii!iii~iii!i:
i)!i!i~i?!i~i~i!:'~!i~i!il
~i!i!?~i~i!i':!!!!!!!~!~!!ii~ ::::::::::::::::::::::::: ::::::::::::::::::::::::::::::: i~iiiiiiiiiiii!iii!iiiiiil
BJ
I J
Sj
F i g u r e 9. For the operators considered in Section 4.2, the vanishing-moment property of the rows of the B-block yields a sparse result (up to a given accuracy ~) when applied to a smooth and dense vector {s j }. for j = 1, 2 , . . . , J - 1 and k E ]F2~-J = {0, 1, 2 , . . . , 2 n-J - 1}, and on the the final, coarse scale,
-
SJ
-
Z Ak+ldk+ J J l + E Sk+lSk+l, g g l l
---- E
FJ+ldJ+ l + E l
Tgk+l8gk-Fl, l
(4.3.3)
(4.3.4)
for k E ]F2~-J. The difficulty in adaptively applying the N S - f o r m of an operator to such functions is the need to apply the B-blocks of the operator to the averages {sJ} in (4.3.1). Since the averages are "smoothed" versions of the function itself, these vectors are not necessarily sparse and may consist of 2 n - j significant coefficients on scale j. Our algorithm uses the fact t h a t for the operator functions considered in Section 4.1, the rows of the B-blocks have M vanishing moments. This means t h a t when the row of a B-block is applied to the "smooth" averages {s j } the resulting vector is sparse (for a given accuracy e), as is illustrated in Figure 9. Since each row of the B-block has the same number of vanishing moments as the filter G, we can use the {d j } coefficients of the wavelet expansion to predict significant contributions to (4.3.1). In this way we can replace the calculations with a dense vector {s} in (4.3.1) by calculations with a sparse vector {~},
--
Bk+l~3k+l , l
l
(4.3.5)
G. Beylkin and J. Keiser
166
for j = 1 , 2 , . . . , J - 1 and k E ]F2--~. In what follows we describe a method for determining the indices of {~J } using the indices of the significant wavelet coefficients {d j }. The formal description of the procedure is as follows. For the functions under consideration the magnitude of many wavelet coefficients {d J} are below a given threshold of accuracy e. The representation of f on V0, (3.2.6), using only coefficients above the threshold e, is J
Sk~g,k(X ), j=l
{k:ld~l>e}
kE]F2n-
whereas for the error we have
[[(Pof)~(z) -(Pof)(x)[[2 --
(4.3.6)
j
/
Z Id~ [2 j=l {k:ld~l_<e}
<eglr/2,
(4.3.7)
where Nr is the number of coefficients below the threshold. The number of significant wavelet coefficients is defined as Ns = N - Nr, where N is the dimension of the space V0. We define the e-accurate subspace for f, denoted D) C V0, as the subspace spanned by only those basis functions present in (4.3.6), D) - V j U {span {r
" 14[ > e},
(4.3.8)
for 1 <_ j < J and k E F2--J. Associated with D~ are subspaces S~ determined using the two-scale difference relation, e.'g., equation (3.1.5i~. Namely, for each j = 0, 1 , . . . , J - 1, S ) , j - {span {~)j,2k+l(X)}
"
Cj+l,k(X) e D)}.
(4.3.9)
For j - J we define the space S),j as S ) , j -- V J .
(4.3.10)
In terms of the coefficients d~+1 ' the space S 'l , J may be defined by S~,j = {span {~j,2k+l(x)}
" [d~+l[ > e}.
(4.3.11)
In this way we can use D~ to 'mask' V0 forming S~j; in practice all we do is manipulate indices. The subset of coefficients {gJ} that contribute to the sum (4.3.5) may now be identified by indices of the coefficients corresponding to basis functions in S~,j.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
167
We now show that significant wavelet coefficients dj+l and contributions of BJ sJ to (4.3.1) both originate from the same coefficients s j. In this way we can use the indices of dj+l to identify the coefficients ~J that contribute to the sum (4.3.5). We begin by expanding f(x + 2Jl) into its Taylor series,
f(x + 2Jl)
- M-1E f(m)m!(X)2Jml m +
f(M)M____~.((z z) - x) M ,
(4.3.12)
m:0
where z = z(x,j, l) lies between x and x + 2J/. We begin by computing
.--~j
9
.
dk -- E l fl3k+tS3k+t using (4.3.12) and obtain -~j dk
2 -j/2
~o(2- j x - k) E co
+
2-J/2 ~
my
m=O L
"
k+l
l=-L
co
E
M! l=-L
~jk+l/_
~( 2-jx - k)f(M)(z)( z - x) Mdx" (4.3.13) c~
By the vanishing-moment property of the B-block, the first term in (4.3.13) is zero and, after a change of variables, we find
2-J/2
L
co
j
l----L
k+l
co
r
f (M) (z)(z - 2j (x + k)) M dx,
(4.3.14)
for k E F2J-j. J To compute the differences dJk+1 -- Y'~-Igls2k'+t, we use the averages
SJ2k,+, -- 2 -j/2
~ ( 2 - J x - 2k')f(x + 2Jl)dx.
(4.3.15)
co
Substituting (4.3.12) into (4.3.15), we obtain +1
2 -j/2 /_~ ~ ( 2 - J x - 2k') M-l ~ f(m)(x) m! (2Jm) ( E g l l m ) Co
m--O
2-J/2
+ M! E g l
/?
l
Co
dx
l
~P(2-Jx - 2k')f(M)(z)(z - x)Mdx"
Using the vanishing moments of the filter G = {gl}, we obtain d~+1 =
2-J/2
M! E gl l
for k ~ E IF2J-(j+I).
/? Co
•(x)f (u) (z)(z - 2j (x + 2k')) Mdx,
(4.3.16)
G. Beylkin and J. Keiser
168
To show that [d~+ll < e implies ladk[ < Ce, we consider two cases. First, if Ida+l[ < e and k is even, i.e., k = 2n for n E ]F2J-(~+I), then we see that ~ n and d~+1 given by (4.3 " 16)only differ in the coefficients g~ and ~J2 n q - l " Since gl and ~J2n+t are of the same order, the differences satisfy I~n[ < Ce for some constant C. On the other hand, if k - 2n + 1 for n =E ]F2J-(j+I), we find ~ n + l --
2_j/2 M!
L
E
l---L
~2n+1+/j
ix)
~(x + 1)f(M)(z)(z- 2J(X + 2n)) Mdx,
oo
which again is of the same order as dj+l. k' E ]F2J-r
(4.3.17) Therefore, if Ida+l[ < e for
then for some constant C, ]~:~k]< Ce, for k E ]F2J-~. w
E v a l u a t i n g f u n c t i o n s in wavelet bases
In this section we describe our adaptive algorithm for evaluating the pointwise product of functions represented in wavelet bases. More generally, our results may be applied to computing functions f(u), where f is an analytic function and u is expanded in a wavelet basis. We note that since pointwise multiplication is a diagonal operator in the 'physical' domain, computing the pointwise product in any other domain appears to be less efficient. A successful and efficient algorithm should at some point compute f(u) in the physical domain using values of u and not expansion coefficients of u. First let us make several observations regarding the calculation of f(u), where u is expanded in an arbitrary basis N
- Z
(5.1)
i--1
where ui are the coefficients and bi(x) are the basis functions. In general, we have N
f(u(x)) ~ ~ f(ui)bi(x).
(5.2)
i--1
For example, if u(x) is expanded in its Fourier series, clearly the Fourier coefficients of the function f(u) do not correspond to the function of the Fourier coefficients: This has led to the development of pseudo-spectral algorithms for numerically solving partial differential equations, see e.g. [23, 24]. In order to explain the algorithm for computing f(u) in the wavelet system of coordinates, we begin with the assumption that u and f(u) are
Pseudo-Wavelet Algorithms for Nonlinear PDEs
169
both elements of Vo, u, f(u) E V0. Then = Z
_ k),
(5.3)
k
where s~ are coefficients defined, as in (3.2.2), by
s~ --
F
u ( x ) ~ ( x - k)dx.
(5.4)
oo
Let us impose an additional assumption that the scaling function is interpolating, so that - u(k).
(5.5)
Since we have assumed that u, .f(u) E Vo, we obtain
f (u) -- ~
f (8~
- k),
(5.6)
k
i.e. f(u) is evaluated by computing the function of the expansion coefficients f(s~). Below we will describe how to relax the requirement that the scaling function be interpolating yet still have property (5.6) as a quantifiable approximation. We point out that typically, f(u) is not in the same subspace as u. In what follows we describe an adaptive algorithm for computing the pointwise square of a function, f(u) = u 2. In this algorithm we split f(u) into projections on different subspaces. Then we consider 'pieces' of the wavelet expansion of u in finer subspaces where we calculate contributions to f(u) using an approximation to (5.6). This is in direct comparison with calculating f(u) in a basis where the entire expansion must first be projected into a 'physical' space where f(u) is then computed, e.g., pseudo-spectral methods. In Section 5.2 we briefly discuss an algorithm for adaptively evaluating an arbitrary function f(u). 5.1
Adaptive calculation of u 2
Since the product of two functions can be expressed as a difference of squares, it is sufficient to explain an algorithm for evaluating u 2. The algorithm we describe is an improvement over that found in [6, 7]. In order to compute u 2 in a wavelet basis, we first recall that the projections of u on subspaces Vj and W j are given by Pju E Vj and Qju E W j for j = 0,1, 2 , . . . , J _< n, respectively (see the discussion in Section 3). Let jf, 1 __<jf _< J (see, e.g., Figure 10 where jf - 5 and J - 8), be the finest scale having significant wavelet coefficients that contribute to the
G. Beylkin and J. Keiser
170 e-accurate approximation of as
u, i.e., the projection of u can be expressed
J
Z E
dJkcj,k(x) +
J=Jl {k:ld{,l>~}
~
sg~g,~(x).
Let us first consider the case where u and (P0u) 2 in a 'telescopic' series,
U2 E
Vo, SO that we can expand
J - ( P j u ) 2 - E (Pj-lU) 2 -(Pju)
(P~
(5.1.1)
kE]F2n-J
2.
(5.1.2)
J=Js Decoupling scale interactions in (5.1.2) using
Pj-1 - Qj + Pj, we arrive at
J
(P~
-
(Pgu)2 + E 2(Pju)(Qju) + (Qju) z.
(5.1.3)
j=j!
Later we will remove the condition that u and u 2 E Vo. R e m a r k . Equation (5.1.3) is written in terms of a finite number of scales. If j ranges over Z, then (5.1.3) can be written as
u2 - E 2(Pju)(Qju) + (Qju) 2,
(5.1.4)
jEz
which is essentially the paraproduct, see [13]. Evaluating (5.1.3) requires computing (Qju) 2 and (Pju)(Qju), where Qju and Pju are elements of subspaces on the same scale and, thus, have basis functions with the same size support. In addition, we need to compute (Pgu) 2, which involves only the coarsest scale and is not computationally expensive. The difficulty in evaluating (5.1.3) is that the terms (Qju) 2 and (Pju)(Qju) do not necessarily belong to the same subspace as the multiplicands. However, since Vj ~ W j
- V j _ l (Z V j - 2 C "'" C V j - j o C " ' ' ,
(5.1.5)
we may think of both Pju E Vj and Qju E Wj as elements of a finer subspace which we denote V j_jo , for some jo _> 1. We compute the coefficients of Pju and Qju in Vj_jo using the reconstruction algorithm, e.g. (3.2.6), and on Vj_jo we can calculate contributions to (5.1.3) using (5.6). The key observation is that, in order to apply (5.6), we may always choose j0 in such a way that, to within a given accuracy e, (Qju) 2 and (Pju)(Qju) belong to V j_jo. It is sufficient to demonstrate this fact for j = 0.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
171
In order to show that such jo ;2_ I exists, we begin by assuming u E Vo C V_jo. This assumption implies that, in the Fourier domain, the support of @(2-J~ "overlaps" the support of fi((). Then, for scaling functions with a sufficient number of vanishing moments, the coefficients s[ j~ and the values u(xz), for some x~, may be made to be within e of each other. In this way we can then apply (5.6). The coefficients s/-j~ of the projection of u on V_jo are given by
s~ j~
2j~
-
F
u(x)~(2J~
(5.1.6)
- 1 ) d x ,
O0
which can be written in terms of fi(~) as
~;-jo _ 2Jo/~
a(2jo~)~(~)~-~d~"
(5.1.7)
(X:)
Replacing the integral in (5.1.7) by that over [ - r , 7r], we have
s~ j~ - 2j~ ~
fi(2J~ (~ + 27rk))~(~ + 27rk)e-i~ld~.
(5.1.8)
kEZ 71"
Since u E Vo, for any e > 0 there is a jo such that the infinite sum in (5.1.8) may be approximated to within e by the first term
f
s~Jo - 2Jo/2
fi(2Jo~)~(~)e-i~ld~"
(5.1.9)
7r
In order to evaluate (5.1.9), we consider scaling functions ~o(x) having M shifted vanishing moments, i.e. f _ ~ ( x - c~)m~o(x)dx - O, where c~ f _ ~ x~(x)dx, see e.g. [8, 22]. We then write
~ (x O0
-
a)mqo(x)dx
1
-
0 TM
__ei~ ( - i ) m O( m
C
r
dx
,~=o
for m = 1 , 2 , . . . , M , and arrive at
m Om
=0, (5.1.10)
(-i)- o-~~(~)~"~ ,~=o = 0 ,
(5.1.11)
@(~)e-i~e I~=o - 1.
(5.1.12)
and
Expanding ~(~)e i ~ in its Taylor series near ~ - O, we arrive at
:M-F1 0M+I ~(~)e ~e - 1 + (M + 1)! 0~"+1 ~(~)e~e
[
e=~'
(5.1.13)
G. Beylkin and J. Keiser
172
where z lies between ~ and zero. Since u E V0, the support of fi(2J~ (~ + 27rk)) occupies a smaller portion of the support of ~(~ + 27rk) as j0 increases, and there exists a sufficiently large jo such that the coefficients (5.1.9) can be computed by considering only a small neighborhood about ~ = 0. Therefore, substituting (5.1.13) in (5.1.9), we arrive at
s~ j~ - 2j~
f
~(2J~
+ EM,jo,
7r
where
EU,jo = (M + 1)!
, ~(2JO~)e-i((/+,)
~U+l(~Uw10U+l~--~eic~(
~ ~--z
(5.1.15) is the error term that is controlled by choosing jo sufficiently large. Remark. In practice jo must be small, and in our numerical experiments jo = 3. We note that for the case of multiwavelets [2, 3] the proof using the Fourier domain does not work since basis functions are discontinuous. However, one can directly use the piecewise polynomial representation of the basis functions instead. For spline wavelets both approaches are available. To describe the algorithm for computing the pointwise product, let 7~3o (-) denote the operator to reconstruct (represent) a vector on the subspace V j or W j in the subspace V j_jo. On V j_jo we can then use the coefficients T~o (Pju) and T~o (Qju) to calculate contributions to the product (5.1.3) using ordinary multiplication as in (5.6). To this end, the contributions to (5.1.3), for j = Jf,Jl + 1 , . . . , J - 1, are computed as
7)j_jo (u 2) - 2(T~o (Pju))(Tt~o (Qju)) + (T~o (Qju)) 2,
(5.1.16)
where 7)if(u) is the contribution to f(u) on subspace Vj (see (5.1.3)). On the final coarse scale J, we compute 7~j_jo (u 2) - (njo (Pju)) 2 + 2(n~o (Pju)) (Ttjjo (Qju)) + (n~o (QJu)) 2. (5.1.17) We then project the representation on the subspaces Vj_jo , for j - Jl,... J, into the wavelet basis. This procedure is completely equivalent to the decomposition one has to perform after applying the NS-form. The algorithm for computing the projection of u 2 in a wavelet basis is illustrated in Figure 10. In analogy with "pseudo-spectral" schemes, as in e.g.[23, 24], we refer to this as an adaptive pseudo-wavelet algorithm. To demonstrate that the algorithm is adaptive, we recall that u has a sparse representation in the wavelet basis. Thus, evaluating (Qju) 2 for j =
Pseudo-Wavelet Algorithms for Nonlinear PDEs
W
V
V
V
2
I
3
f
4
i
M
5 6 7
J iii~i!ill
W
V
f
S !
l
8
173
m ~
ii!!ii!i~
Figure 10. The adaptive pseudo-wavelet algorithm. Averages o n Vj are 'masked' by corresponding differences on Wj. These coefficients are then projected onto a finer subspace Vj_jo , Equation (5.1.16) is evaluated and the result is projected into the wavelet basis.
G. Beylkin and J. Keiser
174
1, 2 , . . . , J requires manipulating only sparse vectors. Evaluating the square of the final coarse scale averages, (Pju) 2, is inexpensive. The .difficulty in evaluating (5.1.16) lies in evaluating the products T~o (Pju)(T~oQju) since the vectors Pju are typically dense. The adaptivity of the algorithm comes from an observation that, in the products appearing in (5.1.16), we may use the coefficients Qju as a 'mask' of the Pju (this is similar to the algorithm for adaptively applying operators to functions). In this way contributions to (5.1.16) are calculated based on the presence of significant.wavelet coefficients Qju and, therefore, significant products T~o (Pju)(T~oQju). The complexity of our algorithm is automatically adaptable to the complexity of the wavelet representation of u. 5.2
R e m a r k s on the adaptive calculation of general
f(u)
This section consists of a number of observations regarding the evaluation of functions other than f(u) = u 2 in wavelet bases. For analytic f(u) we can apply the same approach as in Section 5.1, wherein we assume f(Pou) E V0 and expand the projection f(Pou) in the 'telescopic' series J
f(Pou)- f(Pju) - E f ( P j _ l u ) - f(Pju).
(5.2.1)
j=l
Using Pj-1 = Qj + Pj to decouple scale interactions in (5.2.1) and assuming f(.) to be analytic, we substitute the Taylor series
g f(~)(Pju) f (Qju + Pju) - E n[ (QJu)n+ Ej,g(f, u)
(5.2.2)
rt--O
to arrive at J
N
f(Pou) = f(Pju) + E E f(n)(PJu) n[ (Qju)n § Ej,g(f , u).
(5.2.3)
j=l n=l
For f(u) = u 2, jy = 1 and N = 2 we note that (5.2.3) and (5.1.3) are identical. This approach can be used for functions f(u) that have rapidly converging Taylor series expansions, e.g. f(u) = sin(u), for lul sufficiently small. In this case, for a given accuracy e we fix an N so that IEj,g(f, u)l < e. We note that the partia! differential equation (1.1) typically involves functions f(-) that are not only analytic but in many cases are p-degree polynomials in u. If this is the case then for each fixed j the series in (5.2.2) is of degree p and Ej,g(f, u) = 0 for N > p. In any event we are led to evaluate the double sum in (5.2.3), which can be done using the adaptive pseudo-wavelet algorithm described in Section 5.1.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
175
If the function f is not analytic, e.g., f ( u ) - iul, then the primary concern is how to quantify an appropriate value of jo, i.e., how much refinement (or 'oversampling') is needed to take advantage of the interpolating property (5.5). On the other hand, determining jo may become a significant problem even if f is analytic. For example if the Taylor series expansion of f ( u ) does not converge rapidly, e.g., f ( u ) - e u, we may be led to consider the following alternatives. In the first approach, we begin as above by expanding e u in the 'telescopic' series J
eP~ -- egJlt -- E
egJ-lU -- egju'
(5.2.4)
j--1
and using P j - 1 - Q j + Pj to decouple scale interactions. We then arrive at J e P~ - e PJu + ~ eP~U(e Qju - 1). j=l
(5.2.5)
Since the wavelet coefficients Q j u are sparse, the multiplicand e Q~u - 1 is significant only where Q j u is significant. Therefore, we can evaluate (5.2.5) using the adaptive pseudo-wavelet algorithm described in Section 5.1, where in this case the mask is determined by the significant values of e Q j u - 1. The applicability of such an approach depends on the relative size (or dynamic range) of the variable u. For example, if u ( x ) - a sin(2~x) on 0 <_ x _< 1 then e -~ <__ f ( u ) <_ e ~. It is clear that even for relatively moderate values of a the function e u may range over several orders of magnitude. In order to take the dynamic range into account, we apply a scaling and squaring method. Instead of computing e u directly, one calculates e u2-k and repeatedly squares the result k times. The constant k (which plays the role of j0 in the algorithm for f (u) = u 2) depends on the magnitude of u and is chosen so that the variable u is scaled, for example, as - 1 _ 2 - k u < 1. In this interval, calculating e ~2-~ can be accomplished as described by Equation (5.2.5) and the adaptive pseudo-wavelet algorithm of Section 5.1. One then repeatedly applies the algorithm for the pointwise square to e ''2-~ to arrive at the wavelet expansion of e u.
w
Results of numerical experiments
In this section we present the results of numerical experiments in which we compute approximations to the solutions of the heat equation, Burgers equation, and two generalized Burgers equations. In each of the examples we replace the initial value problem (1.1) with (1.2) and (1.3) by a suitable
176
G. Beylkin and J. Keiser
approximation, e.g., (2.2). The wavelet representation of the operators appearing in this approximation are computed via (4.1.2). In order to illustrate the use of our adaptive algorithm for computing f ( u ) developed in Section 5, we choose the basis having a scaling function with M shifted vanishing moments (see (3.1.7)), the so-called 'coifiets'. This allows us to use the approximate interpolating property, see e.g. (6.2) below. In each experiment we use a cutoff of e = 10 -6, roughly corresponding to single precision accuracy, The number of vanishing moments is then chosen to be M -- 6 and the corresponding length of the quadrature mirror filters LI LI H - {hk }k=l and G - {gk }k=l for 'coiflets' satisfies LI - 3M, see e.g. [22]. The number of scales n in the numerical realization of the multiresolution analysis depends on the most singular behavior of the solution u(x, t). The specific value of n used in our experiments is given with each example. We fix J, the depth of the wavelet decomposition, satisfying 2 n - J > LI, so that there is no 'wrap-around' of the filters H and G on the coarsest scale. Each of our experiments begins by projecting the initial condition (1.2) on V0, which amounts to evaluating so -
uo(x)~(x - 1 ) d x .
(6.1)
For smooth initial conditions we approximate the integral (6.1) (using the shifted vanishing moments of the scaling function ~(.)) to within e via s?
-
(6.2)
(see the discussion in Section 5.1). We note that in this case the discretization of the initial condition is similar to traditional discretizations, where one sets V(xi, to) = uo(iAx) (6.3) for i = 0, 1 , 2 , . . . ,2 n - 1, where Ax = 2 -n and where U(xi,t) is the numerical approximation of the solution at grid point xi = i A x and time t. Since approximations to the integral in (2.1) are implicit in time, we solve an equation of the form U(tj+l ) = E ( U ( t j ) ) + I(U(tj), U(tj+l ))
(6.4)
for U(tj+l) by iteration, where we have dropped the explicit x dependence. In (6.4), E(.) is the explicit part of the approximation to (2.1) and I(.) is the implicit part. One can use specialized techniques for solving (6.4), e.g., accelerating the convergence of the iteration by using preconditioners (which may be
Pseudo-Wavelet Algorithms for Nonlinear PDEs
177
readily constructed in a wavelet basis, see e.g. [7]). However, in our experiments we use a straightforward fixed-point method to compute U(tj+l). We begin by setting
Uo(tj+l) - E(U(tj)) + I(U(tj), U(tj)),
(6.5)
and repeatedly evaluate
Uk+l (tj+l) -- E(U(tj)) + I(U(tj), Uk(tj+l))
(6.6)
for k - 0, 1, 2 .... We terminate the iteration when
llUk+l(ty+l)- Uk(ty+l)ll<
(6.7)
e,
where
2,~ IIUk+l(tj+l) - U k ( t j + l ) P I -
) 1/2
2 -" ~ ( U k + l ( ~ , t j + ~ ) i--1
- V~(~,tj+l))
~ (6.s)
Once (6.7) is satisfied, we update the solution and set
U(tj+l ) = Uk+l (tj+l).
(6.9)
Again we note that one can use a more sophisticated iterative scheme and different stopping criteria for evaluating (6.4) (e.g., simply compute (6.6) for a fixed number of iterations). 6.1
The heat equation
We begin with this simple linear example in order to illustrate several points and provide a bridge to the nonlinear problems discussed below. In particular we show that in the wavelet system of coordinates, higher order schemes do not necessarily require more operations than lower order schemes. We consider the heat equation on the unit interval,
ut-uu~x,
0<x_
0
(6.1.1)
for v > 0, with the initial condition
u(x, O) - u o ( x ) ,
0 _< x _< 1,
(6.1.2)
and the periodic boundary condition u(O,t) = u(1, t). There are several well-known approaches for solving (6.1.1) and more general equations of this type having variable coefficients. Equation (6.1.1) can be viewed as a simple representative of this class of equations and we emphasize that the
178
G. Beylkin and J. Keiser
following remarks are applicable to the variable coefficient case, v = v(x) (see also [32]). For diffusion-type equations, explicit finite difference schemes are conditionally stable with the stability condition v A t / ( A x ) 2 < 1 (see e.g.[19]) where A t = 1/Nt, A x = 1 / N , and Nt is the number of time steps. This condition tends to require prohibitively small time steps. An alternate, implicit approach is the Crank-Nicolson scheme [19], which is unconditionally stable and accurate to O((At) 2 + (Ax)2). At each time step, the CrankNicolson scheme requires solving a system of equations, (6.1.3)
AV(tj+l ) = BV(tj)
for j = 0, 1, 2 , . . . , N t - 1, where we have suppressed the dependence of U(x, t) on x. The matrices A and B are given by A = d i a g ( - y , 1 + a , - ~ ) ~ w h e r e a - v (~)~. At and B - diag(~, l - a , ~), Alternatively, we can write the solution of (6.1.1) as (6.1.4)
u(x, t) = etLuo(x),
where s = uO~x, and compute (6.1.4) by discretizing the time interval [0, 1] into Nt subintervals of length A t = 1/Nt, and by repeatedly applying the NS-form of the operator e AtL via (6.1.5)
U(tj+l ) - e A t L v ( t j )
for j = 0, 1 , 2 , . . . , N t - 1, where U(to) = U(0). The numerical method described by (6.1.5) is explicit and unconditionally stable since the eigenvalues of e AtO~ are less than one. The fact that the Crank-Nicolson scheme is unconditionally stable allows one to choose At independently of Ax; in particular one can choose At to be proportional to Ax. In order to emphasize our point, we set Ax = At and v = 1. Although the Crank-Nicolson scheme is second order accurate and such choices of the parameters Ax, At, and v appear to be reasonable, by analyzing the scheme in the Fourier domain we find that high frequency components in an initial condition decay very slowly. By diagonalizing matrices A and B in (6.1.3), it is easy to find the largest eigenvalue of A - ~ B AN -- ~-2~ 1+2c~ " For the choice of parameters v - 1 and At = Ax, we see that as a becomes large, the eigenvalue AN tends to - 1 . We note that there are various ad hoc remedies (e.g. smoothing) used in conjunction with the Crank-Nicolson scheme to remove these slowly decaying high frequency components. For example, let us consider the following initial condition uo(x) -
{
x, l-x,
1
0 <_ x <_ ~, ~1 _< x < l _ ,
(6.1.6)
Pseudo-Wavelet Algorithms/or Nonlinear PDEs
179
1 Figure 11 illustrates the which has a discontinuous derivative at x - ~. evolution of (6.1.6) via (6.1.3) with At = Ax and u = 1, and the slow decay of high frequency components of the initial condition. We have implemented (6.1.5) and display the result in Figure 12 for the case where = 1, At = Ax = 2 -n = 1/N, and n = 9. We note that there is a proper decay of the sharp peak in the initial condition. In order to illustrate the difference between the results of our wavelet based approach and those of the Crank-Nicolson scheme, we construct the N S-form of the operator A - 1 B and compare it with that of e/xtL. The NS-form of an operator explicitly separates blocks of the operator that act on the high frequency components of u. These finer scale or high frequency blocks are located in the upper left corner of the NS-form. Therefore, the blocks of the NS-form of the operator A - 1 B that are responsible for the high frequency components in the solution are located in the upper left portion of Figure 13. One can compare Figure 13 with Figure 14 illustrating the NS-form of the exponential operator used in (6.1.5). Although the Crank-Nicolson scheme is not typically used for this regime of parameters (i.e. ~ = 1 and At = Ax), a similar phenomena will be observed for any low order method. Namely, for a given cutoff, the NS-form representation of the matrix for the low order scheme will have more entries than that of the corresponding exponential operator in the wavelet basis. Referring to Figures 13 and 14, it is clear that the NS-form of the operator e AtE in our high order scheme is sparser than the NS-form for the operator A - 1 B in the second order Crank-Nicolson scheme. The matrix in Figure 13 has approximately 3.5 times as many entries as the matrix in Figure 14. Let us conclude by reiterating that the wavelet based scheme via (6.1.4) is explicit and unconditionally stable. The accuracy in the spatial variable of our scheme is O((Ax) TM) where M is the number of vanishing moments, Ax = 2 -n and n is the number of scales in the multiresolution analysis. Additionally, our scheme is spectrally accurate in time. Also it is adaptive simply by virtue of using a sparse data structure to represent the operator e vAt~ the adaptive algorithm developed in Section 4.3 and the sparsity of the solution in the wavelet basis. Finally, we note that if we were to consider (6.1.1) with variable coefficients, e.g.
ut - L,(x)uxx,
(6.1.7)
the exponential operator e Atv(x)t: could be computed in O(N) operations using the scaling and squaring method outlined in, e.g., [9] (see also [12]).
G. Beylkin and J. Keiser
180
0.50
....
i'
,
,
"'
'
'
"I'
"
'
'
'
I
'
'
0.40
0..50 x
0.20
0.10
0.00 0.0
0.2
0.4
0.6
0.8
1 .0
X
F i g u r e 11. Solution of the heat equation using the Crank-Nicolson method (6.1.3) with At = Ax = 2 -9 and u = 1.0. Note the slowly decaying peak in the solution, which is due to the eigenvalue )~N = -0.99902344. 0.50
0.40
0.30 x
-4
4
0.20
0.10
0.00
I
0.0
0.2
J
,
,
,
,|
1
0.4
i
|
0.6
i
L
i
0.8
!
1
1 .0
X
F i g u r e 12. Solution of the heat equation using the NS-form of the exponential with At = Ax = 2 -9 and u = 1.0, i.e. Equation (6.1.5).
P s e u d o - W a v e l e t Algorithms for Nonlinear P D E s
NN N
181
% r r.
F i g u r e 13. N S-form representation of the operator A - 1 B used in the CrankNicolson scheme (6.1.3). Entries of absolute value greater t h a n 10 - s are shown in black. T h e wavelet basis is Daubechies with M - 6 vanishing m o m e n t s ( L / 18), the n u m b e r of scales is n -- 9 and J - 7. We have set v - 1.0 and At -- Ax -- 2 -9. Note t h a t the top left portion of the figure contains nonzero entries which indicate high frequency c o m p o n e n t s present in the operator A - 1 B .
P
r
r.
F i g u r e 14. N S - f o r m representation of the operator e ~ s used in (6.1.5). Entries of absolute value greater t h a n 10 - s are shown in black. T h e wavelet basis is Daubechies with M - 6 vanishing m o m e n t s (L/ - 18), the n u m b e r of scales isn-9andJ=7. We have s e t v = l . 0 a n d A t = A x = 2 -9 .
182
6.2
G. Beylkin and J. Keiser
Burgers equation
Our next example is the numerical calculation of solutions of Burgers equation ut + uux = vuzx, O <_ X <_ I, t >_ O, (6.2.1) for u > 0, together with an initial condition, u(x, O) = uo(x),
0 <_ x _ 1,
(6.2.2)
and periodic boundary conditions u(0, t) = u(1, t). Burgers equation is the simplest example of a nonlinear partial differential equation incorporating both linear diffusion and nonlinear advection. Solutions of Burgers equation consist of stationary or moving shocks and capturing such behavior is an important simple test of a new numerical method, see e.g. [34, 29, 4]. Burgers equation may be solved analytically by the Cole-Hopf transformation [27, 17], wherein it is observed that a solution of (6.2.1) may be expressed as u ( x , t ) - - 2 u Cx r
where r = r
(6.2.3)
t) is a solution of the heat equation with initial condition r
O) - e 4~. f u(z,O)d~.
(6.2.4)
Remark. We note that if u is small, e.g. u = 10 -3, then using (6.2.3) as the starting point for a numerical method turns out to be a poor approach. This is due to the large dynamic range of the transformed initial condition (6.2.4) (approximately 70 orders of magnitude for the initial condition u(x, O) = sin(27rx)). Consequently, the finite arithmetic involved in a numerical scheme leads to a loss of accuracy in directly calculating u(x, t) via (6.2.3), most notably within the vicinity of the shock. Our numerical scheme for computing approximations to the solution of (6.2.1) consists of evaluating U(ti+l) - e A t s
1 -- -~0s
[U(ti)OzU(ti+l) + U(ti+l)OzU(ti)],
(6.2.5)
subject to the stopping criterion (6.7). Since the solution is expressed as the sum (6.2.5), and the linear part is equivalent to the operator used in the solution of the heat equation, the linear diffusion in (6.2.1) is accounted for in an essentially exact way. Thus, we may attribute all numerical artifacts in the solution to the nonlinear advection term in (6.2.1). For each of the following examples, we illustrate the accuracy of our approach by comparing the approximate solution Uw with the exact solution
Pseudo-Wavelet Algorithms for Nonlinear PDEs
183
Ue using
2n --1 IIU
- U il -
2
/ t) -
i=0
(6.2.6)
t))
For comparison purposes, we compute the exact solution Ue via cr
f _ ~ e-G(~;x,t)/2vdzl where a ( ~ ; x, t) -
fo
"
F(~')d~' +
,
(x - 77)2
2t
(6.2.7)
'
(6.2.8)
and F0? ) - uo(r]) is the initial condition (6.2.2), see e.g. [35]. The initial conditions have been chosen so that (6.2.8) may be evaluated analytically and we compute the integrals in (6.2.7) using a high order quadrature approximation. E x a m p l e 1" In this example we set n - 15, J - 9, At -- 0.001, v - 0.001 and e - 10 -6. The subspace Vo may be viewed as a discretization of the unit interval into 215 grid points with the step size Ax -- 2 -15. We refer to Figures 15 and 16. Figure 15 illustrates the projection of the solution on V0, and Figure 16 illustrates the error (6.2.6) and the number of significant coefficients per time step. The number of operations needed to update the solution is proportional to the number of significant coefficients. The number of iterations required to satisfy the stopping criterion (6.7) increases during the formation of the shock, yet never exceeds 10 over the entire simulation. The compression ratios of the NS-form representation of the first derivative, exponential and nonlinear operator COL,m are 442.2, 3708.5, and 1364.9, respectively, where the compression ratio is defined as N 2/Ns where N is the dimension of the finest subspace Vo and N8 is the number of significant entries. E x a m p l e 2" In this example we illustrate the wavelet analogue of the Gibbs phenomena encountered when one does not use a sufficiently resolved basis expansion of the solution. In this example n = 10, J = 4, At = 0.001, v = 0.001, and e = 10 -6 , and we refer to Figures 17 and 18. Using n = 10 scales to represent the solution in the wavelet basis is insufficient to represent the high frequency components present in the solution. Figure 17 illustrates the projection of the solution on V0 beyond the point in time where the solution is well represented by n = 10 scales. We see that high frequency oscillations have appeared in the projection which may be viewed as a local analogue of the Gibbs phenomenon. Figure 18 illustrates
G. Beylkin and J. Keiser
184
!
!
0.6
0.8
x ,...___.,,
--0.2
- - 0 . 6
--1
.0
I
0.0
0.2
I
0
4
X
1 .0
F i g u r e 15. The projection on V0 of the solution of Burgers equation at various time steps computed via the iteration (6.2.5). In this experiment n - 15, J -- 9, At - 0.001, u -- 0.001, and e - 10 -6. This figure corresponds to Example 1 of the text.
I
2.OR
1 o
- ~
1 .6x
1 o
- 5
1 .2x
10
-`-%
,
,
i
i
~
/
8 . 0 x l
0
- 6
,4- O w l
0
- 6
~
0
0
4 O O
3
.~
0
2
4
I
T i m e
I 6
--
I
Step
--
-
0
2 0 o
-
1 0 0
-
E z
0
0
I 1 O0
I 2 0 0 Time
I ,:300
Step
I 4 0 0
5 0 0
F i g u r e 16. The error (6.2.6) per sample (Figure 15) and the number of significant wavelet coefficients per time step in the approximation (6.2.5).
Pseudo-Wavelet Algorithms for Nonlinear PDEs
185
the number of significant coefficients and the number of iterations per time step required to satisfy the stopping criterion (6.7). The compression ratios of the N S-form representation of the first derivative, exponential and nonlinear operator OL,m are 14.2, 15.4 and 21.3, respectively. E x a m p l e 3: In this example we compute the solution to Burgers equation using the initial condition 1
u(x, t) - s i n ( 2 7 r x ) + ~ sin(47rx),
(6.2.9)
which leads to the formation of left and right moving shocks. In this example n - 15, J - 9, u - 0.001, At - 0.001, and e - 10 -6 . We refer to Figures 19 and 20. Figure 19 illustrates the projection of the solution on V0. Figure 20 illustrates the error (6.2.6) and the number of significant coefficients needed to represent the solution in the wavelet basis per time step. The number of operations per time step used to update the solution is proportional to the number of significant coefficients in the wavelet representation of the solution.
6.3
Generalized Burgers equation
In this section we consider the numerical solution of the generalized Burgers equation
ut + uZux + Au ~ - vux~,
0<_x_
t_0
(6.3.1)
for constants a , ~ , v > 0 and real A, together with an initial condition u(x, 0), and periodic boundary conditions u(0, t) - u(1, t). This equation is thoroughly studied in [33] and we illustrate the results of a number of experiments which may be compared with [33]. E x a m p l e 4: In this example we set ~ - a - I and A - - 1 , and consider the evolution of a Gaussian initial condition centered on the interval 0 __ x __ 1, e.g., u(x, O) - uoe -(a(x-1/2))2. On the interval, the decay of u(x,O) is sufficiently fast that we can consider the initial condition to be periodic. We set n - 15, J - 4, At - 0.001, and c - 10 -6. For easy comparison with the results of [33], we choose v - 0.0005. The approximation to the solution of ~t t ~
lt~t x -- U --
l]Uxx,
0_<x__l,
t_>0
(6.3.2)
is computed via
U(ti+l ) - eAt(v~
- -~ 1(~0~ ,1 [U(ti)OxU(ti+l ) + U(ti+l )OxU(ti)] , (6.3.3)
G. Beylkin and J. Keiser
186
x
0
--2
I 0 . 0 0
I
0 . 2 5
I
0 . 5 0 X
0 . 7 5
1 .00
F i g u r e 17. The projection on Vo of the solution of Burgers equation at various time steps computed via the iteration (6.2.5). In this experiment n - 10, J - 4, At = 0.001, v = 0.001, and e = 10 -6. An analogue of the Gibbs phenomenon begins because the shock cannot be accurately represented by n - 10 scales. Observe that the scheme remains stable in spite of the oscillations. This figure corresponds to Example 2 of the text.
oo
3 0 0
--
2 0 0
--
~z o ,,._. E z I O 0
--
0
0
.r
816
T i m e
2 0
-
1 5
--
1 0
-
5
-
S
e
50
~
o
I
O
0
41`-.%
816 T i m e
1 1 `--%0 S t e p
I 1 7`--%5
~
/ 2
1 6
F i g u r e 18. The total number of significant wavelet coefficients and the number of iterations needed to satisfy the stopping criterion (6.7) per time step.
Pseudo-Wavelet Algorithms/or Nonlinear PDEs
187
1
-3
o --1
--2
................................. t 0 . 0 0
0 . 2 5
I
I
0 . 5 0
. . . . . .
0 . 7 5
1 . 0 0
X
F i g u r e 19. The projection on V0 of the solution of Burgers equation at various time steps computed via the iteration (6.2.5). In this experiment n = 15, J = 9, v = 0.001, At - 0.001, e = 10 -6, and the initial condition is given by (6.2.9). This figure corresponds to Example 3 of the text.
2 ~_
4 . 0 x
1 0
_ 5
3 . 0 x
10
- 5
2
10
- 5
1
O x
O x l O
-
- 5
0
I 0
u
2
4 Ti ~ m e
Step
6
8
~ o o
o o
E
2 0 0
z
?--~ 0
............ 0
I
~ O0
I 2 0 0 T i m e
1 3 0 0
Step
1 4 0 0
5 0 0
F i g u r e 20. The error (6.2.6) per sample (Figure 19) and the number of significant wavelet coefficients per time step in the approximation (6.2.5).
188
G. B e y l k i n and J. K e i s e r
where 0o~,1 -
e~t(~,o~+I) _ I vO 2 + I '
(6.3.4)
and I is the identity operator. We have chosen to use the operator s in the form s - v02 + I (see the development in, e.g., Section 2). We note that the NS-forms of the operators e At(v~ and (6.3.4) are computed as described in Section 4. Due to the negative damping in (6.3.2), the operator v02 + I is no longer negative definite. Therefore, if the nonlinear term were not present, thus the solution would grow without bound as t increased. The solution of the nonlinear equation (6.3.2) evolves to form a single shock which grows as it moves to the right. Figure 21 illustrates the evolution of the projection of the solution and Figure 22 illustrates the number of significant wavelet coefficients needed to represent the solution over the course of the experiment. On the other hand, the presence of the nonlinearity may affect the growth of the solution, depending on the size of the coefficient v. We have increased the diffusion coefficient to v - 0.005. Figure 23 illustrates the evolution of the projection of the solution and Figure 24 illustrates the number of significant wavelet coefficients. We point out that the number of operations required to update the solution is proportional to the number of significant coefficients. Example 5: As a final example, we compute approximations to the solution of the so-called cubic Burgers equation ut + u2u~ - vu~x,
O <__x <_ l,
t >_ O,
(6.3.5)
via U ( t i + l ) - eZXt~~
1
- ~0o~,1 [U2(ti)O~U(ti+l) + U 2 ( t i + l ) O ~ U ( t i ) ] ,
(6.3.6) where Oa~,l is given by (2.6). The only difference in (6.3.6), as compared with the approximation to Burgers equations (6.2.5), is the presence of the cubic nonlinearity. We have computed approximations to the solution using our algorithms with n = 13, J - 6, At - 0.001, u - 0.001, and e - 10-6. Figures 25 and 26 illustrate the evolution of the solution for a Gaussian initial condition, and Figures 27 and 28 illustrate the evolution of the solution for a sinusoidal initial condition. The Gaussian initial condition evolves to a moving shock, and the sinusoidal initial condition evolves into two right-moving shocks. We note that although the number of grid points in a uniform discretization of such an initial value problem is, in this case, N - 213, we are using only a few hundred significant wavelet coefficients to update the solution.
Pseudo-Wavelet Algorithms/or Nonlinear PDEs
189
1 . , , , _ _ _ 1 . 2
- _ _ _
I
.
o
- _ _ _
o
.
8
- _
,..r
_ _ o
.
~
- _ _ _
0
.
4
- _ _ _
Q . . ~
_ _ _ _
o
.
o . o
F i g u r e 21. The projection on Vo of the solution of (6.3.2) at various time steps. In this experiment n = 15, J - 4, At = 0.001, e = 10 -6, and v -- 0.0005. This figure corresponds to Example 4 of the text. I
I
I
I
I
I
I
I
I
I
I
I
V
~
0
0
I
1,
~
I
~
I
1
I
~,
1
I
~
1
I
~
2 T
I r'1"~
I
~
2
I
4
2
I
7
.:so
I
I
~ - , ,
I
._~e
II
F i g u r e 22. The total number of significant wavelet coefficients per time step. This figure corresponds to Example 4 of the text.
-.~o
G. Beylkin and J. Keiser
190
1 . 2 ~ ,
.oo
1
- -
0.7~,
- -
o.eso
- -
0 . 2 "
- -
o . o o . o
o.~
o . ~
F i g u r e 23. The projection on Vo of the solution of (6.3.2) at various time steps. In this experiment n = 15, J = 4, At -- 0.001, e -- 10 -6, and v - 0.005. This figure corresponds to Example 4 of the text. 7
~
0
e s o o
- -
4 " ~ 0
--
3 0 0
- -
I ' ~ 0
- -
0
0 . 0
I 1 ~..~
| 2 7 . o
I 4 o . 5
F i g u r e 24. The total number of significant wavelet coefficients per time step. This figure corresponds to Example 4 of the text.
1
o
Pseudo-Wavelet Algorithms for Nonlinear PDEs
191
m
m
m
o
.
o
o.~
Q
o.~
I
F i g u r e 25. The projection on Vo of the solution of the cubic Burgers Equation (6.3.5) at various time steps, computed using a Gaussian initial condition. In this experiment n - 13, J = 6 , At =0.001, v = 0 . 0 0 1 , a n d e = 10 -6 . This figure corresponds to Example 5 of the text. 200
I~0
m
-
120
y
~0
0
0
.
I
I
I
I
I
I
I
I
F i g u r e 26. The total number of significant wavelet coefficients per time step. This figure corresponds to Example 5 of the text.
.o
G. Beylkin and J. Keiser
192
I
o.2
I
o.4
I
o.e
I
o.o
I
F i g u r e 27. The projection on Vo of the solution of the cubic Burgers Equation (6.3.5) at various time steps, computed using a sinusoidal initial condition. In this experiment n = 13, J - 6, At - 0.001, v - 0.001, and e - 10 -8. This figure corresponds to Example 5 of the text.
y J
o
o
I
~
I
12
I~
I
2~
L
~o
I
~e
I
~2
I
~a
I
F i g u r e 28. The total number of significant wavelet coefficients per time step. This figure corresponds to Example 5 of the text.
Pseudo-Wavelet Algorithms for Nonlinear PDEs
w
193
Conclusions
In this paper we have synthesized the elements of numerical wavelet analysis into an overall approach for solving nonlinear partial differential equations. We have demonstrated an approach which combines the desirable features of finite difference approaches, spectral methods, and front-tracking or adaptive grid approaches usually applied to such problems. Specifically, we have considered the construction of adaptive calculations with operator functions in wavelet bases, and we have developed an algorithm for the adaptive calculation of nonlinear functions, e.g., f (u) = u 2. We used the semigroup method to replace the nonlinear partial differential equation (1.1) by a nonlinear integral equation (2.1), and outlined our approach for approximating such integrals. These approximations are expressed in terms of functions of differential operators, and we have shown how to expand these operator functions into a wavelet basis, namely how to construct the nonstandard form (NS-form) representation. We then presented a fast, adaptive algorithm for multiplying operators in the N S-form and functions expanded in wavelet bases. Additionally, we have introduced an adaptive algorithm for computing functions f ( u ) , in particular the pointwise product, where u is expanded in a wavelet basis. Both of these algorithms have an operation count which is proportional to the number of significant wavelet coefficients in the expansion of u, and we note that both of these algorithms are necessary ingredients in any basis-expansion approach to numerically solving PDEs. In order to verify our approach, we have included the results of a number of numerical experiments including the approximation to the solutions of the heat equation, Burgers equation, and the generalized Burgers equation. The heat equation was included to illustrate a number of simple observations made available by our approach. Burgers equation and its generalization were included to illustrate the adaptivity inherent in wavelet-based approaches, namely the 'automatic' identification of sharp gradients inherent in the solutions of such equations. Since Burgers equation is the simplest nonlinear example incorporating both diffusion and advection, it is typically a first example researchers investigate when introducing a new numerical method. There are several directions for this course of work which we have left for the future. One may consider nonperiodic boundary conditions instead of the periodic boundary condition (1.3). This may be accomplished by simply using a wavelet (or multi-wavelet) basis on an interval rather than a periodized wavelet basis. Also, we note that variable coefficients in the linear terms of the evolution equation (1.1), see e.g. (6.1.7), may be accommodated by computing the NS-form of the corresponding operators as outlined in, e.g., [9]. Another direction has to do with the choice of the
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wavelet basis. One of the conclusions which we have drawn from this study is that there seem to be a number of advantages to using basis functions which are piecewise polynomial. In particular the spline family of bases appears to be attractive as well as multiwavelets, see e.g. [2]. In both cases there are also disadvantages, and an additional study would help to understand such a tradeoff. Yet another extension, which of course is the ultimate goal, is to consider multidimensional problems, e.g., the NavierStokes equations. Finally, although we did not address in this paper the problem of computing solutions of nonlinear partial differential equations having wave-like solutions, let us indicate the difficulties in using a straightforward approach for such equations. A simple example is the Korteweg-de Vries equation
ut + c~uux + ~uxxx = 0,
(7.1)
where a, ~ are constant. Although our algorithm for computing the nonlinear contribution to the solution can be directly applied to this problem, the N S-form representation of the operator functions associated with this 3 problem, e.g. ezAt~ may be dense even for rather small values of At. Therefore, the adaptivity and efficiency of our algorithm for applying the NS-form of an operator to a function expanded in a wavelet basis are lost, due to the large number of significant coefficients present in the N S-form. Further work is required to find ways of constructing fast, adaptive algorithms for such problems. A c k n o w l e d g m e n t s . This research was partially supported by ONR grant N00014-91-J4037 and ARPA grant F49620-93-1-0474. References
[1] Ablowitz, M. J. and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, PA, 1981. [2] Alpert, B., A class of bases in L 2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993), 246-262. [3] Alpert, B., G. Beylkin, R. R. Coifman, and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Comput. 14 (1993), 159-184. [4] Basdevant, C., M. Deville, P. Haldenwang, J. M. Lacroix, J. Ouzzani, R. Peyret, P. Orlandi, and A. T. Patera, Spectral and finite difference solutions of the Burgers equation, Comput. & Fluids 14 (1986), 23.
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[5] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Numer. Anal. 29 (1992), 1716-1740. [6] Beylkin, G., On the fast algorithm for multiplication of functions in the wavelet bases, in Proc. Int. Conf. "Wavelets and Applications," Toulouse 1992, Y. Meyer, S. Roques (eds.), Editions Fronti~res, Paris, 1993, pp. 53-61. [7] Beylkin, G., Wavelets and fast numerical algorithms, in Proc. Sympos. Appl. Math. 47 (1993), 89-111. [8] Beylkin, G., R. R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math., 44 (1991), 141-183. [9] Beylkin, G., R. R. Coifman, and V. Rokhlin, Wavelets and Their Applications, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, M. B. Ruskai (eds.), Jones and Bartlett, Cambridge, 1992, pp. 181-210. [10] Beylkin, G. and W. Hereman, Unpublished manuscript, 1994. [11] Beylkin, G. and J. M. Keiser, On the adaptive numerical solution of nonlinear partial differential equations in wavelet bases, J. Comput. Phys. (1996), to appear. [12] Beylkin, G., J. M. Keiser, and L. Vozovoi, A new class of stable time discretization schemes for the solution of nonlinear PDEs, Preprint, 1996. [13] Bony, J. M., Calcul symbolique et propagation des singularite!s pour les ~quations aux d~riv~es partielles non-lin~aires, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), 209-246. [14] Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22. [15] Burgers, J. M., A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171. [16] Chui, C. K., An Introduction to Wavelets, Academic Press, Boston, 1992. [17] Cole, J. D., On a quasilinear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225.
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[18] Constantin, P., P. D. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985), 715. [19] Dahlquist, G. and A. BjSrck, Numerical Methods, Prentice-Hall, 1974. [20] Dahmen, W. and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992), 315-344. [21] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996. [22] Daubechies, I., Ten lectures on wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1992. [23] Fornberg, B., On a Fourier method for the integration of hyperbolic equations, SIAM J. Numer. Anal. 12 (1975), 509. [24] Fornberg, B. and G. B. Whitham, A numerical and theoretical study of certain nonlinear wave phenomena, Philos. Trans. Roy. Soc. London Ser. A 289 (1978), 373. [25] Gagnon, L. and J. M. Lina, Wavelets and numerical split-step method: a global adaptive scheme, J. Opt. Soc. Amer. B, to appear. [26] Haar, A., Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. (1910), 331-371. [27] Hopf, E., The partial differential equation ut + uux = #uxx, Comm. Pure Appl. Math. 3 (1950), 201. [28] Jaffard, S., Wavelet methods for fast resolution of elliptic problems. SIAM J. Numer. Anal. 29 (1992), 965-986. [29] Liandrat, J., V. Perrier, and P. Tchamitchian, Numerical resolution of nonlinear partial differential equations using the wavelet approach, in Wavelets and Their Applications, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Raphael, M. B. Ruskai (eds.), Jones and Bartlett, Cambridge, 1992, pp. 227-238 [30] Meyer, Y., Wavelets and Operators, Cambridge Stud. Adv. Math., Cambridge Univ. Press, Cambridge, MA, 1992. [31] Meyer, Y., Le calcul scientifique, les ondelettes et les filtres miroirs en quadrature, Centre de Recherche de Math~matiques de la Ddcision, Report 9007.
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[32] Engquist, B., S. Osher and S. Zhong, Fast wavelet based algorithms for linear evolution equations, 1991, preprint. [33] Sachdev, P. L., Nonlinear Diffusive Waves, Cambridge Univ. Press, Cambridge, MA, 1987. [34] Schult, R. L. and H. W. Wyld, Using wavelets to solve the Burgers' equation: A comparative study, Phys. Rev. A 46 (1992), 12. [35] Whitham, G. B., Linear and Nonlinear Waves, Wiley, New York, 1974. [36] Wickerhauser, M. V., Adapted Wavelet Analysis from Theory to SoTtware, A. K. Peters, Ltd., Wellesley, MA, 1994. [37] Yosida, K., Functional Analysis, Springer, New York, 1980. Gregory Beylkin Department of Applied Mathematics University of Colorado Boulder, CO 80309-0526 beylkin(~newton.colorado.edu James M. Keiser 659 Main St., Apt. B Laurel, MD 20707-4067 [email protected]
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A D y n a m i c a l Adaptive Concept Based on Wavelet Packet Best Bases: Application to Convection Diffusion Partial Differential Equations
Pascal Joly, Yvon Maday, and Val~rie Perrier
Abstract. We exploit in this paper a methodology based on the wavelet packet concept. It allows for solving partial differential equations employing very few degrees of freedom. The main application is the Burgers equation with a small viscosity. The wavelet packet framework allows one to define the notion of a minimal basis that has proven to be an efficient procedure for data compression. The purpose here is to take benefit of this compression to represent accurately and economically the solution of a time-dependent PDE. The time discretization is a standard multistep scheme. The spatial discretization is defined by inferring a reduced basis for the solution at the new time step, from the knowledge of the previous ones. The wavelet packet method is a better approach for adaptivity in the case where the solution to be approximated has many singularities.
w
Introduction
The theory of orthonormal wavelet bases has been developed by Y. Meyer [12] and his group. It allows an efficient representation to characterize isolated defects of some (otherwise regular) function, thanks to a particularly good location both in space and frequency of each element of the basis. The decomposition of such a function is lacunary, in the sense t h a t very few coefficients of its decomposition in a wavelet basis are nonnegligible. This allows for defining compressed approximations by getting rid of the coefficients t h a t are smaller than a prescribed threshold. This leads to a M u l t i s c a l e W a v e l e t M e t h o d s for P D E s W o l f g a n g D a h m e n , A n d r e w J. K u r d i l a , and P e t e r O s w a l d ( e d s . ) , pp. 199-235. C o p y r i g h t ( ~ 1 9 9 7 by A c a d e m i c P r e s s , Inc. All r i g h t s of r e p r o d u c t i o n in any f o r m reserved. ISBN 0-12-200675-5
199
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P. Joly et al.
nonlinear approximation in the sense studied by DeVore et al. in [6]. The natural applications of these bases are in image analysis and data compression. The next candidate for application of this theory are efficient adaptive methods for the approximation of nonstationary partial differential equations. We refer, e.g., to [10, 11] for more details of the definition of the algorithms and their implementation, and to [1] for some details of the numerical analysis of the method. The central idea in the previous papers is to infer the position or the creation of the (isolated) defect of the solution at some time step by knowing the wavelet decomposition of the solution at the previous time steps. This adaptive method is promising in the case when one can predict that the defects in the solution (sharp gradients or discontinuities) are the exception. Indeed, when the method is applied to approximate the solution of the one-dimensional Burgers equation, a significant reduction factor has been observed in the number of degrees of freedom compared to a standard uniform representation. This reduction factor is maintained in higher dimensions. The domain of application of this type of adaptive method is certainly reduced and cannot be used for approximations where large structures coexist with small scales. This is the case, for instance, if one wants to simulate turbulent flows. In this direction one has the work of M. Farge et al. [7] that compares the analysis and the compression of a turbulent field by spectral and wavelet packet decompositions. One can also refer to the recent thesis of E. Goirand [8] that compares the compression properties of wavelets, wavelet packets and local cosine bases for two-dimensional fields. The reduction factor for wavelet bases that has been observed in the case of simple signals does not hold anymore for turbulent ones. Nevertheless, it is recovered through using only the representative coefficients of turbulent fields in the wavelet packet decomposition. The concept of wavelet packets has been introduced by R. R. Coifman et al. [4, 3] as a generalization of the wavelet bases. It relies on the definition of a library of orthonormal bases of the same space, with elements depending on three parameters (space, scale, and frequency), and a procedure to choose, among all of these, the best candidate for the most economical way to represent a given function. This best basis is chosen so as to minimize some given entropy functional evaluated from the coefficients in each basis of the library. The aim of this paper is to take advantage of the good compression properties of the wavelet packets for the solution of partial differential equations in cases where one can guess that wavelets on their own will not be sufficient. In this sense, we want to generalize the notion of an adaptive method to the wavelet packet framework, in the same spirit as [10, 11]. In this paper, we introduce the main ingredients of the adaptive wavelet packet method, and we illustrate it on the Burgers equation and also on a
201
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simple convection diffusion equation in two dimensions (two canonical test cases before tackling the Navier-Stokes equations). More complex cases are under investigation. In Section 2, we recall the construction of the wavelet packets and the definition of the best basis. Although this is not new material, it allows for a better understanding of the following. Section 3 deals with the decomposition-reconstruction algorithm and the choice of entropy, justified by numerical experiments. Section 4 details the space and frequency location of wavelet packets, in order to introduce the concept of neighbors. Section 5 defines the basics of the adaptive algorithm for solving the Burgers equation. Finally, in Section 6, we illustrate the potential of the method by presenting several numerical examples, both in one and two dimensions.
w 2.1
Basic aspects of wavelet packets
Introduction to wavelet packet bases
The presentation of the wavelet packet framework is taken from [4]. The construction of wavelet packet bases is derived from the construction of orthonormal wavelet bases which themselves rely on the concept of multiresolution analysis. (a) C o n s t r u c t i o n of wavelet bases Definition 1. A multiresolution analysis of L2(IR) is a sequence V j, j E Z of closed subspaces of L2(IR) with the following properties: 9
V j C V j+l. +cr
9 N v, - {0}, DOG
(2.1.1) +oo
(2.1.2)
U vJ --
(:X:)
9 V f E L2(IR), V j E Z, f ( x ) E YJ ~
f(2x) E Y j+l.
(2.1.3)
9 There exists a function wo E LI(]R)N L2(jR) such that {wo (x - k) , k E Z}
is an orthonormal basis of V ~ .
(2.1.4)
From (2.1.3) and (2.1.4), we derive that the set {2J/2wo(2Jx- k)}keZ an orthonormal basis of VJ. Using (2.1.1), we introduce the spaces WJ as being the orthogonal complement of VJ in V j+l. In view of (2.1.4), we look for an orthonormal system of W ~ invariant under integer translations. As is proved in [12], we have the following property:
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P. Joly et at.
T h e o r e m 1. There exists a function Wl E L2(]R) such that {Wl(X-- k) , k E Z}
is an orthonormal system of W ~ Proof: In order to prove the existence of such a function, we introduce the decomposition of the function w0 in the basis { v ~ w0(2x - k)}keZ of Y 1" wo(x) - vI2 ~z~ ' hk wo(2x
-
k).
(2.1.5)
kEZ
It is readily checked that the orthonormality of functions w o ( x - k) (property (2.1.4)) implies that hk hk-2t -- 5o,~ 9
E
(2.1.6)
kEZ
Similarly, if wl exists, we have wl E
~(~)
V 1"
- v~~g~
~o(2~- k).
(2.1.7)
kEZ
The mutual orthonormality of the functions wl (x - k) is equivalent to gk g k - 2 1 - do,t,
(2.1.8)
kEZ
while the fact that wo and wl are orthogonal (because V ~ and W ~ are) is equivalent to: E hk gk-21 -- O. (2.1.9) kEZ Taking Fourier transforms, Equations (2.1.5) and (2.1.7) can be written in the following form:
~o(~)
too( ) Wo(~),
(2.1.10)
e~ (~) - m~ (~) ~0(~),
(2.1.11)
-
where 1
e_i~ k
kEZ and 1 E g k e_i~ k . ml (~1 - --~ k~Z
Adaptive Wavelet Packet Method .for PDEs
203
The formulation (2.1.10)-(2.1.11) is convenient since (2.1.6), (2.1.8), and (2.1.9) are then equivalent to the fact that
the matrix
m o ( ~ + ~)
is unitary ]or every ~. (2 1 12)
m~ (~ + ~)
" "
Indeed, relations (2.1.6), (2.1.8), and (2.1.9) imply that columns are orthonormal for all ~. The functions m0 and ml satisfying (2.1.12) are called quadrature mirror filters in the literature. From (2.1.12) we obtain the proof of the existence of Wl since, for instance, one can choose: m~ (~) - r
+ ~).
(2.1.13)
Conversely, property (2.1.12) is equivalent to the fact that the rows of the matrix are orthonormal, which in turn is equivalent to:
{ ~-~k(hkhk-~ + gkgk-l) = 26o,t Y~k (h2kh2k-~ nu g2kg2k-~) -- 60,~.
(2.1.14)
We thus have proved that the system {wl ( x - k), k E Z} is a basis of W ~ as follows from the two last properties. Indeed, this results from
~o(2x)
-
E IE(h2kh2k-t k + g2kg2k-i)l Wo(2X -k
(2.1.15)
h~ Z h ~ _ , wo(2~ - e ) + g~ Z g ~ - , wo(2~ - e) f
k
Theorem I implies in turn that { 2J/2wl (2ix - k) } keZ is an orthonormal basis of WJ and thus that the full collection:
2J/2wl (2ix --
k) } (J,k)e z2
is an orthonormal basis of L2(]R). (b) C o n s t r u c t i o n of wavelet p a c k e t bases Properties (2.1.5) and (2.1.7) of the two basic functions w0 and wl naturally lead to the recursive definition of other basic functions [4]"
w:~(~)
-
v ~ E h, w~(2x - k) kEZ
~+1(~1
-
v~~g~ kEZ
w ~ ( 2 ~ - k).
(2.1.16)
P. Joly et al.
204
These are, following the same methodology as in previous section, linearly independent and satisfy:
W n = s p a n { ( w n ( 2 x - k))aez} = span{(w2n(x - k), W2n+l (X -- k))kEZ} 9 Recursively, we prove that {wn(x - k) ; 0 __ n < 2J, k E Z} is an orthonormal basis of the space VJ. It follows that various orthonormal bases of L2(IR) are at hand, for instance, the collection -
9
In order to select all possible bases, we associate to each couple (n, j) of IN x Z, an interval In,j = [2Jn, 2J (n + 1) ). Let $ be a subset of IN • Z (resp. $g be a subset of IN x (Z N (-oo, J])). Then we have the following result [4]: T h e o r e m 2. Any collection
{WJn'k(x)de-'12J/2Wn(2JX -- k)}
(2.1.17)
with indexes (n, j, k) E $ x Z (resp. (n, j, k) E Sj x Z) is an orthonormal basis of L2(lR) (resp. of V J) if and only if the two following relations hold:
(n,j)ee.
(n,j)eEj
V ( n , j ) , ( n ' , j ' ) 6_ g (resp. gy)
9 In,jNIn,,j,-0
i f ( n , j ) 7/: (n,,j,).
Looking at relation (2.1.16), we derive, as in (2.1.10) and (2.1.11): w2~(~) ~/2n+l(~)
-
m0(~)~n(~),
(2.1.18)
--
ml(~)Wn(~).
(2.1.19)
Recalling that t~ is continuous and assuming ~o(0) = 1, by induction this leads to oo
~,~(~) - m~0(~)me, (2~-~) .- .m6~_~ (2~-~) I I
k=p+l
m~
),
(2.1.20)
where the sequence (ei)i=o,p-1 is defined by the dyadic decomposition of the integer n: n -- So + 2r + ... + 2P-lSp-1,
r E {0, 1} ,
and sp_~ = 1. Figure 1 illustrates the shape of some wavelet packet basis functions, for different values of the parameter n.
Adaptive Wavelet Packet Method/or PDEs
i
205
i .......
i
t
! i
.
.
.
.
.
.
.
.
.
~1 :~2
"
54
_/, ..... 256
162
t,
__,
i._
0 7578
Mult,-Resolut,on Analys,s .
. 62
N b revels 1
9 Scale
0 Frectuency 200 Posit,on
0
--
+ 0 2070
. _ _
128
0.5293
d
i
1oo
F i g u r e 1. Wavelet packet functions plotted in physical space (first column, wn (x)), and in Fourier spectral space (second column, Itb,~(~)1) for different values of n, n - 40 (first row), n 125 (second row), and n - 200 (third row). The symmetry centers of w~ are indicated on each figure.
P. Joly et al.
206 2.2
D e f i n i t i o n of t h e b e s t basis
Let 7)J be the orthogonal projection operator onto V J. Let now u be any fixed function in L2(IR). From the density property (2.1.2) we see that, for J large enough, P J (u) is close to u. In the previous section we constructed a large number of different orthogonal bases for the discrete space V g. In general, there is no reason for prefering one such basis to another one, but it seems natural that, for the particular case 7)g(u), one basis could be better suited with respect to some criterion. For the applications we are interested in, the criterion will be to minimize the information necessary for representing P J (u) well. For image compression, different authors [3] have advocated the Shannon entropy. Our precise objective here is to minimize the number of "large" coe.O~cients that are most useful in the decomposition of P J (u). Indeed, working only with these coefficients will decrease the amount of computation in the algorithms. To be more precise, let ~ be a positive real number, and let u be the function we want to approximate. Let 7)g(u) = E c3~kw~'k , cz~k -- (U;wJ'k), (2.2.1) (n,j,k)EEj •
be its decomposition in some orthogonal basis of V J, indexed by the set ~g • Z satisfying the conditions of Theorem 2. To this decomposition, we
associate the approximation 1-Is(u) defined as follows: =
(2.2.2) (n,j,k)eEj
where
"~.ks { c~ k d~' = 0
•
if otherwise.
The following lemma explains the sense in which IIs (u) is an approximation of P J (u). Lemma 1. For each u E L2(]R), each a > 0, and for each a threshold c such that
s
there exists
117:'J(u) - IL (u)IlL= ~ ~, The fact that the collection {wJ'k}(n,j,k)eCjx z is a Hilbert basis (complete orthogonal system) in Vj means that there exists a finite linear combination of elements of this collection such that
Proof:
IIVJ(u)
c~kw~'kllL2<_a,
-
(n,j,k)eZ~
(2.2.3)
Adaptive Wavelet Packet Method for PDEs
207
where/:~ is a finite subset of { (n, j, k) E $j • Z}. Let ~ - min { IcJ,kl ; (n, j, k) E Zu, cj'k ~ 0}. Then
II~J(u) - II~(u)llL= <_ IIPJ(u)-
~
c:~;k-w~j'kl[L=,
(n,j,k)6Zu
since II~(u) is the projection of u onto the space span{w~ ,k 9 I~kl ___~}, which contains span{w j,k ; (n, j, k) E/7~} by definition of ~. The proof is complete due to (2.2.3). I Let us now introduce the definition of entropy that we will be using. For u and c given, to any linearly independent family {wJ'k; k E Z}(n,j)ei, we associate the number of coefficients of u in the decomposition w. r. t. this family that are larger than ~ in absolute value. This number is defined as the cardinal entropy of the family indexed by I. This is a particular example of entropy; a more standard one is the Shannon entropy. It has been introduced in many papers dealing with the problem of signal compression, as it is optimal for this purpose [3]. The mathematical definitions of these entropies are: - for the cardinal entropy :
H~'I(u) - ~ {Ic:nkl > r (n,j,k) E I • Z } ,
(2.2.4)
- for the Shannon entropy :
HI(u)--
~
14'kl 2 lnl kl.
(2.2.5)
(n,j,k)EI•
For a given entropy, the best basis of u will be defined as the one that has the smallest entropy. Of course, the best basis depends on the chosen entropy. Our interest in this discussion lies in the possibility of minimizing the number of degrees of freedom to approximate a function to given accuracy. If the cardinal entropy is chosen, the number of degrees of freedom (given by the "significant" coefficients that are larger than ~ in absolute value) will be the entropy. If the Shannon entropy is chosen, the size of the coefficients comes into play, and the number of degrees of freedom will be given by the number of significant coefficients in the best (Shannon) basis representation. Thus, for a given ~, the definition of He~ leads to the smallest number of degrees of freedom. This is important for the numerical simulation as it directly influences the size of discretization matrices we have to work with. Another reason for preferring the cardinal entropy is that it is much cheaper to evaluate. In Subsection 3.2 we illustrate numerically the comparison between the two entropies for a less obvious criterion: the accuracy with respect to the number of degrees of freedom (see Figure 2 in Subsection 3.2).
P. Joly et al.
208 2.3
A d a p t a t i o n to t h e p e r i o d i c case
Our interest is to introduce a discrete method based on the wavelet packets to approximate the solution of some initial boundary value problem governed by partial differential equations. The boundary conditions can be either of Dirichlet, Neumann or periodic type. The main feature of the method is to be intrinsically adaptive inside the domain, hence the problem of the correct treatment of the boundary conditions is a separate issue. This allows us to present the method in the case of periodic boundary conditions. The corresponding bases will be derived from the ones on IR described previously by periodization of the generator functions win'k as follows:
f ( x + r).
V f E L2(IR), f(x) - Z
(2.3.1)
rEZ
It is then straightforward to check that VjEIN,
Vk" 0_k<2J-1,
The space V J obtained by periodizing all functions of V J is spanned by
--J,k Wo "O<_k<_ 2 J - 1 ) . These spaces form an increasing sequence of finite dimensional spaces in L2(0, 1), the union of which is dense in L2(0, 1). As in the Subsection 2.1(b), in order to select the proper basis of V J, we associate to each couple (n, j) of IN x IN an interval 7~,j = [ 2in, 2J (n + 1) ). Let C be a subset of IN x IN (resp. Cj be a subset of (IN N [0, 2 g - 1]) x (IN N [0, g])). Then we have a result similar to Theorem 2. Theorem 3. Any collection
{~k(x)
" 0 <_ k <_ 2j - 1, (n, j) E C (resp.(n,j) E s
}
is an orthonormal basis of L 2 (0, 1) (resp. of V J) if and only if the two following relations hold:
[.J I~,j -IR+
(n,j)~c
V
e E ( esp. Ej)
(n,j)es
I ,j N
j, - o
(n, j) # (n', j') .
Adaptive Wavelet Packet Method for PDEs w 3.1
209
N u m e r i c a l aspects of t h e w a v e l e t p a c k e t s
Decomposition-reconstruction
algorithm
In this section we describe an efficient algorithm proposed by Wickerhauser [14], first, to compute the wavelet packet coefficients of a given function f in YJ and, secondly, to select the best basis of f. The data for this function consists in the coefficients of f in the basis {~g,k, k - 0 , . . . , 2 J - 1} (we ,I
shall explain in Subsection 3.1(c) the way to get these coefficients in some particular cases): 2 J --1
f-- E
fJ,k--Jk W O'
.
k=O
From Equation (2.1.16) we derive a similar relation for the periodic basis functions 2j -1 ~-:..j,O k--0 2j --i
=j,o
~+1(~)
-
Zg~
~ ~(~)
(3.1.1)
k=0
where h~ - E
hk+2Jl
and fEZ
~EZ
are the periodic filters. (a) A n a l y s i s We now detail how to determine the wavelet packet coefficients
f~,k _ (f;~k) i.e., for all j - 0 , . . 9, J - 1, k - 0 , . . . , 2 J - 1 n - 1. The recurrence formula (3.1.1) leads to relations between the coefficients f ~ l ' k , J2,+l r j - l , k and fin 'k" in the various bases,
0,...,
2 J-j
--
2j -1
(3.1.2)
-
t----0 2j -1
2J--1, k
" t~----0
fj,~
(3.1.3)
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These are discrete convolutions, thus easily computable through Fast Fourier Transforms (FFTs) in a tree structure of J levels j = J , . . . , 1, each composed of 2 J - j patterns. One of the patterns is given by:
(:~,~) O
FFT(2J) 0
j
$
0<~<2
2j
a(e)s -l(t)
FFT -1 (2j-l)
j-1
2j
a(t)~SJ-l(t)
FFT -1(2 j-l)
S g n l'k
S2nTlJ-i'k
0 < k < 2j-1
0 m < k < 2j-1.
The computation of all coefficients fj, k for j = 0 , . . . , J - 1, k = 0 , . . . , 2J - 1 and n - 0 , . . . , 2 g - j - - 1 makes use of 2 g - j F F T s of order 2 j - 1 at each level j, so that the global computation of the analysis algorithm leads to less than 0(j22 J) operations.
(b) S y n t h e s i s
Conversely, from the coefficients
fj-l,k at
level j - 1, we can recover the
f~,k by a reversed pattern based on the following equality: 2 j-1 --1
fJn'k -- E ~=0
2j - 1 - 1
hJk-2~ fJ2nl'e+ E
gk-2s j J~ej-l'i2nq-i"
(3.1.4)
t=O
Similar to the previous analysis algorithm, the elementary step of the syn-
Adaptive Wavelet Packet Method for PDEs
211
thesis can be decomposed as follows: ~n 1,k 0
f 2J-l,k n+l
O
j-1
$
FFT(2 j-l)
FFT(2J -1)
4 0<~<2
j-1
"j-1
f 2~+~(e)
O < g < 2 j-1
j-1
~ ( e ) - ]~-1 (e mod[2J-1])
~(e)-
f^j-1 2 n + l (g mod[2J_X ])
-
0<~<2
j
FFT-I(2 j) 0
j.
The global cost of this converse synthesis on any prescribed basis is operations.
O(J2 J)
(c) I n t e r p o l a t i o n In practice, we start from sampling values of f at some equidistant points: f(k2-J),k = 0,... ,2 J - 1. From these values we want to compute the element of YJ that interpolates f at the nodes k2 - J , k - 0 , . . . , 2 J - 1. Let L g denote, if it exists, the element of YJ that satisfies
L J( k 2 - J ) - ~ 0 , k ,
k-0,...,2
g-1
(the spline wavelets we use here possess such a property). Then the interpolant in Y'J of f at the nodes (k2 - J ) is defined by 2 J --1
fJ(x) -- ~
f(k2 -J) L J ( x - k2-J).
k=0
This is the interpolant of f which is, however, not yet expressed in the basis l -J'kw 0 , k - 0, . . . , 2 J - 1 }, as assumed for the above decompositiong
reconstruction algorithm. Due to the orthonormality of this basis, we im-
P. Joly et al.
212 mediately get
2 J --1 --
Zfo ,
WO'
k=O
where J~ k _
2J - 1
S(e2
sl+ ,
S J _ (L J(. _ n2-J), Ng, o).
l=O
3.2
Choice of the best basis, choice of the entropy
(a) Recursive algorithm for the best basis Let us now indicate the method introduced in [3] to select the best basis of a given element u in V J. This is, by definition, a basis indexed by a certain set ~J that minimizes the chosen entropy H_~J (u), either Hc or Hs (defined in (2.2.4-2.2.5)). The selection of the best basis is done recursively from the basis indexed with j - 0 up to the one indexed by j - J, in a tree algorithm of order 2. Let us fix (n, j) and denote by P~J the L2-orthogonal projector onto the space spanned by the family { ~ k , 0 <_ k _ 2j - l } . For any v E L2(0, 1), we can define recursively the quantities 7-/{ as follows" j--1 (v) 7-/{(v)- m i n { 7 - ~ n l (v)--1- "~l'~2n+l
,
H{(.,j)} _ (v)},
with 7/~ 1 - +oc. The best representation of u will then be indexed: (3.2.1) Hence, at each branch of the tree algorithm, the best basis of P~(u) is either { ~ k , 0 <_ k _ 2j-1 } or the union of the best basis of p ~ - i (u) and j--1 of P2,~+1 (u).
(b) Comparison of the entropy conditions Recall first that our prefered choice of entropy condition, the cardinal entropy, is designed to minimize the number of degrees of freedom needed to represent the solution. The definition we introduced in (2.2.4) involves a threshold parameter e and differs from the standard Shannon entropy of [3]. Another argument in favor of our definition relies on the better compression property for a g/Yen accuracy. To illustrate this fact, we compare the two entropies together with the plain wavelet approximation. The comparison is performed on various types of functions.
Adaptive Wavelet Packet Method for PDEs
213
For the Shannon entropy, we determine the associated best basis, compute the relative components of the given function, sort them in decreasing order, and plot them, as a function in n, the error between the exact value and the partial sum based on the first n components. For the cardinal entropy, we select, for a given threshold s, the associated best basis, evaluate the number of coefficients larger than ~ in this best basis, and compute the error between the exact solution f and its reconstruction H6(f) obtained by neglecting the coefficients smaller than ~ in absolute value. For various values of c, the error as a function of the number of retained coefficients defines a curve that can be compared to the Shannon error curve (see Figure
2).
Now, since all the wavelet packet bases are orthonormal in L 2, a simple upper bound for the norm difference between the exact f and its partial reconstruction II~ (f) can be derived by bounding each coefficient we cancel by ~. Since there are at most 2 J such coefficients, a crude estimate is given by
III- H=(I)IIL~ _< 2fie. w 4.1
T o p o l o g y of a wavelet packet in position-frequency space
Position-frequency location of the wavelet packets
In this section we want to determine the location in the physical and frequency space of each element w j,k of the basis. This allows us to understand the structure of a given function when written in its best basis as far as its local behavior is concerned. In Figure 3 we have represented the functions mo and m l for the spline wavelets. As a consequence of the hypothesis made on m0
9 mo(O)--l, mo(~')--O, 9 periodicity of m0, it can be understood that m0 is a band pass filter around each 2~7r. Similarly, m l is a band pass filter around each 7r + 2~7r. Since ~b0(~) = I-Ik=l m 0 ( ~ ) , zb0 is a low pass filter while Zbl(~c) - m1(2~) 1-Ik=2 m 0 ( ~ ) is a band pass filter around 7r. It also follows that OG
k=3
is a band pass filter around 37r while
- ml( )ml( ) II m0( k=3
)
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P. Joly et al. B e s t B a s i s E n t r o p y / Shannon Entropy l o g l O(error)
Best Basis
"'-..
.......
Shannon
-4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15
,
50
,
100
t
150
i
!
i
o
~
200
250
300
350
400
........
450 nb c o e f f i c i e n t s
B e s t B a s i s E n t r o p y / Shannon E n t r o p y
0 _ -1 -2
Best Basis
-3
. . . . . . . Shannon
-4
"" . . . . . . . . . . . . .
-5 -6 -7
-9 -10 -11 -12 -13 -14 -15
r ........ -~Z'l
0
50
100
150
200
250
300
350
400
450
500
nb c o e f f i c i e n t s
F i g u r e 2. n 2 e r r o r [If - IIe(f)[[L 2 a~ a function of the number of coefficients retained in He(f). For the Shannon entropy, it corresponds to the bigger coefficients, for the cardinal entropy, it corresponds to H~f. The upper figure corresponds to f(x) - sin 167rx + $o.4~, the lower figure corresponds to the solution of the Burgers equation at time t - 1/47r, with initial condition sin 8rx (see Figure 12).
Adaptive Wavelet Packet Method for PDEs
Mirror
1.2
215
filters i
m0 ml
/ ,
~ ~
,i
0.8
l
,
, nl
0.6
.....
,'
L
,
0.4 I /
0.2
0.2
0.4
0.6
0.8
Figure 3. Plot of the transfer functions Im0(~)l and Im~ (~)1. is a band regard to order. As n written
pass filter around 2;r (see Figure 4). This is quite annoying in clarity, since the wn are not sorted in an increasing frequency already suggested in [14], this leads to a new notation. To any in dyadic form as n -- 6o nt- 261 - t - . . . - b 2P-16p_1,
6i E {0, 1},
with 6p_l - 1, we associate fi defined by -- gO q" 2 g l + . . . - [ - -
2P-lgp-1,
gi
E {0, 1},
where Cp-1 - - 6 p - 1 , 6 p - 2 -
I--6p-2
f
'
Cm-
~ 6m
I 1 --6m
if 6m+l = 0 Vm<_p-2. if em+l = 1
The wn are actually sorted in an increasing order of frequency if we consider the fi ranking. Indeed, wn is a band pass filter around the frequency fizr. It follows now from the definition (2.1.17) that w j is a band pass filter around the frequency f~2JTr. Our interest is to localize the position of this
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B
24
!
!
4O
56
I ,b
256
Figure 4. Modulus of the Fourier transform of functions wo, wx, w3, and w: (left to right). function in the physical space. Recalling that, in the case of the spline framework we are working in, m0 is even and real, we easily deduce that ~k~=p+l m o ( ~ ) is the Fourier transform of a real and even function. From (2.1.13) and (2.1.20) we realize that
~n(E,) - e -i~(~'::~ eq2-q-1) x real even function, p--1 from which we deduce that wn is centered around ~q=O "Cq2-qfunction w j,k is then symmetric with respect to its "center""
1
The
p--1
xt - Xl(j,n,k) -- 2 - j (k + Z
Cq2-q-1 )"
q=O
This result is illustrated in Figure 5 (see also Figure 2; note that the definition of the physical center becomes meaningless when j increases).
4.2
Definition of the neighbors
As explained in the next section, the adaptive procedure will be based on the notion of neighbors. This has to be done carefully, in order to be able to fit - - at best - - the evolution of the elements of the best basis which are
Adaptive Wavelet Packet Method for PDEs
217
Multi-Resolution Analysis " Basis function Scale
4 Frequency
8 Position
0
0.00
Figure 5. Spatial location of function wj'k for n = 8, j = 4, and k = 0. important in representing the solution optimally. This is the place where we shall use the fact that each element w j,k is associated with a special point of discretization x~ - ~ , 0 _< g _< 2 g - 1. The choice of best basis induces a numbering of these points in a proper way. Let (no,jo,ko) be a particular index, the neighbors of which have to be determined. Recall that the frame within which the neighbors will be chosen is the best basis B(u~ of u~ as defined in (3.2.1). In a preliminary step, the elements w j,k, for (n,j) e B(u~ and 0 _< k _< 2j - 1 are grouped by Fourier location [4] in sets bq, 0 <_ q <_ 2p - 1:
b q - {w j'k,
(n, j ) E B(u~
n2 j . - q}.
We remark that some of the bq can be empty, the other ones group elements of B(u~ localized around the same point in Fourier space (compare Figure 6 for the Fourier location of the elements of b128, that is 64). These two considerations allow us to define, in a position-frequency diagram, an influence rectangle for each element win,k of the basis, centered around the point (xl(j,n,k),n2 j) which symbolizes its space-frequency support. In Figure 7, we show an example this type of representation. Let us now define the neighbors of w j~176 Let q0 - n02 j~ be the Fourier
218
P. Joly et al. Multi-Resolution Analysis Scale I
t
7 Frequency i ,,
B9 a s i s f u n c t i o n 1 Position
64
~1~.~
} Multi-Resolution Analysis Scale i
I
,
p
o
;
,
o
5 Frequency ~ i
o
B 9asis function 4 Position
16
;
o
F i g u r e 6. Plot of wavelet packets w j'k in physical space (first column) and in Fourier space (second column) for the parameters n and j such that n2 j = 128, for example n = 1, j = 7 (first plot), n = 4, j = 5 (second plot), n = 16, j = 3 (third plot), and n = 128, j = 0 (last plot).
Adaptive Wavelet Packet Method/or PDEs
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Multi-Resolution Analysis " Basis function I
o
,
t
o
,
I
~
,
Scale
o
16 Position
3 Frequency I
,
,
4
Jl
Multi-Resolution 1~ B a s i s function II 4 1 Analysis Scale
0 Frequency
128 Position
0
F i g u r e 6. Plot of wavelet packets w~'k in physical space (first column) and in Fourier space (second column) for the parameters n and j such that n2j = 128, for example n = 1, j = 7 (first plot), n = 4, j = 5 (second plot), n = 16, j = 3 (third plot), and n = 128, j = 0 (last plot). (Continued)
P. Joly et al.
220
location of this basis function. The neighbors of this basis function are those elements of B(u~ which are either 9 in the same group bqo" w j~176 and on both sides of Xl(jo,no,ko),
win~176
the center of which are
9 in the previous (nonempty) group bq," corresponding to the one or both elements w~,k, the center of which is (are) the closest to
Xl(jo,no,ko) 9 in the next (nonempty) group bp,," corresponding to the one or b o t h elements w~,k, the center of which is (are) the closest to xt(jo,,~o,ko ). We notice t h a t the number of neighbors is _< 6 and _> 4. Wavelets packets / Position-Frequency Analysis
time 9 0.0000
Frequency
lm Position
F i g u r e 7. Representation in a position-frequency diagram of the best basis associated to the function plotted at the top. The black rectangles (called Heisenberg cellulars) are those for which the related coefficients are larger than 10 -4 in absolute value. The chosen function is sin 16~rx + 50.45 discretized at 256 points.
4.3
Adaptive filtering procedure
The definition of the cardinal entropy is done so as to associate to each given function u its minimal representation in the space V J. Indeed, after
Adaptive Wavelet Packet Method for PDEs
221
the choice of the best basis B(u) for an associated threshold e, we can drop the coefficients that are smaller than e, defining this way the element II~(u). The representation of u in V J is then (much) reduced and can be written as II~(u)~ u~ k wj'k , (4.3.1) (j,n,k)E ft(u) where and
A ( u ) - {(j, n,k) E A(u);
lu kl
~}
A(u) - {(j,n,k); (n,j) E B(u),O <_ k <_ 2j - 1}.
(4.3.2) (4.3.3)
The adaptive procedure consists in adding to .4(u) the neighbors of the indexes (j, n, k) in A(u). It can be noticed that even though the number of neighbors of each index is _> 4, the cardinality of the updated set is not much larger than the cardinality of Jl(u) since most of the neighbors are already elements of A(u).
4.4
Numerical illustration: Stability of the best basis
By applying a small perturbation to a function u, yielding fi, its best basis may be drastically different from that of u. However the entropy of fi in the best basis of u does not increase significantly. We will use this property in our algorithm and illustrate it in two examples. The stability of the best basis is illustrated here for both translation and sharpening of gradients. We have first defined the best basis B(uto) of the solution Uto at time to = 0.1343 of the Burgers equation with initial = sin(27rx). Then we decompose Uto ( x - T) in B(uto). The condition u~ number of coefficients larger than e is about the same as for the original solution, though the best basis is different; in addition, the coefficients involved are all neighbors in position of those of Uto (see Figure 8). The same conclusion holds for the analysis of the solution utl of the Burgers equation at time tl = 0.1766 that presents a larger gradient than Uto: the coefficients which appear in the decomposition are neighbors in space and frequency of those of uto (see Figure 9). Note that the difference in time between the two functions Uto and utl corresponds to 425At in our experiment. This confirms the stability of the best basis for these two types of evolution. The two experiments also show the viability of our approach, based on the concept of neighbors which we have introduced.
Adaptive Wavelet Packet Method for PDEs
221
the choice of the best basis B(u) for an associated threshold e, we can drop the coefficients that are smaller than e, defining this way the element II~(u). The representation of u in V J is then (much) reduced and can be written as II~(u)~ u~ k wj'k , (4.3.1) (j,n,k)E ft(u) where and
A ( u ) - {(j, n,k) E A(u);
lu kl
~}
A(u) - {(j,n,k); (n,j) E B(u),O <_ k <_ 2j - 1}.
(4.3.2) (4.3.3)
The adaptive procedure consists in adding to .4(u) the neighbors of the indexes (j, n, k) in A(u). It can be noticed that even though the number of neighbors of each index is _> 4, the cardinality of the updated set is not much larger than the cardinality of Jl(u) since most of the neighbors are already elements of A(u).
4.4
Numerical illustration: Stability of the best basis
By applying a small perturbation to a function u, yielding fi, its best basis may be drastically different from that of u. However the entropy of fi in the best basis of u does not increase significantly. We will use this property in our algorithm and illustrate it in two examples. The stability of the best basis is illustrated here for both translation and sharpening of gradients. We have first defined the best basis B(uto) of the solution Uto at time to = 0.1343 of the Burgers equation with initial = sin(27rx). Then we decompose Uto ( x - T) in B(uto). The condition u~ number of coefficients larger than e is about the same as for the original solution, though the best basis is different; in addition, the coefficients involved are all neighbors in position of those of Uto (see Figure 8). The same conclusion holds for the analysis of the solution utl of the Burgers equation at time tl = 0.1766 that presents a larger gradient than Uto: the coefficients which appear in the decomposition are neighbors in space and frequency of those of uto (see Figure 9). Note that the difference in time between the two functions Uto and utl corresponds to 425At in our experiment. This confirms the stability of the best basis for these two types of evolution. The two experiments also show the viability of our approach, based on the concept of neighbors which we have introduced.
Adaptive Wavelet Packet M e t h o d / o r PDEs w 5.1
223
D e f i n i t i o n of t h e n u m e r i c a l s c h e m e
D e f i n i t i o n of the adaptive algorithm
The model equation is the Burgers equation with periodic boundary conditions: Find u such that
Ou Ou 02u 0---t-+ U~xx - v~--2x2'
t _> 0, x e [0, 1],
(5.1.1)
with initial condition
u(0,
= u~
1 . The approximation involves a discrete and viscosity parameter v - 400~ parameter in time, At, that is a positive real number, and a discrete parameter in space, J. The solution u ( m A t , .) is approximated by an element u m - u ~ of V J defined by the following algorithm.
ALGORITHM: Initial step: Let A(u~ be the indices of the best basis associated to u~ (see equations (4.3.2-4.3.3)), the notation ~o and A(u~) is as explained in Section 4.3. Set A ~ - fi.(u ~g). (m + 1) th step: Let u TM be given by:
Z
(j,n,k)EA m
where A TM C A ( u ~ 9 i) Define first ~m and ~m as explained in (4.3.1). 9 ii) Define A m+l as follows: A m+l - { ( j , n , k ) e A(u~
that are neighbors of ( j ' , n ' , k ' ) e rim},
(for the definition of neighbors see Subsection 4.3). 9 iii) Compute u m+l written as" U m + l --"
~
Ol.~k(~'~+ 1) Win'k ,
(j,n,k)EA m+l
such that it satisfies the discrete problem: V ( j , n , k ) E A m+l,
P. Joly et al.
224 Urn+l_
fo
At
(tm
w~'k dx
= js
dftm* dwJn,k 1 f01 d(fi,~) d x - -~ dx 2 w j'k dx, dx dx
-u whereu,m = ~3 U m - ~ 1 U m -
1.
The time discretization that is presented here is the simplest one and, of course, can be easily improved, but this is not the point of this paper. Remark. The previous algorithm will certainly tend to increase the dimension of A m+l . Thus, in order to keep the size of the basis for the representation of u minimal, we have from time to time to compute again the best basis of u m and use the set of indexes A(u m) in place of A(u~ for further time steps (see the next section).
5.2
Details on the implementation
In order to solve (5.1), we need to construct the stiffness matrix related to this adaptive problem. Since the linear set spanned by the selected best basis functions (with indices in A TM ) which has to be used to perform the Galerkin procedure changes at each time step, and since from time to time the definition of the best basis may also evolve, it is worthwhile to have some computation and storage done in a preliminary stage. For this purpose, we consider the stiffness matrix with respect to the entire basis (~J'k)o
Adaptive Wavelet Packet Method/or PDEs
225
Range of coefrci~ts ll o OE+O0. 01E-03. [0.1E-02, 0.IE-01, { 0.1E+O0 0 IE,O! 0 IE.02 0 fE~,03 [ 0. IE*04 I 0 ~g*o5
I
Wevelets packers / Der;vatlon
0 YE.031 779 0.IE~021 744 O.IE-OI [ 1160 O. IE+O0[ 2472 0 IE,,OI [ 3312 0.IE,,02[ 2768 0 tE+03[ 2409 0 IE,04 [ 2012 0 IE+051 632 0 f E , yT{ 96
M a rr~x : mat2
Rar<je of coefficients [ OOE+O0. ( 01E-03, [ 0, IE-02, [0IE-01. 0tE+O0. [ 0 IE§ 0 1E.02, I o ~E.O3, I o IE.O4, I o 7E.o5,
0 1 E - 0 3 [ 2462 0 IE-02[ 16 0, I E 0 1 { 688 O. fE§ 2134 0tE§ [ 2878 0 tE,~02[ 2924 0 fE~03I 3012 O, YE~ 1866 0 IE,O5{ 3o~ o I E . 1 7 { 96
Figure 10. Shape of the second derivative matrix computed in the waveletpacket best basis of the function sin 2~rx (top) and function sin 8~rx (bottom).
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P. Joly et al.
that are not in Am). The values of (tim)2 are then computed at each point, which allows to derive the values of the coefficients of the interpolant of (~m)2 with respect to the best basis, by another fast transform. The overall cost is thus (9(J 2 J) operations, which is much less than the construction of the stiffness matrix. Since the chosen best basis has to be recomputed after some time, the appropriate moment to optimize the complexity of the algorithm is every 2J-th time step. Assume that the best basis needs no change. Then the computation of these nonlinear contributions could be done in a different way, by constructing a partial collocation procedure that allows to compute the only values of the coefficients of the interpolant in A m+l of (?~m)2 from the values of (~m)2 over the corresponding centers Xt(j,n,k) in O(p 2) operations, where p is the cardinality of A m+l [5]. Finally, the derivative matrix has to be applied to the coefficients of the term (~m)2. The construction of this matrix is done following the same lines as for the stiffness matrix. The global cost of each time step is thus (.9(p2) + O ( J 2 J) operations. An implicit treatment of the diffusion operator would certainly improve the stability of the scheme 5.1, although at the price of the inversion of a stiffness matrix which is nonconstant in time. This could be done again as quoted for the treatment of the nonlinear term [5]. w 6.1
Numerical results
Sine waves
In this section we present numerical results for the resolution of the onedimensional Burgers equation (5.1.1). As an initial condition, we consider the sine waves uo(x) - sin 27rx (first simulation) or uo(x) - sin 87rx (second simulation). The adaptive algorithm presented in Section 5 is implemented by using spline wavelets of order 4. The parameters in the experiments are 1 the time-step At -- 32007r' 1 the as follows: the viscosity is equal to v - 4007r' total number of grid points 2 J - 1024, and the threshold on the wavelet packet coefficients is taken at ~ - 10 -l~ As is well known, a shock appears during these simulations, at time ts = 1. for the first one, and at time ts - ~ for the second. The first simulation necessitates 5 redefinitions of the best basis of the solution during the run (between t - 0 and t s) while the second simulation needs 8 redefinitions. Figures 11 and 12 display respectively the time evolution of the solution for the first and the second experiment, and the corresponding spacefrequency diagram showing the structure of the solution best basis. Finally, Figure 13 compares in time the number of degrees of freedom in the (adaptive) wavelet-Galerkin algorithm versus our (adaptive) wavelet-packet algo-
Adaptive Wavelet Packet Method for PDEs
227
rithm. The result confirms the better compression factor obtained through the best basis representation. 6.2
E x t e n s i o n to m u l t i d i m e n s i o n a l s i m u l a t i o n s
The adaptive method we have presented here has a natural extension to multidimensional computations. Indeed, there are first several ways to define wavelet packets over the periodic cube ( - 1 , 1) d. For instance (say for d - 2) one can use the basis obtained from the one-dimensional basis by a standard tensor product construction:
~j,k(x) w~e(y); 0 <_ k <_ 2 j - l ,
(n,j) C $y,
0 <_ t~ <_ 2 i - l ,
(re, i) E ~ j ,
where g y and ~ y are index sets defined as in Theorem 3. Another way of defining two-dimensional bases is, as in ([7,13]), to consider the generation of "same level" bases by a tree of 2 d branches, similarly to the two branch tree in one dimension. This basis has the following form" N~,k(x ) ~ e ( y )
; 0 _< k, t~ < 2j - 1, ( n , m , j ) E Gj ,
where the set G j has to verify
[_J I ,j • Im,j -- [0, (n,m,j)EG:7 Note that, as appears in the analysis done on some test functions hereafter, this basis combines good compression properties with efficiency and economy of implementation. Indeed, the choice of the best basis following the chosen definition of the entropy allows to define a set Gsr that minimizes the entropy (say He). This selection is done recursively, from the basis indexed with j - 0 up to the one indexed by j - J, in a tree algorithm of order 4. Let us fix an index (n, m, j) and define by H{c (n'm'j)} (v) the entropy of the projection PJ,mv of v over the space spanned by the family
~j,k(x ) u
, 0 <_ k <__2j - 1.
This allows to define then recursively the quantity 7-l(n,m,j)(v) as follows:
?-l(n,m,j) (v)
- min{'H(2n,2m,j-1)(v) + ~-~(2n+l,2m,j-1)(V) "-[-~rg(2n,2m+l,j_ l ) (V ) Jr- "]-{(2n+l,2m+ l,j_l ) (V), H{ (n'm'J) } (v) }.
Hence, similar to the one-dimensional case, at each branch of the tree, the best basis of P~,mV is either
{~j,k(x ) ~ e ( y )
; 0 <_ k <_ 2j - 1}
P. Joly et al.
228
Wavelets packets / Position-Frequency Analysis "time 0.0000
Wavelets packets / Position-Frequency Analysis time 9 0.3125
Signal
Frequency
Position
F i g u r e 1 1. Time evolution of the Burgers solution and its best basis representation, for the initial condition uo(x) = sin 21rx, at times t = 0, t = o.3125 t = ! andt= i.
t Wavelets
~.Signal
packets/ Position-Frequency Analysis"time 0.5000
..................... ~
............................................
I F~quency
Position
Wavelets
packets/ Position-Frequency Analysis time 9 1.0000
Signal ------._____._____.,_ .............................................
F i g u r e 11. Time evolution of the Burgers solution and its best basis representation, for the initial condition uo(x) = sin 27rx, at times t = 0, t = o.3125 t = !2 1 r ~ ~r ~ and t = ~.(Continued)
P. Joly et al.
230 Wavelets packets / Position-Frequency Analysls : time 0.0000
!
..
,I
J.
i
,
I
i III IIil If! I1!, I I lli I lit;li i _.
I
I illilltIrl~il IIIIII I
J
........
'
,
g
,
' ,
i ,
i
~ilI!llltli
........ I
.......Ill
I
i
'
L
v
Position
Wavelets packets / Positlon-Frequency Anatysls : time 0.0697
t
Frequency
Wavelets packets / Position-Frequency Analysis : time 0.2500 i na
f 'Frequency
i
,
l
.........."
i
'
!' ' ~
.
,
I
~
'............
i
i
i
~
'
,i,uji~!
t
i
~ ; Position
F i g u r e 12. Time evolution of the Burgers solution and its best basis represen1 and tation, for the initial condition uo(x) - sin 8~rx, at times t = O, t = Tg-#, t= 1 411- ~
Adaptive Wavelet Packet Method for PDEs
Solution
1200
I
of
the B u r g e r s i
231
equation
(sin*8Pix)
!
I
Best Wavelet
Basis Basis
.....
i000
800
600
400 J 200
0
I
I
0.05
0.i
Time
I
I
0.15
0.2
0.25
Figure 13. Comparison of the time evolution of the number of degrees of freedom necessary to represent the solution, in the wavelet decomposition (dotted line) and in the best basis decomposition (solid line), of the Burgers equation for the initial condition uo (x) - sin 8~rx. or the union of the best basis of four projections: j--1
j--1
P2n,2m v,j-1 P2n+l,2mV,J-1
P2,~,2m+lv and P2n+l,2m+l v. Once the definition of the best basis is done, the adaptive algorithm can be defined following the same lines as in the one-dimensional case (definition of the neighbors allowing for a refinement, truncation of the negligible coefficients allowing for a "coarsening", Galerkin approximation on the reduced basis). A first simulation of this type is reported here; more will be detailed in a future work. In particular, we have to improve the algorithm complexity in order to take better into account the important reduction factor of the best basis (see also [7]). The following results deal with the analysis of the compression properties following the cardinal entropy for two types of two-dimensional functions. In order to compare the compression properties with respect to onedimensional situations, we have indicated the numerical results for both
P. Joly et al.
232 f (x) and f (x, y). First signal" sin(27rx) and sin(2~rx + 2~ry)
Level
Total number of coefficients 1D 128 256 512 1024
7 8 9 10
Number of coefficients larger than ~ - 10 -1~ 1D 2D
2D 16384 65536 262144 1048576
23 24 24 24
158 162 162 162
Second signal: Gaussians u(x) and u(x, y)
Level
Total number of coefficients 1D
7 8 9 10
128 256 512 1024
Number of coefficients larger than ~ - 10 -s 1D 2D
2D 16384 65536 262144 1048576
41 64 81 89
6004 7542 7832 7833
Here the Gaussians are defined by:
1 ( ( x - x~) 2 ~(~) - ~ o~ N
1'~ exp - ( x - Xc) 2 ) 2a 2
with a - 90 and Xc - 400, and 1
{ (x - xc) 2
u(~,y) - 2(o~ + ~i) ~
~
+
(Y-yc)2 ) a2 - 1 e x p ( -(x-xc)22a 2 + -(Y-Yc)2)2a 2 9
Note that the compression factor in dimension two is about the square of the one-dimensional compression factor for a similar function. We have also implemented a two-dimensional adaptive scheme, and the preliminary results are consistent with the analysis. Indeed, for a convection-diffusion equation
Ou
o--i-
vAu + 1Vu 2-0
Adaptive Wavelet Packet Method/or PDEs
233
with initial condition uo(x, y) - sin(27rx)sin(27ry) and v - 4oo~, 1 the simulation with N = 64 points in each direction and a threshold parameter of size 10 -6 leads to a number of coefficients equal to 112 at time t = 0, and less than 500 at time t - 2_ (results are similar for N - 128) 71" w
Conclusion
In this paper we have introduced a new discrete method for the adaptive solution of some partial differential equations. It relies on the good approximation properties of the wavelet packet bases that allow to define a self-adaptive procedure following the concept of best basis. This best basis is chosen so as to minimize some entropy. The standard entropy (Shannon) that was formerly introduced appears to be less interesting for our purpose than a simpler definition based on the number of large coefficients. The scheme we have introduced is illustrated on the simple Burgers equation and on a two-dimensional convection-diffusion equation. It appears that the adaptive concept does reduce the number of degrees of freedom significantly. Some improvements have still to be done on the treatment of the nonlinear terms in order to reduce the storage. We have sketched some ideas in this direction that need further testing (see [5]). Note that the tests we have performed serve as test examples before tackling the Navier-Stokes equations. The range of solution behavior we are interested in is more complex than the ones that we have treated here. But we already know that the wavelet packets are better suited for compression than the pure wavelet bases ([8]). We have illustrated this point by analyzing a one-dimensional section of a two-dimensional turbulent flow field extracted from a numerical simulation. For a threshold between 10 -2 and 10 -3, the number of coefficients retained in the wavelet packet decomposition is about 2 of the number of coefficients in the wavelet decomposition. Finally, an interesting feature of the wavelet packet bases (that we have not entirely used here) lies also in the localization of its elements. This can certainly help in the stabilization of the space-time discretization, as the elements of the best basis can be viewed as naturally tuned up bubble functions ([2]) that our adaptive process introduces where required (thanks to the definition of the neighbors). A c k n o w l e d g m e n t s . All computations have been performed on the Cray C90 of IDRIS (Orsay, France).
References [1] Bertoluzza, S., Y. Maday, and J. C. Ravel, Numerical analysis of the dynamically adaptative wavelet method for solving partial differential equations, Comput. Methods Appl. Mech. Engrg. 116 (1994), 293-299.
234
P. Joly et al.
[2] Brezzi, F., M. O. Bristeau, L. P. Franca, M. Mallet, and G. Rog6, A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg. 96 (1992), 117-129.
[3]
Coifman, R. R., Y. Meyer, S. R. Quake, and M. V. Wickerhauser, Signal processing and compression with wavelet packets, in Progress in Wavelet Analysis and Applications, Y. Meyer, S. Roques (eds.), Editions Fronti~res, Paris, 1993, pp. 77-93.
[4]
Coifman, R. R., Y. Meyer, and M. V. Wickerhauser, Size properties of the wavelet packets, in Wavelets and Their Applications, G. Beylkin, R. R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. A. Raphael, M. B. Ruskai (eds.), Jones and Bartlett, Cambridge, MA ,1992, pp. 453-470.
[5] Danchin, R., Th~se de Doctorat de l'Universit6 Paris VI, in preparation.
[6]
DeVore, R.A., B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), 737-785.
[7]
Farge, M., E. Goirand, Y. Meyer, F. Pascal, and M. V. Wickerhauser, Improved predictability of two-dimensional turbulent flows using wavelet packet compression, Fluid Dynam. Res. 10 (1992), 229-250.
[8]
Goirand, E., Paquets d'ondelettes 9algorithmes et m6thodes associ~es, parall61isation et applications g la turbulence, Th~se de Doctorat de l'Universit6 Paris VI, in preparation.
[9] Joly, P., Y. Maday, and V. Pettier, Towards a method for solving partial differential equations by using wavelet packet bases, Comput. Methods Appl. Mech. Engrg. 116 (1994), 301-307.
[10]
Liandrat, J., V. Perrier and P. Tchamitchian, Numerical resolution of nonlinear partial differential equations using the wavelet approach, in Wavelets and Their Applications, G. Beylkin, R. R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. A. Raphael, M. B. Ruskai (eds.), Jones and Bartlett, Cambridge, MA, 1992, pp. 227-238.
[11]
Maday, Y., V. Perrier and J. C. Ravel, Adaptativit~ dynamique sur base d'ondelettes pour l'approximation d'6quations aux d6riv6es partielles, C. R. Acad. Sci. Paris Sdr. I Math. 312 (1991), 405-410.
[12] Meyer, Y., Ondelettes et Op6rateurs I, Hermann, Paris, 1990. [13] Perrier, V. and C. Basdevant, Periodical wavelet analyses, a tool for inhomogeneous field investigation - theory and algorithms, Rech. A6rospat. 3 (1989), 53-67. [14] Wickerhauser, M. V., Adapted Wavelet Analysis from Theory to Software, A.K. Peters, Ltd., Wellesley, MA, 1994.
Adaptive Wavelet Packet Method for PDEs Pascal Joly Laboratoire d'Analyse Num~rique Tour 55-65, 5~me ~tage Universit~ Pierre et Marie Curie 4, Place Jussieu 75252 Paris Cedex 05, FRANCE [email protected] Yvon Maday ASCI, UPR 9029, Bat. 506, Universit~ Paris Sud, 91405 Orsay Cedex, FRANCE & Laboratoire d'Analyse Num~rique Tour 55-65, 5~me ~tage Universit~ Pierre et Marie Curie [email protected] Valdrie Perrier Laboratoire d'Analyse, G~om~trie et Applications URA 742, Universit~ Paris Nord 93430 Villetaneuse & Laboratoire de M~t~orologie Dynamique Ecole Normale Sup~rieure 24, rue Lhomond, 75231 Paris Cedex 05, FRANCE perrier~lmd.ens.fr
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Nonlinear
Approximation
a n d A d a p t i v e T e c h n i q u e s for S o l v i n g
Elliptic Operator
Equations
Stephan Dahlke, W01fgang Dahmen, and R0nald A. DeV0re
A b s t r a c t . This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions, the best possible approximation rate which a function can have for a given number of degrees of freedom is characterized in terms of its regularity in a certain scale of Besov spaces. Therefore, after demonstrating the gain of nonlinear approximation over linear approximation measured in a Sobolev scale, we review some recent results on the Sobolev and Besov regularity of solutions to elliptic boundary value problems. On the other hand, nonlinear approximation requires information that is generally not available in practice. Instead one has to resort to the concept of adaptive approximation. "We briefly summarize some recent results on wavelet based adaptive schemes for elliptic operator equations. In contrast to more conventional approaches one can show that these schemes converge without prior assumptions on the solution, such as the saturation property. One central objective of this paper is to contribute to interrelating nonlinear approximation and adaptive methods in the context of elliptic operator equations. w
Introduction
Adaptive methods are an i m p o r t a n t tool for numerically solving elliptic equations. Their origins appear in the adaptive grid refinements in finite element methods. Heuristically, adaptive methods are effective when trying to recover solutions u which exhibit singularities. Much impetus to the theory of adaptive finite element methods was provided by the introduction Multiscale Wolfgang Copyright
Wavelet Dahmen,
Methods
for PDEs
Andrew
3. K u r d i l a ,
(~)1997 b y A c a d e m i c
All r i g h t s o f r e p r o d u c t i o n ISBN 0-12-200675-5
237 and Peter
Press, Inc.
in a n y f o r m r e s e r v e d .
Oswald (eds.), pp. 237-283.
238
S. Dahlke et al.
and analysis of p and h - p methods by Babuska and his collaborators (see e.g. [2] and [3]). A lot of further results on this subject have been developed in the last years (see e.g. [4, 33, 52]). For an overview on the theory of adaptive finite elements, the reader is referred to [28] and [53]. On the other hand, for most adaptive algorithms, there exist no proofs of convergence. The purpose of this paper is to phrase the problem of designing and analyzing adaptive methods in the context of approximation of functions. In this way, we shall introduce some analytical tools which may prove useful for constructing and analyzing adaptive algorithms. In particular, we shall utilize heavily the theory of multilevel methods and wavelets. An outline of this paper is as follows. In Section 2, we introduce the elliptic problems that we shall consider. They take the form Au = f
(1.1)
where A is a symmetric positive definite operator which is boundedly invertible on some Sobolev space. Thus, (1.1) includes both integral equations and boundary value problems. In Section 3, we briefly recall the theory of Sobolev and Besov spaces which we shall need for this paper. While these spaces have their classical definitions in terms of derivatives and smoothness, we introduce them from the viewpoint of wavelet decompositions. This gives a simple criterion for membership in these spaces in terms of certain sequence norms applied to the wavelet coefficients. This also gives us an opportunity to introduce wavelet decompositions for various types of domains which will be important for both our numerical and analytic considerations. We conclude this section with summarizing some relevant facts about wavelet discretizations of (1.1). In Sections 4 and 5, we discuss some approximation concepts which are relevant for numerically approximating the solution u to (1.1). We draw distinctions between two cases: 9 L i n e a r methods where the approximation comes from a linear space. 9 N o n l i n e a r methods where the approximation takes place from a non-
linear set. Adaptive methods are a form of nonlinear approximation. In these sections, we shall discuss the smoothness required of a function v in order that it can be approximated with a certain efficiency by linear or nonlinear methods. This is important, visa vis (1.1), since it tells us the smoothness (regularity) the solution u must have in order for it to be approximated with a given efficiency. There will be two scales of regularity: one for linear methods, the other for nonlinear methods. The regularity for linear methods is given by the
Nonlinear Approximation and Adaptive Techniques
239
usual scale H a of Sobolev spaces. This regularity is well known and often used in error estimates for finite element methods. The corresponding regularity for nonlinear methods takes place in a certain scale of Besov spaces. This scale of spaces does not seem to be as fully understood in the finite element literature. This type of regularity needs to be used (in place of Sobolev regularity) when analyzing nonlinear methods such as adaptive finite elements. In particular, this type of regularity needs to be kept in mind when constructing adaptive numerical methods and when analyzing their performance (establishing error estimates). In Section 6, we discuss the regularity of solutions to (1.1) from the viewpoint of the Sobolev and Besov scales noted above. The regularity of the solution tells the maximum efficiency a numerical method can achieve. However, the problem still exists to construct such numerical methods and establish their convergence and error estimates in specific settings. In Section 7, we discuss how this might be accomplished in the context of wavelet decompositions. w
A class of elliptic p r o b l e m s
In this section, we shall introduce the generic elliptic problems which we wish to analyze. We shall consider the model case of a linear operator equation
Au = f
(2.1)
where A : H -+ H* is a boundedly invertible operator from some Hilbert space H into its dual H*, i.e.,
IIAVllH* • i[VlIH,
v e H.
(2.2)
We use the notation A • B to express that A ~ B a n d B < A. Here A < B means that A < C B for some constant C. In the case that A and B depend on parameters or variables, the constant C is to be independent of these parameters or variables unless explicitly stated otherwise. To simplify the exposition we will confine the discussion to self-adjoint operators A, i.e.,
a(u, v) "- (Au, v),
(2.3)
is a bilinear symmetric form and (2.2) means that the energy norm []u[I - a(u, u) 89
(2.4)
I1"11 II'ilH,
(2.5)
satisfies
S. Dahlke et al.
240
In variational (weak) form, the solution u to (2.1) is a function u E H which satisfies a(u, v) - (f, v), v e H. (2.6) The following examples indicate the scope of problems we have in mind. Suppose that ~ C ]R d is a domain. We shall always assume that ~ is a bounded, open, and connected Lipschitz domain. This covers all domains of practical interest. If k is positive integer, the Sobolev space H k ( ~ ) :-- W k ( L 2 ( ~ ) ) consists of all functions f C L2(~) whose distributional derivatives D ~ f , lui = k, satisfy Ifl~/k (~) "2 . - ~ 11D . fliL2(f~) I~l=k
(2.7)
is finite. The square root of (2.7) is the semi-norm for H k ( ~ ) and adding to it []flin2(~) gives the norm [[f][gk(a) in Hk(~). The spaces Ha(Ft) can then be defined for noninteger values a _ 0 in several equivalent ways, for example, using either interpolation theory, or viewing them as special cases of Besov spaces. The Besov spaces on domains are defined using moduli of smoothness (see e.g. [25]). For negative a one can employ duality. Simple examples for A are in this case Au - - A u or Au - - A u + cu d 02 where A - ~ j = l ~ is the Laplacian and c > 0. Here H = H~ (fl) or H H1 (Vt), respectively, where H ~ ( f l ) is the closure of C ~ (Ft) with respect to the [!" IIg m-n~ Of course, when A is the biharmonic operator, one has H - H02(~t) or similar spaces incorporating mixed boundary conditions. The second example represents a different type of problems which still fits into the present framework. In order to solve an exterior boundary value problem AU-0
in IR3 \ f l ,
U-f
on 0~t,
(2.8)
it is tempting to transform this problem into a boundary integral equation. For instance, the indirect method leads to the equation on the boundary
Au - f,
A - I + 2K
(2.9)
where K involves the double layer potential operator
Ku(x) "-[1/2-
Oa(x)]u(x) +
( x - y) iy - xl (y)dy,
(2.10)
and 0~(x) denotes the interior angle at x E 0 ~ located on an edge of 0~. In this case it is known that H - L2(F), F := 0gt. Solving (2.9) for u leads to the solution of (2.8) by evaluating a singular integral.
Nonlinear Approximation and Adaptive Techniques
241
Alternatively, the direct method yields an integral equation of the first kind
Au - Vu-
1
(-~ + K ) f,
where V is the single layer potential operator
47rl x _ y] dy.
Vu(x) 012
It is known that this fits into the above framework with H - H-1/2(F), H* - HI/2(F). Of course the latter context requires a proper definition of Sobolev spaces on surfaces or manifolds. This is usually done by "lifting" Sobolev spaces from domains in IRd with the aid of an atlas and a partition of unity (see e.g. [9, 19]). We shall comment on this issue later in more detail. Thus in what follows, H typically stands for a Sobolev space H t = Ht(f~) (or some subspace determined by boundary conditions) while for t _> 0, H -t is to be understood as the dual of H t. We conclude by pointing out one more property of the class of operators under consideration. Note that in the above examples A has support of measure zero or has a Schwarz kernel with certain asymptotic properties. More precisely, we will assume that
(Av)(x) - f K(x, y)v(y)dy, f~
where K;(x, y) is smooth off the diagonal x - y and where we require that whenever d + p + Ipl + [r'l > 0 v tOx/.t CgyK~(x, Y)I <_ c,,,dist (x, y)-(d+p+l,l+l-I)
(2.11)
holds with constants c,,. depending only on the multi-indices p, u C Z~_. Estimates of the type (2.11) are known to hold for a wide range of cases including classical pseudo-differential operators and Calderdn-Zygmund operators (see e.g. [18, 50]). Thus the single and double layer potential operator above as well as classical differential operators fall into this category. w
S m o o t h n e s s spaces and wavelet d e c o m p o s i t i o n s
We wish to treat the above type of problems by means of wavelet methods. To this end, we have to explain first what is meant by wavelets defined on the various types of domains appearing in the previous section and how wavelet expansions are related to smoothness spaces on such domains. The
S. Dahlke et al.
242
simplest setting is ~ = IR d. Although this is of limited use for problems of the above nature it is particularly well suited for bringing out the essentials of wavelet analysis and serves as a core ingredient for the construction of wavelets on other domains. 3.1
W a v e l e t s on E u c l i d e a n s p a c e
We begin by discussing smoothness spaces on ]R d. As above, if k is a positive integer, the Sobolev space Hk(IR d) "- Wk(L2(IRd)) is defined as before with ~ - IRd (compare with (2.7)). The Sobolev spaces H~(IR d) for other values of a C ]R are usually defined by Fourier transforms. In particular, H~ d) - L2(IR d) and H-a(IR d) is the dual space of H~(]Rd). There is an equivalent definition of the Sobolev spaces in terms of wavelet decompositions which is of primary importance in the present context and which we now describe. Let D denote the set of all dyadic cubes in IRd and let 7Pj be the collection of all dyadic cubes at level j. Then, I C Dj if and only if I - 2-J(k + [0, 1]d), for some j C Z, k C Z d. If r/C L2(lRd), we define T]I(X) " - - 1 I [ - 1 / 2 1 7 ( 2 J x
-- ~),
I --
2-J(~ -Jr-[0, 1]a).
(3.1.1)
Then T/i is a scaled, shifted dilate of ~ and [I~//I[L2(R~) = [l~]ln2(Rd) for all ICTP. We begin our discussion of wavelet bases with the biorthogonal bases of compactly supported basis functions. We recall briefly how such bases are constructed from multiresolution analysis. A function r/is said to satisfy a refinement equation (sometimes called a two-scale relation) if
~(x) - Z
ak~(2x- k).
(3.1.2)
kEZ d
We shall only deal with compactly supported functions r/. In this case, only a finite number of the coefficients ak are nonzero. The starting point for constructing biorthogonal wavelets is a pair of univariate functions ~ and @ of compact support, each satisfying a refinement equation, and both are in duality
f
-
k) - 5(k),
k e z,
R
with ~ as the Kronecker delta function on Z. From ~ and ~ we construct the := ~ ( x l ) . . . ~(Xd) multivariate functions r := ~ ( x l ) ' " ~(Xd) and r which are also in duality: / r162
Rd
- k) - 5(k),
k e Z d.
(3.1.4)
Nonlinear Approximation and Adaptive Techniques
243
Let Vo be the L2(IR d) closure of the linear span of the shifts r k), k E Z d, of r From the existence of a compactly supported dual function q~ it follows that the functions r k), k E Z d, are a Riesz basis for V0 and each element v C V0 has the representation
v - E {v, r
k))r
k),
(3.1.5)
kEZ d
with (f, g> - (f, g>L=(R") --- / f (x)9(x) dx 1=[d
the inner product for L2(]Rd). By dilation, we obtain the spaces .-
9v e
Then Vj is spanned by the functions r representation
I
(3.1.6)
v0}.
C ~)j,
and each v E Vj has the
V -- E (V, ~I>r Ic~j
(3.1.7)
Because r satisfies a refinement equation, the spaces Vj are nested Vj C Vj+l,
j C Z.
(3.1.8)
It follows [7] that Uj~/~ is dense in L2(IRd). Let Pj be the projector from L2(IRa) onto Vj given by
Zjf - Z
(3.1.9)
The projectors Pj are uniformly bounded on L2(IRd) and for each f C L2 (]Rd), I I f - PjfI]L=(Rd) ~ O. (3.1.10) The operators Qy "- Pj+~ - Py are also uniformly bounded on L2(IR a) and their range Wj C L2(]R d) is called a wavelet space. The spaces Wj inherit the same structure as the Vj. For example, each Wy is the dilate (by 2J) of Wo. Also, Wo is a shift invariant space generated by a set tI,~ of 2 a - 1 functions r That is, W0 is the closed linear span of the functions r k C Z d, r C ~~ Moreover, there is a d u a l set ~~ of 2 d - 1 functions ~ such that
(r162
j, k E Z d,r176
(3.1.11)
S. Dahlke et al.
244
Now suppose that r r are exact of order N, N, respectively, i.e., x ~ - E k ~ z ~ ( ( . ) ", ~ ( . 9 " - E k ~ z , ((.) ", r
k))r
- k),
k ) ) 5 ( x - k),
x E I R d, ] a l < N ,
I~1
x ~ ~d
(3.1.12) <
where ]al denotes the sum of the components of the multi-indices a E Z~_. An immediate consequence of (3.1.11) and (3.1.12) is that the r E @~ have
vanishing moments, i.e., P(x)r
- O,
r E @~
(3.1.13)
Rd
for all polynomials P of coordinate degree less than .N. An analogous statement holds for ~ and /V replaced by r and N, respectively. It is known that for any N, N E IN such that N +/V is even, there exist compactly supported dual pairs ~o, ~5 whose order of polynomial exactness is N, N, respectively [11]. Thus the biorthogonal setting not only offers more flexibility in constructing compactly supported wavelets, where all filters have finite support and therefore give rise to fast reconstruction and decomposition algorithms, but also allows one to construct wavelets with an arbitrarily high number of vanishing moments, which is crucial for treating integral equations. It follows that each function in Le(IR d) has the wavelet decomposition
S- E E (f'r162
(3.1.14)
IET) CE~ ~
An alternative wavelet decomposition and the one preferred in numerical considerations starts at a finite dyadic level (which we, for notational convenience, v'r take as level 0). Then, for each f E L2(IRd), we have
-
PoIS/+Z Z I s , ~ , / ~ , IET)+ CE~ ~
:
E Is, ~ ( - k))~(- k)+ E kEZ d
Z (s,;~I/~i,
(3.1.15)
lET)+ CE~ ~
with D + the set of dyadic cubes with measure _< 1. The set of functions {r162 is a Riesz basis for L2(]R d) (sometimes called a stable basis). This means that llSll ~~ ( ~ )
~ Z Z LIs, ~,/I ~
v
I E T ) CEq~ ~
(3.1.16)
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As mentioned earlier, the constants in (3.1.16) are independent of f. We will use similar notation throughout this paper. The set {r is also a Riesz basis for H a, and I l f l ] ~ ( n d) • E
E
1II-2a/dl{f'~I)]2'
(3.1.17)
IET~ C E ~ ~
for a certain range - ~ < a < 7 that depends on the sets t~ ~ and t~ ~ The equivalence (3.1.17) gives us a simple way to compute equivalent H a norms in terms of wavelet coefficients. Also note that in this form, we see that we scale up or scale down the Sobolev spaces by simply multiplying wavelet coefficients by II[ s/d. Namely, let
Zs(f)
"-- E E Ills~d
~I>r
(3.1.18)
Then ~ (f) E H a+s (IR d) if and only if f E Ha(IRd), s + a E (-~, 7) and
(3.1.19)
IlZs/IIH,+o(R,') • II/IIHo( ").
The Besov spaces Bq(Lp(IRd)), a > O, 0 < p,q < ~ , are smoothness spaces in Lp(lRd). The index a is the primary index and gives the order of smoothness (analogous to the number of derivatives). The second parameter q gives a finer scaling. For example, the space B~(L2(IRd)) = Ha(IR d) and B~(Lp(IRd)) is the Lipschitz space of order a in Lp(IRd), provided a > 0 is not an integer. Besov spaces are usually defined by Fourier transforms or moduli of smoothness (see e.g. [25]). There is, however, an equivalent definition in terms of wavelet decompositions which we shall employ. In fact, for the above range of a one has ~" ~ IIfl] B;(/,(nd)) q "~ IET) r
III-q(a/d+l/U-1/P)l(f,r
E
< co,
(3.1.20)
o
whenever .f E Bq(Lp(IRd)). We shall mainly be concerned with a particular scale of these spaces which will replace the role of the Sobolev spaces when treating nonlinear approximation. If a > O, we let
7 "-(aid + 1/2) -1
(3.1.21)
so that 7 < 2. Then the Besov space Ba(IR a) "- B~(L.,.(IRd)) is the set of all functions in L2(IR a) which have a wavelet decomposition (3.1.14) and the wavelet coefficients of f satisfy
IISII o(, ) •
I(f,r
< cr
(3.1.22)
S. Dahlke et al.
246
Then, (3.1.22) gives an equivalent quasi-norm for B a. Note that B~ d) = L2(IR d) with equivalent norms. As a gets larger, the spaces Ba(IR d) get smaller" Ba(IR d) C BZ(IRd), a _ ~. Wavelets of the above type are still of limited use for the numerical treatment of operator equations. Below we will briefly indicate how to obtain wavelet bases with the above properties (3.1.16), (3.1.17), (3.1.22) in several other cases of practical relevance. 3.2
W a v e l e t s on t h e i n t e r v a l
As the simplest example of a bounded domain let us consider first gt = [0, 1]. This case deserves particular attention because it will also serve as a core ingredient in constructions for more complex situations. The common starting point (see e.g. [1, 8, 12, 17]) is to construct collections Ok = {r : m E Ak} C L2([0,1]) such that the spaces Sk := span Ok are nested and contain all polynomials up to a certain desired degree. Taking some dual pair ~,~5 as in (3.1.3) and fixing such that for k > ko we have supp~(2 k. -m),supp~5(2 k. - m ) C (0, 1), m = t~,..., 2 k - e, the collections Ok are comprised of these interior translates 2k/2~(2 k. --m) together with certain boundary functions which are needed to preserve the desired degree of polynomial exactness. If ~ has exactness order N, these boundary functions are simply obtained by truncating the expansions (3.1.12). For instance, for the left end of the interval one adds the N functions L
~k,t-~+ ~(x) "-
E
<(')~'@(" - m)>2}/2~(2k " - m ) I[o,1],
(3.2.1)
m=--c~
r -- 0 , . . . , N - 1, and analogously at the right end. One easily infers from (3.1.12) that these boundary functions together with the interior translates reconstruct all polynomials up to degree N - 1 on [0, 1]. Thus the resulting spaces inherit the approximation properties of their shift-invariant counterparts defined on all of IR. Moreover, since ~k,~_n+~(X) L behaves near 0 like 2k/2xr, it is relatively easy to incorporate homogeneous boundary conditions. The construction of dual collections ~)k differs somewhat in the above mentioned papers. In [17], it is shown that also (~k can be made to be exact of order N in an analogous fashion and that the resulting sets can indeed be biorthogonalized (compare with (3.1.11) and (3.1.12)). Let us denote the biorthogonalized sets again by (I)k, ~k with elements Ck,m, Ck,m. Except for a finite number of boundary functions, the Ck,m, Ck,m still have the form 2k/2~(2 k. - m ) , ~ E {~, ~5}, respectively. Moreover, compactly supported biorthogonal wavelets Ck,m, Ck,m, m -- 1 , . . . , 2 k, are constructed, which
Nonlinear Approximation and Adaptive Techniques
247
form Riesz bases for L2([0, 1]). Due to the modifications of the basis functions near the end points of the interval, the simple recipe from (3.1.1) of taking translates of dilated functions is no longer applicable. Nevertheless, it will be convenient to accept the slight abuse of notation and still write r r In fact, setting in this case De "- {k0} • { g - n , . . . , 2 k~ - g + n}, Z)k "-- {k} x { 1 , . . . 2 k - l } , k > k0 we can still identify the indices (k,m) with dyadic cubes I = 2 - k ( m + [0, 1]). Defining 7) + "- Uk>koZ)k and 79 "- De U Z)+, we obtain essentially the same format as above:
f -- E (f' ~I)r -Jr Icv~
E
(r
•
~)I)~)I,
(3.2.2)
111-2~l(f, ~,)12,
(3.2.3)
(f,
as well as
Ilfll~([o,~]) ~ ~
I(f,,~,)l 2 +
IC79c
~ (r
•
or
I]fl[~([o,1]) x ~ [(f, r I~v~
~+
~ (~,i)c.o •
I(f, r
T,
(3.2.4)
where 7 := (c~ + 1/2) -1 . Of course, in this case one has # ~ ~ = 1, but in anticipation of the tensor product case below this redundance is useful. Also one should note that for notational simplicity, we have suppressed the fact that, due to boundary modifications, ~~ actually depends on I. Again the range for which (3.2.3) is valid is ( - ~ , 7), where "7"-sup{a'~eH
a(IR)},
~.-sup{a.qSeH
a(]R)}
(see [17]). The case c~ - 0, of course, recovers the Riesz basis property. Since by construction the spaces Sk "-- span ~k are exact of order/V, one has
/
1
xrr
- O,
I e D +,
r - O,...,N -1.
(3.2.5)
0
3.3
T h e i s o p a r a m e t r i c case
Taking tensor products of wavelets on [0, 1] immediately yields biorthogonal wavelet bases on the unit d-cube [::1 "- [0, 1] d with analogues of (3.2.3), (3.2.4), (3.2.5). One can push this line a little further in the following direction. Suppose that for some d ~ _> d, ~ is a regular mapping from IRd into IRd', i.e., ~ is smooth and its J acobian is bounded away from 0. Let
S. Dahlke et al.
248
f~ . - n(r-1). Sobolev spaces or Besov spaces on Ft can be defined by lifting corresponding spaces from [2 with the aid of n. In fact,
(f , g) .-- / f(~(x))g(~ (x)) Idet ~' (~ -1 (x))ldx []
is a natural inner product which can be used to define Sobolev norms. On the other hand, / .
(f, g) "- / f(a(x))g(a(x))dx
(3.3.1)
. I
[]
induces equivalent norms whenever a is sufficiently regular. Taking tensor products of the above mentioned wavelets on the interval readily yields biortho~gonal wavelet bases 9 - { r 9 I E Dc} U {r " r E ~o, I E D+}, - {r " r E ~} on D. Here we have used the convention r "- r for I E De. Of course, in this case one has # ~ o _ 2 d - 1 and the structure of the sets De, D + is clear from the tensor product construction. Then the collections .-
.-
o
e
(3.3.2) t~f) .-- {~/~ .-- ~ I o / ~ - 1 . ~ I E i~},
are obviously biorthogonal relative to the inner product (.,.) in (3.3.1), which again satisfy (3.2.3) and (3.2.4). The moment conditions take the form (P,r 0, ( r E ~o • D+, (3.3.3) whenever P o n-1 is a polynomial of coordinate degree less than .N. Here and in the following we reserve the notation De for those dyadic cubes associated to the scaling functions on the coarsest level. The importance of this case will become clearer below. 3.4
W a v e l e t s on m a n i f o l d s
When d = d' = 2 the above construction yields, for instance, wavelet bases on various planar domains. However, the case d' > d is important too. In fact, the examples in Section 2 show that one needs wavelets defined on manifolds which are embedded in some higher dimensional Euclidean space. The simplest case is the d-torus. Functions defined on the d-torus correspond in a one-to-one way to l-periodic functions f(x + rn) = f(x), m E Z d. Clearly every compactly supported function r/in L2(IR d) is easily periodized by [r/](x) "- ~ ~(x + k). (3.4.1) kEZ d
Nonlinear Approximation and Adaptive Techniques
249
Moreover this is easily seen to preserve orthogonality relations,
f g(x)f(x)dx
-
0
]Rd
~
f[f](x)[g](x)dx
i.e.,
=0.
El
Thus, given wavelets r ~I on IRd, the functions [r [~I] form corresponding wavelet bases on the d-torus. The ease of this construction is exploited in many papers. Again, the case d = 1, the circle, deserves special attention. Suppose C is any smooth closed curve (without self-intersection) in IR2. Then C can be written as a parametric image C = ~([0, 1]) of a smooth 1-periodic mapping ~. Thus combining periodization with the isoparametric approach from Section 3.3 immediately provides wavelet bases on C giving rise to analogous norm equivalences and moment conditions. These wavelets can be used to discretize, for instance, boundary integral equations of the type mentioned in Section 2, arising from exterior boundary value problems for planar domains with smooth boundaries. When the curve is not smooth but has corners, it may have to be subdivided into smooth sections and wavelet bases can be obtained by piecing together parametric images of wavelets on the interval. This gives stable bases for L2. However, for the characterization of smoothness spaces, this is not sufficient. Here the transition between adjacent segments requires special care. We will briefly indicate a systematic approach to this problem below in the context of a more general situation. Note that example (2.10) requires wavelets defined on two-dimensional closed surfaces in IR3. In such a case periodization does not help. Instead one can use the tools developed in computer aided geometric design where such surfaces are modeled as a union of parametric patches. Thus assume that F is a piecewise smooth d-dimensional manifold of the form M
F-
[..Jf'i,
FiNFt-0,
i~l,
(3.4.2)
i--1
where Fi = tci(O) and tci are regular sufficiently smooth parametrizations. Again one can consider function spaces 9V(F) where 9V(F) = H~(F) or F ( F ) - B~(Lp(F)) and the range of c~ depends on the global regularity of F. For instance, when F is at least Lipschitz it makes sense to consider Sobolev spaces with index c~ < 1. For practical purposes and for the sake of constructing wavelets on F the characterization of 9v via an atlas and a partition of unity is rather useless. An interesting alternative was offered in [9] where a characterization of $'(F) is directly based on a decomposition of F into patches Fi. The following brief indication of the basic ideas is taken from [19] where an attempt is made to make the existence statements
S. Dahlke et al.
250
from [9] constructive and where the details of the following comments are given. First one orders the patches Fi in a certain fashion. If Fi n Fl "-- ei,l is a common face and i < l, then ei,l is called an outflow (inflow) face for Fi (F1). 0F~, 0F~ are called the outflow and inflow boundaries of the patch Fi. Let F~ denote an extension of Fi in F which contains the outflow boundary OFt in its relative interior, and whose boundary contains the inflow boundary 0F~ of Fi. Now suppose that Ei is an extension operator from the domain Fi to F~ such that m
IIE~flI~(F,) ~< IlfllJ=(F,), II(E~'f)l"ll:=(r,), <~ Ilfll~=(r,*), where fl" ( x ) " -
f(x), O,
m
(3.4.3)
xCFi, x e F~ \ Fi,
and 9~(Fi) 1" "- {f e 9~(Fi) 9fl" e ~(F~)} consists of those elements in the local space ~(Fi) whose trace vanishes on the outflow boundary 0F~. Such extensions can be constructed explicitly as tensor products of Hestenes-type extensions [9, 19]. Then, denoting by Xa the characteristic function of Ft and defining 7)1f
"-- E1
(Xr~f),
Pif "- Ei(xr,(f - E l
TPlf))' i- 2,...,N,
one can prove that the mapping
T" f ~ {Xr~Pif} i--1 N
(3.4.4)
defines a topological isomorphism acting from ~'(F) onto the product space IIiN 1 ~(Fi) * ~ where the spaces 9t-(Fi) * are defined analogously to ~-(Fi) * Since, in view of (3.4.3), an analogous statement holds for the mapping
R" f ~ {XriT);f} N i--1 which takes $'(F) onto the product space IIiN1 fi'(Fi) 1", T is also isomorphic for the dual ~-*(F), i.e., N
Ilfll~(F) x ~ IIV~fllY(F,)*,
(3.4.5)
i--1
and likewise for ~(F), 5r(Fi) $ replaced by the duals ~'*(F), 9t-*(Fi) t, respectively.
Nonlinear Approximation and Adaptive Techniques
251
Recall that the component spaces Y(Fi) $ are Sobolev or Besov spaces with certain boundary conditions (while their duals satisfy complementary boundary conditions) which can be viewed as liftings of analogous spaces defined on the unit cube [] as described in Subsection 3.3. Biorthogonal wavelet bases for these spaces, in turn, can be constructed via tensor products of suitable wavelet bases on the interval [0, 1] satisfying certain boundary conditions [19]. Lifting such bases for each patch Fi and then applying T -1 produces biorthogonal wavelet bases on F which, due to (3.4.5), gives again rise to norm equivalences of the type (3.2.3), but this time for F. Note that the latter step is only done for the sake of analysis. In practical calculations one would avoid executing T -1 but rather transfer all the computations to the component spaces and thereby to functions defined on the unit cube. Likewise moment conditions are formulated as in (3.3.3) ultimately on D. For corresponding consequences with regard to domain decomposition, see again [19].
3.5
Lipschitz domains
The above techniques are of limited use when dealing with bounded domains in IRd with complicated boundaries. We shall always assume that is an open and connected Lipschitz domain. This covers all domains of practical interest. In the following we shall briefly recall the results from [10]. One can also realize the Sobolev and Besov spaces by extension operatots. The conditions we assume on ~t guarantee that there is an extension operator E which simultaneously extends Sobolev and Besov spaces. For example, in the cases of interest to us, if r is any positive real number, there is an extension operator E = Er such that
E" Ha(Q)--+
Ha(]Rd),
O < a <_ r O<
E fI~ - f,
f C L2(f't),
(3.5.1)
]]EfliT(R~) ~ IIfi]y-(~),
whereY-H ~ory-B ~. In principle, the extension E can be used to generate a wavelet basis for 12 from a wavelet basis on IRd. To this end, suppose that ~ - {r ' I E De} U {r " ~ C ~ o , i E D+}, and analogously 9 are biorthogonal wavelet bases for L2(IRd). Given f E L2(~) it follows that E f has a wavelet expansion (3.1.14). Let P0 be the projector defined in (3.1.9). Then, for each f E Le (~t), we have on Ft,
s - P0(ef)+ Z lED+
Z (Ef, CE~
o
(3.5.2)
S. Dahlke et al.
252
with 7) + the set of dyadic cubes with measure _ 1. The sum in (3.5.2) can be restricted to those r whose support nontrivially intersects ~. The function f is in Ha(~t) (respectively Ba(~t)) if and only if for the series in (3.5.2) the expressions (3.1.17) (respectively (3.1.22)) are finite. Denoting again by E* the adjoint of E this can be interpreted as
Ilfil~(a)
x
~
I(f,E*~i)ai 2
(3.5.3)
IET:)c
+
~
~_,[I[2"/dl(f,E*@I)ai 2
f e H~(a),
lED+ CE~ ~
where (f,g)a := f f(x)g(z)dx.
This requires evaluating E*, which is in
general not feasible numerically. Moreover, the above procedure does not necessarily preserve biorthogonality. Therefore this approach is useful as an analytical tool, but not for practical purposes. For somewhat more specialized domains, it is possible to develop an extension strategy which is more numerically accessible. Indeed, we can utilize multiresolution analysis to construct E and E*. For this, we shall use the approach described in [10]. The class of suitable domains is specified there. Roughly speaking, these domains are coordinatewise Lipschitz. Given such a domain and a dual pair r r as in (3.1.4), biorthogonal collections ~k -- {r " m C Ak}, ~k -- {r " m E Ak} on ~t were constructed where the Ck,m are adapted to the boundary so as to ensure polynomial exactness while the Ck,m involve only translates r k. --m) which are fully supported in ~. Now, we define
Pkf "-
~ (f,r mEA~
It was shown in [10] that for 0 _< ~ < 3' o~
Ilfll 2H~(n)
x
llPkofllL~.(a) + k=ko+l
22~
-
Pk-1)fll L2(~)'
(3.5.4)
where again 3' - sup {s > 0 9r E HS(]Rd)}. Each Ck,m is either of the form 2dk/2r or is a linear combination of such functions restricted to f~. Therefore the functions Ck,m possess a canonical extension r to IRd. Note that since the Ck,m are supported in f~, the collections (I)e, ~)k are still biorthogonal. Thus, the operators
P~f'- ~ (f,qSk,m>f~r mEAk
Nonlinear Approzirnation and Adaptive Techniques
253
take L2 (ft) into L2 (IRd) and its adjoint
(Pf~)* f "- E
(f ' Cek,m)~pk'm
rnEAk
takes L2(IR d) into L2(f~). Moreover, the mapping OO
E f " - p eko s "Jr Z
e - P ke) f (P/~+I
(3.5 . 5)
k=ko is an extension satisfying (3.5.1) for any r < 7. Due to biorthogonality, evaluating E*(bi requires computing the inner products (~I, q5ek,m)rte for levels k larger than the level of I. In numerical implementations, by using decay properties, this can in turn be restricted to finitely many levels depending on the required accuracy. In view of the above comments, we shall assume in the following that we always have a pair of biorthogonal bases 9 - {@I "~) C ff~, I C De U D + } and ~ - { ~ I " ~ E ~ , I C Dc tO D +} where Dc corresponds to functions on the coarsest level, while for I E 79+ the ~t, ~ I play the role of wavelets. Again the sets ~} will generally depend on I but will always contain at most finitely many functions. Setting as above D := 79c tO79+, on one hand moment conditions of the form (3.2.5) or (3.3.3) hold, while on the other hand relations like
IlSll r (r,(a))
• ~ ~ III-q(~/e+l/2-~/V)l<S,~I>l q I ED ~bEq~
(3.5.6)
and
Ilfll (n)
IIl- /al(f,6,)t
c~ C (-5', 7),
(3.5.7)
are valid when ft is a domain or manifold of dimension d as discussed above. As in all the above examples, it will be convenient to identify always the indices I with dyadic cubes of volume III. So far, we have outlined several principles to construct wavelets on various types of domains and manifolds. If one wants to employ such wavelet bases for solving an operator equation, the issue of boundary conditions is, of course, important. When dealing with boundary integral equations on a closed manifold this problem does not arise. It may also not be severe in connection with natural boundary conditions for elliptic problems on bounded domains. Appending essential boundary conditions is, in principle, a possibility to avoid incorporating boundary conditions in the trial spaces and to preserve possibly
254
S. Dahlke et al.
many favorable properties of wavelets defined on simple domains [41]. For domains which can be represented as a union of parametric images of a cube, the approach outlined above also facilitates incorporating essential boundary conditions in the wavelet spaces. We dispense here with elaborating further on this issue and refer to [19, 36] for details of corresponding recent progress in this problem.
3.6
Wavelet discretization of operator equations
We return now to the operator equation (2.1) where in the following H = H t and H* = H - t , where either H t = H t (gt) when f~ is a closed surface, or when natural boundary conditions are assumed, or H t is a closed subspace of H t (Ft) determined by boundary conditions so that A is injective on H t. In fact, we will assume that (2.2) holds with H = H t. The standard Galerkin method for approximating the solution u of (2.1) begins with a finite dimensional space S c H t and finds the function u s E S such that ( A u s , s) = ( f , s), s e S. (3.6.1) By choosing a basis {sk } for S, (3.6.1) becomes a system of linear equations (a(si, s j ) ) i , j c = f,
(3.6.2)
with f := (fi) and fi := (f, si), c the vector of coefficients of u s with respect to this basis and the matrix (a(si, sj))i,j the stiffness matrix. In the sections that follow, we shall be interested in the efficiency, in which u s approximates the exact solution u of (2.1). The typical choices for S in the standard finite element theory are spaces of piecewise polynomials on some partition associated to f~. An analogous choice in the context of wavelets are spaces S = S j : - span{r : r C ~ , III < 2-Jd}, or more generally S = SA := span {r : (r I) C A}, where A is some finite subset of V := { ( r r E ~ , I E :De U :D+}. The efficiency of Galerkin methods depends on: (i) the approximation power of the spaces S, (ii) properties of the stiffness matrix (condition number and sparsity). We shall see in the following sections how the accuracy of the approximation of u s to u depends on the regularity of u. The properties of the stiffness matrix, including its amenability to preconditioning, is a central theme in finite element methods amply reported on e.g. in [16, 18, 48, 49]. Wavelet discretizations offer the following advantages with regard to (ii). To describe this, for A C V as above, let PAY "-
(Y, (r
(3.6.3)
Nonlinear Approximation and Adaptive Techniques
255
Note that under the assumption (2.2), which we will quantify as
c'~llAvll.-, < Ilvll., <__411Av[l.-,,
v e H t,
(3.6.4)
and the self-adjointness of A, the Galerkin schemes are stable. In this case, this means in terms of the projectors PA that
IIPXAvllH-, ~ llvllw,
v e &.
(3.6.5)
A first important relation between the wavelet bases and A is It[ < 'r, ft.
(3.6.6)
To explain its relevance for preconditioning, consider the general version of the scale shift (3.1.18)
z,f
.-
~_~
IZl~/d(f,~z)r
(3.6.7)
(r so that by (3.5.7)
(3.6.8) Thus considering w := Z_tv for v C SA, (3.6.8) and (3.6.5) yield, under the assumption (3.6.6),
ilwllL= ~ [tvllH, x IIP~AvlIH-, x III~PXAPA~tWIIL~,
(3.6.9)
where we have used that I S 1 "-- ~--S,
since ~, ~ are biorthogonal. Clearly (3.6.9) means that the operator
13A := Z~ P~ APhZt satisfies cond2(BA) "--IIBAIle= IfBillle= - o(1),
# A -+ oc,
(3.6.10)
where I1" Ite2 denotes the spectral norm. It is not hard to verify that the matrix representation of/3A is given by AA - (A((I, r
(J, r/)))(i,r
A(I, J) - Illt/d(A~j, r
t/d. (3.6.11)
Thus, a suitable diagonal scaling applied to the stiffness matrix relative to the wavelet basis results in a matrix with uniformly bounded condition numbers [18].
S. Dahlke et al.
256
This suggests that we reformulate the equation (2.1) as an infinite discrete system by representing u and S with respect to the primal and dual wavelet basis, respectively, i.e.,
(r
(r
In view of (3.6.10) and (3.6.11), it is useful to introduce the following rescaling. Let f "- (]r with ]r := [I[t/d(s,r let fi := (fie,i) with ~r "- ]II-t/d(u, (hi), and let ,4 be the infinite matrix with entries A((I, r (J, 7)) "- IIlt/da(r ~TJ)[J]t/d. Then, (2.1) becomes A f t - f.
(3.6.12)
Clearly AA is a finite submatrix of ,4, and A is boundedly invertible on Another important advantage of the wavelet basis is that the matrix A is almost diagonal in the sense that
A((r
I),
-
J)) <
1 + m x- Yi:lJl) /
min(
Igl)
' III
'
(3.6.13)
where ~I is a point in the support of r and where r > d/2 depends on the regularity of the wavelets. The parameter N again denotes the number of vanishing moments of the wavelets Cx, I E 7:)+. Thus in the present biorthogonal framework, r and fi/can be made a large as one wishes by proper choice of the wavelet basis. We shall make a distinction in what follows between the cases of linear methods in which the space S is chosen independent of u and nonlinear or adaptive methods in which S depends on u and previous choices for S. We wish in particular to understand what, if any, are the advantages of adaptive methods. w
Linear
approximation
For simplicity we confine the following discussion to the case that gt C ~d is a bounded and connected Lipschitz domain. We have noted above that in standard finite element theory one seeks an approximation to the solution u of (2.1) from a finite dimensional linear space S. It is well understood in approximation theory that for standard spaces S, consisting for example of polynomials, splines, or wavelets, the efficiency of the approximation to u by elements of S is related to the regularity of u in the scale of Sobolev spaces H a . To briefly recall this theory, we shall restrict ourselves to the case where S is chosen from a sequence (Sn) of spaces of the following
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two general types: (i) S~ is a linear space of piecewise polynomials on a partition related to Ft, (ii) Sn is a subspace of a multiresolution space Vj with the Vj as described in Section 3. We assume that Sn has dimension nd. In case (i), this corresponds to partitions with cell size h ,-~ n -1 and in case (ii) this corresponds to taking S~ = Vm, n = 2 TM, or a subspace of Vm reflecting boundary conditions. We let En(f)
"- E(f, Sn) "-
inf [If - Sl]L2(~ ).
sESn
(4.1)
The following is a generic theorem for approximation by the elements of Sn. It requires additional conditions on Sn which we shall discuss after the theorem is stated. T h e o r e m 1. For a ) 0 and a function g C L2(~), the following are equivalent: (i) g e H ~ ( ~ ) , (ii) ~ n=l [n~En(g)] 21n is finite. The sum in (ii) is equivalent to the semi-norm for H ~ ( ~ ) . Moreover, a simiIar resuIts holds for approximation in H t (~), provided En (g) is replaced by the error in approximation by functions from Sn in the norm of H t, and H a is replaced by H ~+t in (ii).
R e m a r k 1. The condition (i) is slightly stronger than requiring En(g) = (,0(n-a). The class of functions which satisfy the latter condition is precisely the Besov space B ~ (L2 (~)) in the case of approximation in L2 (~t). A similar result holds for approximation in H t ( ~ ) . We are purposefully not being precise about the conditions needed on the spaces Sn so that Theorem 1 is valid. In the case that Sn is a wavelet space Vm with n = 2 TM, there is a real number r such that Theorem 1 holds for 0 < a < r. The number r is related to the smoothness and polynomial exactness of the wavelet basis. It is the same number r such that the Sobolev spaces H a are characterized by the wavelet coefficients as in (3.1.17) for 0 < a < r. To see precise conditions under which Theorem 1 holds for wavelets, we refer the reader to any of the standard treatments on wavelets such as Meyer [47], DeVore and Lucier [23], or Frazier and Jawerth [29]. In the case of spline approximation, necessary and sufficient conditions for the validity of Theorem 1 can be quite subtle (see e.g. Jia [34]). We refer the reader to one of the standard texts on finite elements, e.g. Oswald [48]. We should also mention that if boundary conditions are to be incorporated in H a then these boundary conditions must be incorporated into Sn, and this must be incorporated into the analysis.
S. Dahlke et al.
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We can use Theorem 1 to infer the potential accuracy of numerical methods for solving (2.1) based on Galerkin solutions. Since the numerical solution us,, comes from the space Sn, it will provide efficiency of approximation of order O(n -~) in the L2 norm in the sense of (i) of Theorem 1 only if the solution u has smoothness of order a in the scale of Sobolev spaces. Thus the maximum regularity of u in this scale determines the maximum efficiency a linear numerical scheme can have. We shall see in the next section that using nonlinear methods changes the scale of smoothness spaces in the generic theorem.
w
Nonlinear wavelet approximation in L2(~t)
We are ultimately interested in adaptive methods for solving (2.1). Adaptive methods are a form of nonlinear approximation. For the purposes of orientation, it will be useful to consider the following simpler (but related) form of nonlinear wavelet approximation called n-term approximation. Recall that V - {(r I) " r E ~ , I C De U 7:)+}. Let En denote the set of all functions S ~ c~,r (r
where A C V and # A _ n. Then E~ is a nonlinear space. We let
fin(f)
9-
inf IIf-SlIL=(~) SEEn
(5.1)
be the error in approximating f by the elements of En. We are interested in characterizing the functions f C L2(~) for which fin(f) tends to zero like O(n -a/d) for some a > 0. It is not difficult (see [24]) to prove the following theorem.
Theorem 2. For each 0 ~ a < r, the following two statements are equivalent: (i) f C B a, (ii) ~ n--1 [/ta /d fin (f)]T n1 < CX:). Condition (ii) in Theorem 2 is the analogue of (ii) of Theorem 1. Notice that in Theorem 1 the dimension of Sn is O(n d) but in Theorem 2, En is of dimension n. This explains the difference in the form of (ii) in these two theorems. Thus both theorems talk about the same approximation rate in terms of space dimension. As noted before, (ii) of Theorem 2 is close to =
We shall now make several remarks which will bring out the differences between Theorem 1 for linear approximation and Theorem 2 for nonlinear approximation. Both theorems characterize functions with a prescribed
Nonlinear Approximation and Adaptive Techniques
259
accuracy of approximation by smoothness conditions. But these smoothness conditions are of a quite different nature. In Theorem 1 the function g should be in H a and thus have c~ orders of smoothness in L2. In contrast, Theorem 2 requires (for the same approximation rate) only t h a t g E B ~. Recall that B ~ measures smoothness of order c~ in a space LT, with ~- := (c~/d + 1/2) -1. Since 7- is generally much smaller than 2, this is a much weaker smoothness condition. Another view of the spaces B ~ comes from the Sobolev embedding theorem. These are in some sense the smallest spaces of smoothness c~ which are embedded in L2; for example, for # < 7-, B ~ ( L , ) is not embedded in L2. The spaces B ~ contain functions which are very unsmooth in the classical sense. For example when d = 1, any piecewise analytic function is contained in all of the spaces B ~ but only in H a if c~ < 2. Since the wavelet basis is a Riesz basis, the following simple algorithm asymptotically realizes the best n-term approximation. R e m a r k 2. We take A to be a set of n pairs (I, r largest. Then,
Sn -" PAf --
E
for which I<.f, 't/~I}l
is
(f ' ~I)~)I
(I,e)6a
is in E~. and Ilf - S~]IL2(~) ~ cry(f), with constants depending only on the constants in (3.1.16). It follows that that (iii) The property (ND
-
-
n=l
--<~00 7~
is equivalent to (i) and (ii) in Theorem 2. Note t h a t this algorithm requires knowledge of all of the wavelet coefficients of f. The n-term approximation is not directly applicable to numerical methods for operator equations since the wavelet coefficients of the solution u are not available to us. Instead, one constructs nonlinear approximations to the solution u using adaptive algorithms. An adaptive wavelet method for approximating the solution u of (2.1) would select the wavelet terms to be retained in the approximation from prior computations combined with any additional information that may be available. We discuss specific adaptive methods later in this paper. For the present, we want only to draw out the distinction in the form of the approximation between these adaptive methods and the n-term approximation just described.
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We shall discuss adaptive wavelet approximation based on the wavelet decomposition (3.1.15). It is notationally convenient to combine all wavelet terms that correspond to a fixed dyadic cube. Therefore, for I E De U D +, we define
:=
(S,
(5.2)
CE~ ~
and CE~ ~
If we assume to work with wavelets adapted in some way to the underlying domain the sets Dc, D + of dyadic cubes is to be understood in the above sense. In what follows we could likewise employ wavelets defined on all of ]R d when ~ is a bounded domain in ]Rd. In this case one can always assume that there is a fixed dyadic cube Q such that supp r gl~ - 0 unless I C_ Q. For notational convenience, we assume that Q - [0,1] d. We shall adopt the following notational conventions. We let De U D + denote the set of dyadic cubes I C_ Q. If I c De U D +, and supp r fl ~ - 0, then define fI - 0 and c i ( f ) - O. In this way, we can always consider the wavelet decomposition in (3.1.15) to be indexed on all the cubes of D "- De U D +. An adaptive procedure usually creates approximations of the form
Po(f) + ~
fI,
(5.4)
IEJ
where J is a set of dyadic cubes which have a certain tree structure. If I E D, we say that J is a child of I (and I is a parent of J), iff IJI - 2-dlII and J C I. We denote the set of all children of I by C(I). We say J is a sibling of I if they are both children of the same parent. Trees typically arise in an adaptive algorithm where at each inductive stage cubes are refined by adding all their children to the tree. The trees ff C D that arise in adaptive algorithms have the following two properties: P 1 . If I E ff with II! < 1, then its parent is in ft. P 2 . If I C J , then all of its siblings are in J . We call a set ff C D+ which satisfies PI~ P 2 an admissible tree. For an admissible tree J , we let ~ ' ( J ) C J denote the set of final leaves of if, i.e., the set of those 'I E ff such that I has no children in ft. It is of interest to know what is the overhead (in efficiency when compared to n - t e r m approximation) in forcing such a tree structure in the approximant. In [10] a simple adaptive wavelet algorithm was given that shows that this cost is quite minimal. To describe this algorithm, we define for any ff C D, PSI'-
~fI, IEJ
Nonlinear Approximation and Adaptive Techniques
261
where f I is defined by (5.2). It follows that
[If-
P J fl[ 2L2(n) ~
~
[ci(f)l 2
(5.5)
IED\J
For a dyadic cube I E 7), we let T ( I ) be the tower of I which is the collection of all J C 7) such that J C I, J ~: I. We let
E
1
JeT(I)
Algorithm 1. Fix e > 0 and choose an initial admissible tree J0. Set Bs = { J E J:(Jo) : R ( J ) > e}, Js = J0. If B6 = 0 stop. Otherwise, for I E B~ do: 9 replace J~ by J~ UC(I). 9 replace Bs by (B~ \ {I})U {J e C(I) : R ( J ) > e}. Since [ ] P j ~ f - fl[L2(n) ~ 0, n --+ oc, Jn -- {I E 7;)'11 [ < 2 -nd} the above algorithm terminates for every r :> 0 after finitely many refinement steps, i.e., eventually one obtains BE = q), and the resulting tree J~ has the property that R ( J ) <_ e for J e 5r(J~). (5.7) The following theorem from [10] estimates the approximation error of the adaptive algorithm. Theorem 3. Let a > 0 and T := ( a i d + 1/2) -1 be as in Theorem 2. If g e B ~ ( L ~ ( ~ ) ) for any/~ > a and # > 7, then
IIg- PJ.flIL ( ) <
(5.s)
with a constant depending only on d and a.
When compared to Theorem 2, this theorem shows that with only a slightly stronger assumption on g, we obtain the same approximation order as in n-term approximation. There is an analysis, similar to the above, for adaptive approximation based on piecewise polynomials. In [26], this was carried out for adaptive algorithms which use partitions into cubes. It should be possible to carry over the arguments in [26] to more general adaptive partitions, for example, to triangulations, provided the refining triangulations are always done in the same manner and lead to shape preserving triangulations.
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262
The above analysis shows that it is the regularity of the solution u in the Besov scale B ~ which determines its approximability by nonlinear methods. Adaptive methods therefore should be evaluated against the optimal value that is theoretically possible. In this context the Besov spaces B ~ replace the role of the Sobolev spaces H a when analyzing adaptive numerical methods. w
R e g u l a r i t y of s o l u t i o n s to P D E ' s and a p p r o x i m a t i o n order
In the preceeding sections, we have already seen that the maximal possible efficiency that a numerical method to recover the solution of (2.1) can have is determined by the regularity of the exact solution of (2.1) in specific smoothness spaces. It was emphasized that the approximation order of linear methods is related to the Sobolev scale Ha(f~) (compare with Theorem 1), whereas the efficiency of nonlinear and adaptive methods is determined by the Besov scale B ~ (compare with Theorem 2 and Theorem 3). Therefore, in this section, we shall give a short survey of the regularity theorems for partial differential equations for both kinds of smoothness spaces. Let L be an elliptic differential operator of order 2m on a bounded and connected Lipschitz domain f~, L-
E
E(
--
1)lllDl a k , l
(x)Dk
,
ak,l e L~(a).
(6.1)
Ikl<_mlll<_m
For simplicity, we shall restrict ourselves to Dirichlet boundary conditions, i.e., we consider the problem" find u E H g ( ~ ) such that
a(u,v)-
E
I ak,l(x)(Dku)(Dlv) d x - - / f(x)v(x) dx
Ikl,lZl-<mh
(6.2)
n
holds for all v E H~(f~). Let us start with the usual Sobolev scale H ~. We want to investigate how the regularity of the solution u of (6.2) depends on the coefficients ak,l, the right-hand side f and on the shape of the domain f~. Most of the time, this question is formulated in the following form. Suppose that f is contained in H~-m(f~) for some a _> 0. What are the conditions which imply that the variational solution u E H~(ft) is in fact in Hm+~(f~), and satisfies
[[ttllHm+" ~ [If[lH~,-m -!-[lullHm .9
(6.3)
A boundary value problem with these properties is called a-regular. A first answer is that a-regularity holds if the coefficients and the domain are sufficiently smooth.
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Nonlinear Approximation and Adaptive Techniques
Theorem 4. Let f~ E C s+m for some s >_ O. Let a >_ 0 satisfy a + 1/2 r { 1 , 2 , . . . , m } ; 0 <__a <_ s, i f s E IN; 0 <_ a < s, i f s r For the coemcients let the following hold: D n ak,l E Lcc(fl) for all k,1, n with Inl <_ max(0, s + I l l - m)
if s E IN,
ak,l E cs+ttl-m(~) for Ill - m, ak,1 E L~(f~) otherwise, if s r IN. Then the solution u E H~(gt) of (6.2) with f E H-m+~(f~) belongs to Hm+a(fl) N H~n(f~) and satisfies
Ilull. +o < IISII-o-
+ II ll .
Results of this form were obtained for example by Lions and Magenes [42], see also Hackbusch [32]. Theorem 4 implies that for problems satisfying the conditions of that theorem, linear methods are sufficient in the sense that they can provide a convergence rate of order O(n -(m+a)) (compare with Theorem 1 and Remark 1). Such a result does not hold for nonsmooth domains, e.g. for domains with edges and corners, even if the coefficients are arbitrarily smooth. In this case, the regularity is only preserved strictly in the interior [32]. Theorem 5. Let f~ be a Lipschitz domain and let s C f~l C f~ and a >_ O. Let us assume that the coetticients satisfy the conditions of Theorem 4 with f~ replaced by f~l and with s >_ a where a E IN, s > a where a ~ IN. Suppose that the restriction f~l belongs to H-m+~(f~l). Then the restriction of u to f~o belongs to Hm+~(f~o) and satisfies tlul]Hm+~(~o) ~ IlfllH--m+~(~l) + II lIH ( ). In general the smoothness of the solution u will decrease significantly as one approaches the boundary. Therefore, the estimation of the Sobolev regularity on nonsmooth domains is a delicate task. Roughly speaking, the results can be divided into three types" results on specific operators and specific domains, see e.g. Grisvard [30, 31], results on specific operators and general domains, see e.g. Jerison and Kenig [35], and results on general operators on general domains, see e.g. Dauge [21], Kondrat'ev [37, 38, 39, 40] and Maz'ja and Plamenevskii [43, 44, 45, 46]. Among other things, Grisvard has intensively studied the Poisson equation, i.e., L - - A on polyhedral domains in IRe and IR3, respectively, primarily with f E L2. Let us first describe a typical result in IRe. Let ~ be a polygonal domain with vertices ~j, j - 1, 2, ..., and let wj denote the measure of the interior angle at ~j. We introduce polar coordihates rj, 9j in the vicinity of each vertex ~j. Furthermore, let ~j denote a suitable truncation function which depends only on the distance rj to ~j. Then the following holds.
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264
T h e o r e m 6. Assume that f~ is a bounded polygonal open subset of ]R2. For f E L2(f~) let u denote the unique solution of
V u . Vv d x - / f v dx for every v E H~ (~). Then there exist unique numbers cj such that u- ~
cjOj(rj')rj'/~J sin(TrOj/wj) E H 2(Ft).
(6.4)
w j >Tr
The explicit expression (6.4) enables one to determine exactly the Sobolev regularity of the solution u. Since only the nonconvex corners contribute, we obtain that u E H 2 for convex polygonal domains. In the worst case, u will only be contained in H 3/2+~ for some e > 0 which can be arbitrary small. This result implies that for general polygonal domains, linear methods can only provide an approximation of order (.9(n-3/2). A similar result holds for polyhedral domains in ]R3. However, the treatment of this case is much more complicated since one has to deal with different types of singularities according to edges and vertices. The regularity along edges can in principle be reduced to a two-dimensional problem with parameters, see again [30] for details. To keep the technical difficulties at a reasonable level, we shall not discuss this case here and confine ourselves to a typical result concerning the behavior of the solution at one particular vertex. We need some further notation. Let Fi, Fl be two faces of ~ and let ei,z denote the edge between Fi and Fl whenever Fi and Fl intersect. The measure of the interior angle of the edge ei,l is denoted by wi,t. We set m
T "- inf{m~/wi,t E]0, 1[; F, i'IFl ~' 0, m _> 1}. For convenience, we translate the typical vertex to zero. Thus, in a neighborhood V of 0, ~ coincides with a cone C whose intersection with the unit sphere S 2 is denoted by G. Thus G is an open subset of the unit sphere whose boundary is the union of a finite number of arcs of great circles. We introduce spherical coordinates Q,a and denote by A' the LaplaceBeltrami operator on S 2. It can be shown that the spectrum of A' is an infinite sequence of real numbers -~l, l = 1 , 2 , . . . where ~l >__ 0, with no limit points. We denote by vl, 1 = 1, 2 , . . . the orthonormalized sequence of related eigenfunctions, i.e., - A%
-
r
The following theorem was shown in [30].
Nonlinear Approximation and Adaptive Techniques
265
T h e o r e m 7. Let f~ be a bounded polyhedral open subset of ]R 3. For f E L2(f~) let u denote the solution of
f V u . Vv d x -
f f v dx
f2
f2
for all v C Hlo (f~). Then there exist unique numbers ct such that U -- Z
C1Q-1/2+~/((t+l/4)Vi(O')
C Ha(V)
(6.5)
1
for every c~ <_ 2 such that c~ < T + 1, where the sum is over the 1 such that
(t
(c).- {vl
Z
Vq(a)log q 0, Vq C H ~ ( G ) } .
(6.6)
O<_q<_Q
We say that Lx is injective modulo polynomials on S ; ( C ) if whenever v C Sr is such that Lxv is polynomial, v is a polynomial itself. For regular conical domains, the following theorem holds.
S. Dahlke et al.
266 T h e o r e m 9. Let a >_ O,a q[ { 1 / 2 , . . . , m and only if for all ~ E C satisfying
1/2}.
Then
(6.2) is a-regular if
E [m - d/2, a + m - d/2],
~
(6.7)
Lx is injective modulo polynomials on 8r (C). There exist also a lot of regularity theorems for Besov and non-Hilbertian Sobolev spaces. They are concerned with questions of the form: Given f C Bp-m(Lp(f't)), what are the conditions that imply that u E Bp+m(Lp(Ft))? Consequently, these theorems provide us with information concerning the approximation order of linear methods as measured in L v. We shall state two typical results for the spaces W~(Lv(Ft)) and B~(Lv(Ft)) , respectively. For instance, Theorem 6 has the following extension to non-Hilbertian Sobolev spaces. T h e o r e m 10. Let f~ be a bounded polygonal open subset of IR2. For each f C Lv(~ ), 1 < p < co, there exists a unique solution u of
f Vu. Vv dx-
/fv
f~
f~
dx
for every v E H~ (fl). In addition there exist numbers cj such that IZ m
E
cjtgj ( r j ") r j '~/'J sin(TrOj/wj) E W2(Lp(fl))
provided that none of the numbers 7r/wj is equal to 2 - 2/p. For the case of Lipschitz domains, the most general results were again obtained by Jerison and Kenig [35]. We shall restrict ourselves here to the case d >_ 3; a similar result holds for d - 2. T h e o r e m 11. exists e, 0 < such that for W a + l (Lp(f~))
Let fl be a bounded Lipschitz domain in ]Rd, d _> 3. There e _< 1, depending only on the Lipschitz constant of fl, every f E W a-X(Lp(~)) there is a unique solution u E to -Au-
u
-
provided one of the following holds:
1 - - 1 < o l < lp (a) Po < P < P'o and -~
/
on
~,
0
on
Of~,
(6.8)
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Nonlinear Approximation and Adaptive Techniques
(b) 1 < p < po and 3p
-- 2--s
<
O~ <
(c) p~) <_ p < oo and k, - - l < c t < where 1/po estimate
1/2 + r
p1
3--1+c -~
and 1/p~o -
1/2- r
Moreover, we have the
for a11 f e W a - l ( L p ( ~ ) ) .
So far, we have seen that linear methods are only suitable for smooth and convex domains, even if the coefficients are arbitrarily smooth. Therefore, the hope is to gain efficiency by employing an adaptive numerical scheme. According to Theorem 2, the use on nonlinear methods is justified if the weak solution u is contained in the scale B ~, 0 __ a < a*, where the maximal index a* is significantly higher than the one for the usual Sobolev scale H a. Therefore, the first step of a systematic study of adaptive schemes should consist of the derivation of regularity theorems for u with respect to B ~. It seems that this kind of study is still in its infancy. 1 First regularity theorems for the specific scale B~(L,(f~)), 71 - (~c~ + 7) were given for certain model problems by Dahlke and DeVore [15]. We give the following example for the Poisson equation taken from [13]. T h e o r e m 12. Let ~ be a bounded Lipschitz domain in R d. Let u denote the solution of (6.8) with f C B~-I(L2(~)) for some a > - 1 / 2 . Then the following holds:
uCB~_(L,-(~)),
l_(s
r
a +2
1)
'
0<s<min
{
3d
2(d-l)
'a +1
}
.
We observe that for a large range of the parameter c~ we have a jump of two for the smoothness of the solution in the special scale B~ (L~ (~)), iT = (a8 + 89 For instance, for d - 2, we obtain the condition a < 2, whereas for the usual scale H a = B ~ ( L 2 ( ~ ) ) we have the jump of two only for a < 1/2; compare with Theorem 11. Therefore, the maximal index for the spaces B ~ is in general much larger than the one for H a. Consequently, Theorem 12 can be interpreted as a justification for adaptive and nonlinear methods. Indeed, this theorem implies that adaptive methods on Lipschitz domains can perform as well as linear methods on smooth domains, provided that the right-hand side f is contained in a suitable smoothness space.
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268 w
A n a d a p t i v e s c h e m e for elliptic e q u a t i o n s
In this section, we discuss possible connections between the above concept of nonlinear approximation and certain adaptive schemes for elliptic problems of the type considered above. We adhere to the assumptions made in Sections 2 and 3. While nonlinear approximation uses idealized information about the approximand, any adaptive solver has to contend itself with information acquired during the computation, combined perhaps with some information about the given data. In fact, the basic idea of adaptive schemes is to refine step by step the discretization only at those places where the behaviour of the searched object requires a higher resolution, so that the error is more or less balanced throughout the domain. In the presence of singularities this results in highly nonuniform meshes. In the present context we do not have to think in terms of refined meshes but rather in terms of refined spaces. By this we mean the following. How can one find possibly few further wavelets which, when added to the current trial space, guarantee a prescribed decay of the error of the corresponding Galerkin approximation? One can therefore perhaps not expect to obtain equally strong theoretically founded results in this latter context. Nevertheless, an issue of central importance will be to interrelate both concepts.
7.1
Prelimary remarks
In the context of elliptic problems it is natural to measure errors in the energy norm ]]. ]] defined in (2.4) or equivalently in I]" IIH~, while the concept of nonlinear approximation has so far been formulated for the L2-norm ll" ILL2. It is easy though to carry this over to measuring errors in Sobolev norms. This will be indicated first. To this end, it will be convenient to economize our notation a little further by subsuming all information on a wavelet in one index ~ containing its type (if applicable), location and scale. For V - { (r I) " r E ~ , I E 7:)c U 7:)+ } as before let
.-(r
-111
so that the biorthogonal bases are briefly denoted by 9 - {r - { ~ ' )~ E V} and Is takes the form
9A E V},
)~EV
In analogy to (5.1) let
an,t(g) "- inf { Iig - E )~EA
d~r
" d~ E IR, ~ E A C V, # A -
n! .
Nonlinear Approximation and Adaptive Techniques
269
We have the following analog to Remark 2. R e m a r k 3. Let g C H t. We take An to be a set of n indices )~ for which li~l-t[(g, ~b~}l is largest. Then, for PA defined by (3.6.3) one has (7.1.1) Thus picking the n first largest weighted coemcients realizes asymptotically the best n-term approximation relative to the norm I1" I[H t and hence, by (2.5), also relative to the energy norm Ii " II.
Proof: Under the assumption (3.6.6), the assertion is an immediate consequence of the norm equivalence (3.5.7) which implies that (Tn,t(g) ~ (Tn,O(Z_tg)
"-- ( 7 n ( Z _ t g ) ;
(7.1.2)
compare with (3.6.8). Again the Besov regularity of a function g can be characterized in terms of its best n-term approximation relative to I1" IIH'. Proposition 1. Assume that c~ - t < "y, and for t < ct let 1
r--;-"=
c~-t
1
d
' 2"
(7.1.3)
Then one has (X)
E
(n(~-t)/dan,t(g))
T*
< oo
(7.1.4)
n=l
if and only if g C B~. (Lr*(fl)).
Proof: Combining Theorem 2 with (7.1.2) ensures that Z - t g belongs to B~j-t(L~-,(~t)) if and only if OO
So it remains to verify that Is not only shifts between Sobolev but also between Besov scales. In fact, one easily infers from (3.1.20) that the statements I - t g C B~i-t(L~,(~)), g E BT, (L~, (Ft)), and l(g, s
are equivalent.
< oo
270
S. Dahlke et al.
Proposition 1 has an interesting application to the Poisson equation (6.8). We have already discussed the efficiency of the best n-term approximation when applied to the solution of (6.8); compare with Theorem 12. However, these results were formulated with respect to approximation in L2(f~). For elliptic equations, the energy norm is slightly more natural. A combination of Proposition 1 and Theorem 12 provides us with the following result concerning approximation relative to I1" ilH1. Proposition 2. Let u denote the solution of (6.8) with f E B~-I(L2(f~)), > 1. Then O0
(n~/dan,1 (u) )r < oc
for a11 0 < s < s*/3,
(7.1.5)
n=l
where s* - min{ 2(d_1), 34 c~ + 1} and r -- (S -- 1)/d + 1/2.
Proof" Since a _> 1, the right-hand side f is contained in L2(fl). Therefore, On the other hand, Theorem 8 implies that u E H3/2(f~) - ~ -~3/2(L2(fl)). 2 we know from Theorem 12 that u E B~(L,(a)),
By interpolation and embeddings of Besov spaces, we can conclude that u is in a family of Besov spaces B~(Lq(f~)) for a certain range of parameters q and s, i.e., u E Bq(Lq(f~)) whenever ( 1 / q , s ) i s in the interior of the quadrilateral with vertices (1/2, 0), (1/2, 3/2), (s*/d + 1/2, 0), (s*/d + 1/2, s*). Therefore, to compute the range of parameters s for which u is contained in B~(L~(f~)), T = (s -- 1)/d + 1/2, we have to determine the intersection of the lines 1 1 2s* and
1 q
1 2
s-1 d
which is the point ( s * / ( 3 d ) + 1/2, s * / 3 + 1). An application of Proposition 1 with t 1 yields the result. To illustrate this result, we consider the example where d - 2. If c~ >_ 2, then s* - 3. Hence, in this case, the nonlinear method gives an H 1approximation to u of order up to n -I/d, whereas a linear method using n terms could only give n - 1 / 2 d in the worst case. In general, a priori knowledge about the Besov regularity of the solution u to (2.1) would give lower bounds for the errors produced by any adaptive method. Conversely, if we knew that a particular adaptive scheme
271
N o n l i n e a r A p p r o x i m a t i o n and Adaptive Techniques
is asymptotically as efficient as best n-term approximation in I[" IIH', its performance would allow us to infer the regularity of u. Of course, since the wavelet coefficients of the solution u are not known a priori, one cannot apply Remark 3 directly. There are several possible ways of dealing with this problem. Let d~(g) denote the sequence of wavelet coefficients of g relative to 9 , i.e., d~,~(g) - (g,r A C V, and analogously d,i,(g ). By (3.6.12) the solution u of (2.1) is determined by dq~(Z_tu) = A - l d c , ( Z t f ) .
(7.1.6)
Recall from ( 7 . 1 . 2 ) t h a t the best n-term approximation of u in I1" IIH' corresponds to the best n-term approximation of Z _ t u in the L2-norm I1" IIL~ which, by Remark 2, corresponds to selecting the n largest terms Id,~,;~(:T_._tu)l = IAI-tld,~,)~(u)l. It is known that in certain cases the decay properties of (3.6.13) of the infinite matrix ,4 imply similar decay properties for A -1, perhaps with different parameters, see e.g. [51]. In such a case the largest coefficients of Z - t u are expected to appear in a 'neighborhood' of the (accessible) largest coefficients f~ - IAItd,i,, ~ (f). The effect of the smearing caused by the application of A -1 can in principle be estimated by the same methods as used in connection with matrix compression [18]. However, this assumes that the singular behavior of u is completely governed by the righthand side f. Next we shall describe a somewhat different approach from that of [14]. To motivate this let us briefly recall first a basic strategy employed by many adaptive finite element schemes. A key observation is the equivalence between the validity of two-sided error estimates and the so-called saturation property. This issue is discussed in [5] in the context of finite element methods. The basic reasoning can be sketched as follows. Suppose that S C V C H t are two trial spaces with respective Galerkin solutions u s , u v . By orthogonality one has
l i l y - ~sll _ I1~- ~sll, where I1" II denotes again the energy norm. Moreover, one easily checks that one has II ~ - uvll < ~11'-'- usll
(7.1.7)
for some/3 < 1 if and only if
(1 - 9~)'/~11~ - ~sll < lily - ~sll.
(7.1.8)
Thus, if the refined solution u v captures a sufficiently large portion of the remainder (7.1.8), the global energy error is guaranteed to decrease by a factor/3 when passing to the refined solution u v and one has the bounds
It~v - ~ s l l _
I1~- ~ s l l _ (1 - / ~ ) - ' / ~ l l ~ v
- ~sll,
(7.1.9)
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S. Dahlke et al.
which are computable. In practice, one controls the local behavior of u v us and refines the mesh at places where (an estimate for) this difference is largest. This results in balancing the error bounds. Although this has been observed to work well in many cases, the principal problem remains that something like (7.1.8) has to be assumed to prove convergence of the overall adaptive algorithm. It is perhaps worth stressing that wavelet analysis allows us to remedy this conceptual deficiency and derive much stronger information about remainders. In fact, we shall see below that the assumption (7.1.8) about the unknown solution u can be replaced (quite in the spirit of the previous comments) by some (rather weak) information on the accessible data f. To this end, let us first relate the type of estimates (7.1.9) to n-term approximation. Instead of minimizing the error for a given allowance of n terms one can minimize the number of terms needed to meet a given error tolerance. Specifically, given any strictly decreasing sequence {~'i}ie~, we can look for a sequence {A(Ti)}ielN of index sets A(Ti) C X7 such that a#i(r,),t(U) • Ti,
i e iN.
(7.1.10)
The following observation is an immediate consequence of Remark 3. R e m a r k 4. One has I 1 ~ - PA(~,)~II • ~ ,
i e IN,
(7.1.11)
and the sets A(Ti) can be chosen to be nested, i.e., A(Ti) C A(Ti+I),
i e IN.
(7.1.12)
Let, with a slight abuse of notation, UA denote the solution of Galerkin problem (3.6.1) with S := SA := span {r
: A E A}.
If A C/~ we have II~ - ~x II~ - I I ~ - ~All ~ - I I ~ A - ~xll ~,
since the Galerkin approximation is an orthogonal projection relative to the energy inner product. Therefore, we obtain R e m a r k 5. Consider the following sequence {Ai}ie~: (i) Fix some A 1 C V and a < 1. Define T1 := IlU- UA1 II.
Nonlinear Approximation
and Adaptive Techniques
273
(ii) Given A i, choose A i+1 c V, A i C A i+1 such that II~A'
-
ZtA'+l
II _> ~llu - UA, It,
(7.1.13)
while for any A C V with A i g A and # ( A \ A ~) < # ( A i+1 \ A i) one has (7.1.14) [lUm, -- UAII < ~ l l u UA, II, Set ~ + 1 -
(iii)
(1 - ~2)1/211u - UA, II,
Replace i + 1 by i and go to (ii).
Then one has I l u - ~A~ II • ~ # A o + c , , ( ~ ) ,
n ~ IN,
(7.1.15)
where c is some constant. In practice, it will generally not be possible to realize the above strategy of capturing a significant portion of the remainder by a possibly small set of additional indices, since the exact estimation required in (7.1.13) and (7.1.14) is generally not possible. However, it will be possible to bound quantities of the form IlUA --us for A C A, from below and above by computable local quantities times constants which are independent of the sets A, A but different from one. 7.2
A posteriori error estimates
Suppose that for some A C V, SA is the current trial space and that we have computed the solution Uh of (3.6.1) (within some appropriate tolerance). According to Remark 5, the next step is to estimate the error I l u - Uhll in the energy norm in a way that indicates how to select next a bigger set A C V, A C A, of wavelet indices so that on one hand, A stays still possibly small while on the other hand, the error Ilu - u AII is guaranteed to decrease by a certain amount. As mentioned before, selecting the index sets A implicitly corresponds to creating possibly nonuniform meshes. In fact, the spaces S n - - span{r : 1~1 _< 2-n} correspond to uniformly refined meshes, and taking only subsets of the complement bases {r : I~1 = e} corresponds to a nonuniform refinement. To this end, we exploit the commonly used fact that the error in the energy norm can be estimated by the residual in a dual norm which, at least in principle, can be evaluated. In fact, since rA "-- AUA -- f -- A ( u A -- u),
by (3.6.4) and (2.5), one has C1
IIrAIrH-, ~
Ilu - ~All ~ c~ IIrAIIH-, 9
(7.2.1)
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S. Dahlke et al.
Expanding the residual rA by the dual basis ~ and taking the Galerkin conditions P2 AUA -- P i f (7.2.2) into account, yields
rA-xev
xeV\A
Bearing (3.6.6) in mind, and quantifying the constants in (3.5.7), ensures the existence of finite positive constants c3, c4 such that
C3( E ]'~12tI(rA'@X)I2) 89-~ IIrAl[H-t -~ C4( E IAI2t I(rA, %Dx)12)1 XCV\A XCV\A (7.2.3) Thus, in principle, the nonnegative quantities -
.-
:
* I
ev\a,
are in some sense the desired local quantities bounding the error Ilu - UAl[ from below and above. However, in the present form, (7.2.3) is still useless since the bounds involve generally infinitely many terms (ix. To understand these bounds a little better, suppose that ux, denote the wavelet coefficients of the current solution
~tA--
E~X' ~)X' 9
,VCA
Straightforward calculations then yield fx - E
(A~bx,, ~bx)ux,
X'CA
(7.2.4)
where as above fx := (f, Cx) denote the wavelet coefficients of the righthand side f relative to the dual basis ~. (7.2.4) shows that the size of 6x is influenced by two quantities. First, if the right-hand side f itself has singularities, this will result in large wavelet coefficients fx. Second, the sum ~X'CA (ACx,, Cx}ux, gives the contribution of the current solution which, for instance, could reflect the influence of the boundary. Thus replacing the bounds in (7.2.3) by finitely many computable but still sufficiently accurate terms requires a) estimating the smearing effect of A b) some a priori knowledge about f.
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Nonlinear Approximation and Adaptive Techniques
So far we have only used the ellipticity (2.2) or (2.5) of A and the norm equivalence (3.5.7). To deal with problem a) one has to make essential use of the decay estimates (3.6.13). These estimates are usually deduced from (2.11) with the aid of moment conditions (see (3.1.13)). We describe now how they can be utilized. Let again N denote the order of vanishing moments of the wavelets ~ and let ~ < r - d/2, where r is the constant in (3.6.13). Choose for any c > 0 positive numbers C1, C2 such that C2/~+2t -~- 2
5
(7.2.5)
~2 < C.
For each A E V, we define the influence sets V~,c
9-
{~' e v
9 Illog ~1 - [log ~'11 ~< log 2 ~2 x and m i n { l $ 1 - 1 , I~'1-1} d i s t ( ~ ,
~x,)
< c11 },
where f ~ denotes the support of ~ . The sets V~,~ describe that portion of the sum E~,i - Z (Ar r A'EA
appearing in the residual weights ~ (7.2.4), which is significant. In fact, the remainder e,x "E (Af,x,, r ~'EA\V~,~
can be estimated as follows [14]. Proposition 3. For e x and ~ , ~ as above there exists a constant c5 independent of f and A such that
( ~
~V\A
I~Xl=~I~,1=)-~ ~ c5~ II~AII.
('7.2.6)
Note that, again by (3.5.7),
II lr II ll.,
(Z
)~EA
so that the right-hand side in (7.2.6) can be evaluated by means of the wavelet coefficients of the current solution UA. Moreover, one can even give an a priori bound. In fact, the stability of the Galerkin scheme assured by (3.6.5) says, on account of the uniform boundedness of the P~ in H -t, that
IluAII ~< IIP~flIH-, ~< IlfllH-*.
(7.2.7)
276
S. Dahlke et al.
As for b) above, by construction, the significant neighborhood of A in V\A NA,~ "-- {A E V \ A 9A N Vx,~ r 0} (7.2.8) is finite: #NA,~ < co. Outside NA,~ the quantities 6~ in (7.2.4) are essentially influenced by wavelet coefficients of f. But this portion is essentially a remainder of f. In fact, by (3.5.7), 1
XeV\(huN^,~) _< c6
inf
VE,.~AO NA, ~:
IIf-
VlIH-, ~ c6
in f I I f -
VESA
vllH-,,
for some c6 < oo. This suggests defining I
d),(A, e ) " - ]Altl I
~
I
(Ar
,~,)u~,, I'
A E V \ A.
I
A'EAAVx,~
Note that, in view of (7.2.8), dx(A,e)-0,
AEV\A,
(7.2.9)
ACNx,~.
The main result can now be formulated as follows [14]. Theorem 13. Under the above assumptions, one has 1
vllH-,)
+ c;~ [IfllH-, + c6 in_f IIf vESa
)~ENA,e
as well as, ( Z ~ENA,e
d~'(A'e)2) 89
1 Ilu - uAll +
C1C3
c'se [IfllH-~
+ c6 in_f Ilf - VIIH-~" vESA
Moreover, for any A C V, A C A, one has
( E
1
C1C3
[luA
-
UAll+c~ c IlfllH-~+c6 in f Ill yES^
-
VllH-,,
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Nonlinear Approximation and Adaptive Techniques
This result provides, up to the controllable tolerance
r(A,
~) " -
c~c II/lln-* + c6 in_f II/-- villi-,, vESA
computable lower and upper bounds for the error I l u - uill. Usually under more specialized assumptions, results of a similar nature have been obtained also in the finite element context (see e.g. [27]). Furthermore, one expects that nonlinear problems can be handled by combining such estimates with known abstract results. 7.3
C o n v e r g e n c e of an a d a p t i v e r e f i n e m e n t s c h e m e
In the present setting, it can be shown with the aid of Theorem 13 that under mild assumptions on the right-hand side f a suitable adaptive choice of A enforces the validity of the saturation property (7.1.8). We continue with the notation of Subsection 7.2. The following theorem was proved in [14]. Theorem 14. Let tol > 0 be a given tolerance and fix 0 E (0, 1). Define 9-
--+
ClC3
,
2C2C4
(7.3.1)
choose iz > 0 such that 1-0
#C* <_ 2 ( 2 - 0)c2c4' and set
(7.3.2)
# tol (7.3.3)
c : = 2c~ I I / l l H - , "
Suppose that for A C V, one has 1
c6 in_f IIf - viiH-, < ~# tol. vESA
Then, whenever A C V, A C A is chosen so that 1
(
E /kE/~CINA,s
d'~(A'c)2) ~-> ( 1 - 0 ) (
1
E
d,~(A,e)2) ~
/kENA,~
there exists a constant ~ 6 (0, 1) depending only on the constants #, 8, ci, i = 1 , . . . , 6 , such that either
278
S. Dahlke et al.
or
< tol.
Of course, the idea is to choose/~ D A as small as possible, i.e., in any case/~ \ A C Nh,e. This leads to the following Algorithm 2. 0. Choose tol > 0, ~ E (0, 1) and compute C*, # according to (7.3.1), (7.3.2). 1. Compute s = ~(#, tol) by (7.3.3). 2. Determine A C V such that 1
c6 in_f ] I f - v i l l i - , < ~# tol. VESA
3. Solve ( A u i , v) = ( f , v),
Vv E SA.
4. Compute )~ENA,e
If r]A,e < tol stop, accept Uh as solution.
5. Determine A with A C/~ C A U NA,~ such that 1
(1-
Set/~ ~ A and go to (3). Although quite different with regard to its technical ingredients, the above algorithm is very similar in spirit to the adaptive scheme proposed in [27] for bivariate piecewise linear finite element discretizations of Poisson's equation. As above the coarsest grid is chosen in [27] in such a way that all errors stemming from data are kept below any desired tolerance. In that sense, the approach in [27] has motivated part of the developments described above and in [14]. We wish to add a few more comments on the above scheme. It may not be practically efficient to shoot for the final accuracy in the first step. One would rather select a sequence tolt - ~1 t - 1 , ~ - 1 , . . . , N, say where tolg --: Tol is the final accuracy. One would then proceed as follows:
Nonlinear Approximation and Adaptive Techniques Algorithm 3.
279
0. Choose Tol = 2-Ntol0. Set tol = tol0.
1. Apply Algorithm 2 with tol. tol --+ tol and 2. If tol _< Tol stop, accept Uh as solution. Otherwise set -ygo to (1).
A brief comment on Step 3 in Algorithm 2 is in order. By (3.6.10), the principal sections of the matrix ,4 are well conditioned. This can be used to update a current Galerkin approximation UA as follows. Let UA := d~(uh) be the vector of wavelet coefficients of u~. To compute the coefficient vector uA of u A we choose an initial approximation v according to v~ -
{u~,
AEA
w~, A c ~ \ A ,
(7.3.4)
where W/~\A -- d~(wS.\A ) are the coefficients of the Galerkin solution of the complement system
(AwA\i,v) - (f,v),
W/~\A
v E Sh\ A,
where Ss h "- span{r 9 A E A \ A}. The corresponding matrix entries have to be determined anyway for the adaptive refinement. Since by (3.6.10), the corresponding section Ah\ i of A is well-conditioned only a few conjugate gradient iterations are expected to be necessary to approximate ws i well enough to provide a good starting approximation of the form (7.3.4), which will then have to be improved by (a few) further iterations on the system matrix .AA. References
[1] Andersson, L., N. Hall, B. Jawerth, and G. Peters, Wavelets on closed subsets of the real line, in Topics in the Theory and Applications of Wavelets, L. L. Schumaker and G. Webb (eds.), Academic Press, Boston, 1994, pp. 1-61. [2] Babuska, I., Advances in the p and h-p versions of the finite element method, in Proc. Singapore Cons ISNM 86, Birkhguser, Basel, 1988, pp. 31-46. [3] Babuska, I. and W. C. Rheinboldt, A posteriori error estimates for finite element computations, SIAM J. Numer. Anal. 15 (1978), 736754.
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Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics 1341, Springer, Berlin, 1988.
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Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.
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Tchamitchian, P., Wavelets, functions, and operators, in Wavelets: Theory and Application, G. Erlebacher, M. Y. Hussaini, and L. Jameson, (eds.), ICASE/LaRC Series in Computational Science and Engineering, Oxford University Press, New York, 1996.
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Verfiirth, R., A posteriori error estimation and adaptive mesh refinement techniques, Comput. and Appl. Math. 50 (1994), 67-83.
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Stephan Dahlke, Wolfgang Dahrnen Institut f/ir Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55, 52056 Aachen Germany [email protected], [email protected] Ronald A. De Vote Department of Mathematics University of South Carolina Columbia, S.C. 29208 U.S.A. [email protected]
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III. Wavelet Solvers for Integral Equations
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Fully Discrete Multiscale Galerkin BEM
Tobias von Petersdorff and Christoph Schwab
A b s t r a c t . We analyze multiscale Galerkin methods for strongly elliptic boundary integral equations of order zero on closed surfaces in lRa. Piecewise polynomial, discontinuous multiwavelet bases of any polynomial degree are constructed explicitly. We show that optimal convergence rates in the boundary energy norm and in certain negative norms can be achieved with "compressed" stiffness matrices containing O(N (log N) 2) nonvanishing entries, where N denotes the number of degrees of freedom on the boundary manifold. We analyze a quadrature scheme giving rise to fully discrete methods. We show that the fully discrete scheme preserves the asymptotic accuracy of the Galerkin scheme with exact integration and without compression. The overall computational complexity of our algorithm is O(N (log N) 4) kernel evaluations. The implications of the results for the numerical solution of elliptic boundary value problems in or exterior to bounded, three-dimensional domains are discussed.
w
Introduction
Strongly elliptic boundary value problems in smooth and bounded domains ~'~ C ]R3 can be reduced to equivalent integral equations on the boundary manifold F = 0 ~ [4, 37]. For second order elliptic systems, the solution is represented as a combination of so-called single and double layer potentials, and boundary integral equations are obtained by passing to the boundary with the source point. The resulting boundary integral operators are often strongly elliptic pseudo-differential operators on the boundary manifold F [12]. The discretization of these integral equations by finite elements on the boundary manifold leads to the so-called boundary element methods. In the Multiscale Wolfgang Copyright
Wavelet Dahmen,
Methods
for PDEs
Andrew
J. K u r d i l a ,
(~)1997 b y A c a d e m i c
All rights of reproduction ISBN 0-12-200675-5
287 and Peter
Press, Inc.
in a n y f o r m r e s e r v e d .
Oswald
(eds.), pp. 287-346.
288
T. yon Petersdor]:] and Ch. Schwab
present paper we analyze Galerkin discretizations of a class of boundary integral operators of order zero which contains in particular classical Fredholm equations of the second kind. We admit closed, piecewise analytic surfaces in IR3 and assume that the parametric representations are explicitly available (here, we do not discuss the approximation of the surface). Furthermore, we require strong ellipticity of the boundary integral operators in the form of a Gs inequality in L 2(F). The Galerkin method analyzed here is based on subspaces V L C L 2 (F) of discontinuous piecewise polynomials of degree d _> 0. We use a particular, fully orthogonal multiwavelet basis of V L, the construction of which we perform explicitly for subdivisions based on triangles or quadrilaterals. Special cases include the Haar wavelets (d = 0) and the piecewise linear multiwavelets introduced in [27]. Following [24, 27] we show that the stiffness matrix in the wavelet basis can be compressed to O(NL(lOg NL) 2) "essential" elements practically without affecting the asymptotic convergence rate of the scheme. This analysis assumes, however, the exact evaluation of the entries in the stiffness matrix which is unrealistic on general, curved surfaces. Hence we give here, apart from a self-contained exposition of the boundary reduction of elliptic partial differential equations, the construction of multiwavelets and the consistency analysis of the stiffness matrix compression, a new and general scheme for the approximation of the nonzero entries of the compressed Galerkin stiffness matrix by numerical quadrature. We show that with tensor product Gaussian quadratures of judiciously chosen orders and possibly geometric subdivisions of the region of integration, the asymptotic convergence rates of the Galerkin scheme based on the compressed stiffness matrix can be retained with only a slight increase in computational complexity. Our scheme is in contrast to algorithms inspired by image compression, where first the whole Galerkin stiffness matrix is evaluated (corresponding to a full, digitized image), then a fast wavelet transform is applied to the rows and columns to change into the wavelet basis and only then small, nonzero entries are dropped (this so-called "~-truncation" was first proposed in the context of integral operators in [3]). Clearly, this approach is still of O(N~) complexity. In particular the quadrature bottleneck to generate the dense Galerkin stiffness matrix makes it unattractive for large scale applications. In contrast, our quadrature error estimations in conjunction with the consistency analysis of the compression allows us to determine a priori which entries of the compressed Galerkin stiffness matrix must be calculated to which accuracy and which entries can be dropped altogether. This allows to generate a priori an appropriate sparse matrix storage scheme which handles only the O(NL (log NL)2) essential entries of the compressed
Fully Discrete Multiscale BEM
289
stiffness matrix. Further, since our multiwavelets are piecewise polynomial in local coordinates, standard quadratures can be applied. As far as we know, the present paper is the first analysis of a fully discrete multiscale Galerkin scheme in lR3 that is not confined to a particular integral equation, but rather covers a whole class of boundary integral operators. Recent related work on the analysis of fully discrete, fast discretization schemes includes Rathsfeld [28] who considers the double layer potential equation on a polygonal boundary and a different approach to numerical quadrature. Although we confine ourselves in the present paper essentially to classical boundary integral equations of the second kind, i.e. operators of order zero, we point out that the quadrature techniques used here are quite flexible and apply directly to weakly singular integrals and also to hypersingular ones after analytic regularization, thus solving in effect the quadrature problem at least for piecewise polynomial (multi-)wavelets. Furthermore, the concepts of multiresolution analysis and the consistency analysis of the compressed Galerkin scheme can be generalized to boundary integral operators of nonzero order, provided a stable, piecewise polynomial multiwavelet basis with the proper number of vanishing moments is available (the construction of specific, necessarily biorthogonal bases and the proof of their stability appear to be the principal issues here). For integral operators of orders +1 on polygons this was done in [24]. The compression and quadrature error analysis in Sections 3 and 4 of the present paper applies also for piecewise smooth surfaces such as polyhedra. There, however, the strong ellipticity in L 2(F) of the boundary integral equations considered here (i.e., the validity of a G&rding inequality in L 2 on the boundary manifold) is a delicate problem and stability of Galerkin schemes must be proved by some other means (see [8]). The main result of the paper, namely the consistent quadrature approximation of the compressed Galerkin stiffness matrix in essentially optimal complexity, is likewise not confined to zero order operators on smooth surfaces. The paper is organized as follows. In Section 2 we briefly review the reduction of elliptic boundary value problems to strongly elliptic boundary integral equations following [4, 37]. We focus on the classical, so-called indirect method (see, e.g., [4]). In this case, no general principle ensures strong ellipticity of the resulting boundary integral equations which are now Fredholm equations of the second kind. Thus, strong ellipticity in the form of a G&rding inequality in L 2(F) for these boundary integral operators must be checked on a case by case basis via the associated principal symbol. As is well known, strong ellipticity implies quasi-optimal asymptotic convergence rates of Galerkin discretizations [11]. We give several examples for the boundary reduction which result in strongly elliptic boundary
290
T. yon Petersdorff and Ch. Schwab
integral equations (for the general theory we refer to [4]). The description and the analysis of the multiscale Galerkin discretization schemes is divided into two parts, Sections 3 and 4. In Section 3 we present, following [6, 24, 27] (all motivated by the seminal paper [5]), a multiwavelet Galerkin discretization for strongly elliptic boundary integral operators of order zero. This class includes in particular all examples presented in Section 2. We give a consistency analysis showing that most of the O((NL)2) entries in the Galerkin stiffness matrix can be neglected while essentially retaining the optimal asymptotic convergence rates of the full Galerkin scheme. This is also true in negative norms, which implies superconvergence of field values at interior points obtained from inserting the Galerkin approximations to the boundary densities into the representation formula used in the boundary reduction. In this part of the analysis, we assume that all O(NL(log NL) 2) entries that are kept in the "compressed" Galerkin stiffness matrix are computed exactly - a rather unrealistic assumption. The second part of our analysis, i.e., the derivation of a quadrature scheme for the direct evaluation of the compressed Galerkin stiffness matrix, is presented in Section 4. We show how a consistent (i.e., preserving all asymptotic convergence properties established in Section 3 for the compressed Galerkin scheme), fully discrete multiwavelet Galerkin scheme can be obtained by using tensor product Gaussian quadratures of appropriate orders and, where necessary, element subdivisions [32], to approximate the entries of the compressed matrix. Special attention is paid to the evaluation of the singular integrals. We show that the Galerkin scheme together with certain regularizing coordinate transformations due to [10] allow for a stable and accurate quadrature. In particular, these transformations allow us to remove the integrand singularities completely so that only analytic functions have to be integrated numerically. The total work necessary to obtain the consistent quadrature approximation to the compressed Galerkin stiffness matrix is shown to be O(NL(logNL) 3) kernel evaluations for the nonsingular and O(NL (log NL) 4) kernel evaluations for the singular integrals. The present work has appeared in slightly different form in the preprint [25].
291
Fully Discrete Multiscale B E M w
S t r o n g l y elliptic b o u n d a r y
integral equations
We consider boundary value problems for elliptic systems of second order in variational form: Given f E L2(~), find U E H~(~t) such that s
in~
(2.1)
on F.
(2.2)
subject to the boundary conditions BU-
f
We will focus in this work on the case when s is a self-adjoint second order N • N matrix differential operator and B a boundary differential operator, either the trace operator ~/0 for the Dirichlet problem or the boundary operator 71 for the Neumann problem. We assume that the boundary value problem (2.1)-(2.2) admits a unique weak solution in H I ( ~ ) . For the operators s B, we may also consider exterior boundary value problems posed in ~c _ IR3\~. Here the boundary conditions must be appended by suitable radiation conditions at infinity in order to ensure the unique solvability of the boundary value problem in H~oc(~tc) (see, in particular, [13] for exterior problems in elasticity). Throughout, n(y) denotes the unit normal vector at y E F pointing into ~c. We describe now the boundary integral equation reformulation for the boundary value problem (2.1)-(2.2). We assume that we are given a fundamental solution of the differential operator s in (2.1) which is a matrix function so that
(2.3)
G(x - .) = 5(x - . ) I
holds in the sense of distributions. For the boundary reduction, we look for w in the form of a potential with an unknown density function u on F. In our examples, this approach will lead to boundary integral operators of order zero and is also known as the indirect method. We use the following potentials. Double layer ansatz for the Dirichlet problem: U(x)
-
jfr[Tx, G(x -
x e ft.
(2.4)
Taking traces in (2.4) and using the Dirichlet boundary condition, we arrive with the jump relations for the double layer potential (2.4) at the boundary
T. yon Petersdor~ and Ch. Schwab
292 integral equation
Au -
(1) -~I + g
u = f on F,
(gu)(x) -
~/1,yG(x - y)Tu(y)d%.
(2.5) Once the boundary integral equation (2.5) is solved, the density function u(y) is inserted into (2.4) and the solution U(x) is obtained in ~. Single layer ansatz for the Neumann problem: x e ~.
U(x) = fr G(x - y)u(y)dsy,
(2.6)
Applying the natural boundary conditions ')'1 in (2.2) and letting x --+ F, we get the boundary integral equation
Au-
(~I-K')
u - f on F,
( K ' u ) ( x ) - fr[71,xG(x-y)]u(y)dsy.
(2.7) Once the boundary integral equation (2.7) is solved, the density function u(y) is inserted into (2.6) and the solution U(x) is obtained in F~. The unique solvability of the boundary integral equations (2.5), (2.7) can be ensured provided the boundary operator A is injective and satisfies, in each case, the following conditions:
1. Continuity:
(2.8)
IIAullo _ C(A)Ilullo.
2. G~rding Inequality: There exists a compact operator T : L2(F) --+ L2(F) and a positive constant C such that Re((A + T)u, u) >_ C [lu[I2n:(r)
rue
L 2(r).
(2.9)
3. Injectivity: Au = 0 =~ u = O.
(2.10)
A sufficient condition of the validity of (2.9) is the positive definiteness of the principal symbol of the operator A which is defined, for example, in [12, 16].. We close this section with some examples of boundary value problems (2.1)-(2.2), exhibiting in each case the boundary operator "yl, the fundamental solution G and the principal symbol of the boundary integral operators A in (2.5) and (2.7). Note that since the boundary integral operators in (2.5), (2.7) are mutually adjoint, their principal symbol matrices coincide up to Hermitean transposition.
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Fully Discrete Multiscale B E M
We illustrate these considerations by some particular boundary value problems. In each case, we exhibit the fundamental solution and the boundary operator 71 together with the double layer kernel of the operator K in (2.5). Example A: Boundary value problem for the Laplace equation. Here
s
B U = 7x U =
= -AU, 1
OU ~n
[7~,yG(x - y)] - n(y) 4 7 r l. x(x- y-l 3y) "
G ( x - y) - 47r Ix - Yl '
As is well known, on smooth surfaces the double layer operator K and its adjoint K' are compact operators in L2(F) whence it follows that the principal symbol of the boundary integral operator A is 1/2. This boundary value problem arises in many areas of engineering, so for example in electrostatic field calculations where U is the electric potential and u(y) in (2.6) is the charge distribution on the electrode F. Example B: Boundary value problem for the Lam6-Navier equations of linearized, three-dimensional elasticity. Here the infinitesimal displacement field U" ft --+ IR3 is determined from (2.1)-(2.2) with
s
()~ + #)grad(divU)
= -pAU-
and the Lam6-constants ~ and # are given parameters characterizing the homogeneous and isotropic material constituting the deforming body. Here 71 is the traction operator which is given explicitly on F by OU 13U - 71U - )~ (divU) n(y) + 2 # ~ n + #n • curl U. The fundamental solution is
c(~-
y) =
A+3#
{
s ~ ( ~ + 2#)
1
)~ + # (x - y) (x - y) r }
I~ - yl I ~ ~ + 3#
I~ - yl ~
and ~ [71,yG(x
-
y)] - 4= (~ + 2#)
{ n(y)T (z _ y) z + Ix - yl a
n(y) (~ - y) ~ - (~ - y ) ~ ( y ) ~ ix_yl 3
2(~ + ~) n(y) T (~ - y)
+
~
Ix-yl 5
To present the principal symbol, for x E F let N ( x ) = (tl (x), t2(x), n(x))
(~-y)(~-y)
T}
T. yon Petersdorff and Ch. Schwab
294
denote the matrix consisting of two mutually orthogonal unit tangent vectors at x to F and the exterior unit normal vector n(y). Then the principal symbol ao(x, ~) of A in (2.5) is given by [37]
o'0(x~ ~)
I
Y-l(X) lgl
0
i7~1 i7~2
el~l,
Y(z)
where 7 - x + ~ and e - 1 for interior and e - - 1 for exterior problems. Hence we see that the boundary integral equations (2.5), (2.7) are strongly elliptic in this case, too. Example (3: Oblique derivative problem. Assume that f~ C IR3 is a smooth and bounded domain. The oblique derivative problem for L = - A in f~c consists in solving s - 0 in ft c subject to the boundary condition
BU(x) = b(x) . grad U(x) + a(x)U(x) = f ( x )
x e F
(2.11)
and the radiation condition lim U(x) = O. Here b = b(x) : F --+ ]R3 is a given direction field of length one, i.e.b T b = 1, depending smoothly on x and f ( x ) is a given, sufficiently smooth function on F. It was proved by Giraud that this problem has a unique solution if, for example, a(x) 3> 0 and b(x)Tn(x) > 0 V x E F (see, e.g., [19]). Using the single layer ansatz (2.6) and inserting into (2.11), we get the following Cauchy-singular boundary integral equation for the unkown density u(y) (see [19]):
2a(x---~-F
fr
- y) u(y)dsy + a(x) fr G ( x -
Ob(x)
y)u(y)dsy = f (x)
x e F.
(2.12) Here a(x) = b(x)Tn(x). This equation arises for example in physical geodesy for the determination of the earth's shape from gravity measurements [20]. In [19], the principal symbol of the boundary integral operator in (2.12) is derived. We have
1
ao(x) - -~ (n(x)Tb(x) + ir(x)Tb(x))
(2.13)
where r(x)Tn(x) = O, i.e. r(x) is any direction in the tangent plane. Since
inf ib(x)Tn(x)] a e a 0 ( x ) >_ ~1 zer
Fully Discrete Multiscale BEM
295
the boundary integral operator in (2.12) is strongly elliptic if the direction b(x) is not tangential anywhere on F. Example D: Exterior Stokes flow. Here we are interested, for in determining the velocity field and the pressure distribution Newtonian, incompressible viscous flow exterior to a smooth and surface F in IRa. For illustration we consider the exterior Dirichlet Other cases can be handled similarly, see, e.g., [17]. The governing equations are -uAU+gradp
U
= -
example, (U,p) of bounded problem.
0, d i v U = 0 i n f ~ C , f on F
(2.14)
where f is a prescribed velocity field on the surface of the body satisfying fr f" n ds - 0 and u > 0 denotes the viscosity of the fluid. We require in addition that the fluid is at rest at infinity, i.e. -1
I U ( x ) l - o(1),
I g r a d U ( x ) l - o(Ix I -)2) I p ( x ) l - o(IxL-~), I g r a d p ( x ) l - o(Ixl
for Ixl ~ ce.
The fundamental velocity tensor, the so-called Stokeslet, is given by
{
1
G(z-v)-~
I~-vl
-1
z+
-
(x - y)(x 13 y)7-
I~-v
}"
We represent (U,p) as double layer potentials of an unknown density u : F ~ IR3 as follows: 3
E
U~(x)
j,k=l
/,
/.. uj(y)Tijk(y -- x)nk(y)ds v,
(2.15)
,]1
3
p x,
uj (y)Iljk (y -- X)nk (y)ds v
where
T~k(~) --
(2.16)
3 ~ci~cj~Ck
4~ I:~15
and
Iljk(~)--~
--~+31~15
,
Letting in (2.15) the point x tend to F, we obtain the boundary integral equation (2.5) with the hydrodystatic double layer potential 3
(Ku)i (~)
E
j,k=l
/,
[_ uj(v)T,j~(v Jl"
- ~)nk(v)a~,
9 e
F.
T. yon Petersdorff and Ch. Schwab
296
We observe that due to the classical estimate I( 9 - Y)" n(Y)l < c ( r ) I x - yl ~
x, y e r 3
valid for smooth, bounded surfaces F, the double layer kernel Y~k=l Tijk (X-y)nk(y) admits the estimate 3
Z Tijk(x - y)nk(y) <_C(F)Ix - y1-1 , k--1
i.e.it is weakly singular and thus integrable. Moreover, on smooth surfaces the hydrodynamic double layer potential K is a compact operator on [L2(F)] 3. Thus, the principal symbol of the operator A = 89 + g is equal to 89 and hence A is a strongly elliptic boundary integral operator
in [L2(F)] 3. This can also be seen by letting formally A --+ c~ in the fundamental solution and the principal symbol for the elasticity problem. The proof of the injectivity of A is given in [17, Theorem 3.1]. We remark that corresponding boundary integral equations are also obtained for the time harmonic variants of the above boundary value problems. The positivity of the principal symbols is unchanged then since the mass term -pw2U in the differential operator does not contribute to the leading derivatives. In summary, in each of the above examples we can reduce the boundary value problem (2.1), (2.2) to a boundary integral equation
Au = f
(2.17)
for the unknown density function u E L 2 (F) with a strongly elliptic pseudodifferential operator A of order zero. It is for such problems that we develop and analyze a wavelet based Galerkin discretization scheme. {}3 3.1
Multiscale
Galerkin boundary
elements
Preliminaries
Let f~ C ~3 be a bounded domain with a piecewise analytic, orientable Lipschitz boundary manifold F = aft. More precisely, F admits a partition into No disjoint open sets Fj, j = 1, ..., No and there exists a covering of F by a collection of larger, open sets Fj with Fj C Fj C F, i.e.
r-
U
rj -
l<_j<_No
[.J
rj n
- 0 j # k.
(3.1.1)
l<_j<_No
We assume that there exist local charts ~j E C ~ 2) which map F j bijectively onto certain reference domains/~0 C ~2 and that the set { (F j, ~j)} forms a Lipschitz atlas of F.
Fully Discrete Multiscale BEM
297
We assume that each Fj is a curvilinear, either quadrilateral or triangular surface piece in ]R3. We can therefore in particular assume that for all quadrilateral (resp. triangular) F j there exists a common reference domain U ~ C/~o such that
~;I(~A(0 ) __ i~j,
g;1 is analytic on ~-6,
j _ 1, ..., No.
(3.1.2)
Therefore U ~ is either the unit triangle {(~1, ~2) : - 1 < ~1 < 1 , - 1 < ~2 < ~1} or the unit square {(~1,~2) : - 1 < ~i < 1, i = 1,2} in ]R2. Admissible boundaries include therefore closed C~-manifolds as well as polyhedra. By ds we denote the surface measure defined almost everywhere on r. We consider the space L2(F) of functions u : F -+ CN which are square integrable with respect to ds. An inner product on L2(F) is given by
(u, v) - fr u~ds.
(3.1.3)
We also consider the Sobolev spaces HS(Fj) of functions with pullback in H~(U ~ endowed with the norm H~(U ~ transported to Fj. The space of functions u C L2(F) with Ulr j E HS(Fj) for s > 0 is denoted by n j No = l H ~(Fj) . Evidently, the expression
Ilul18 -
\j=l
IlUl]s.(r~)
No H 8 (Fj) is a norm in nj=l An inner product ( . , . ) , equivalent to ( . , . ) alent norms) in L2(F), can then be defined by
(3.1.4)
(i.e., giving rise to equiv-
No
(3.1.5) Here a* is the usual pullback operator, i.e.a*~ := ~ ( a - l ( u ) ) . We are interested in the numerical solution of the operator equation (2.17) in the weak form
u e L2(r)
(Au, v) - (1, v)
V v e L2(r).
(3.1.6)
Here the operator A is a boundary integral operator which can be represented in the form
(Au) (x) - c(x)u(x) + p.v. fr K(x, y)u(y)dsy.
(3.1.7)
T. yon Petersdorff and Ch. Schwab
298
Here we have, for x, y in a sufficiently small tubular neighborhood of F, K(x, y) - Ko(x, y) + 1C(x, y, x - y), with an analytic function Ko(x, y) and /C of the form
IC(x, y, z) -
s~(x, y)z ~ Iz] -2-k ,
E
(3.1.8)
k<_]c~]<_k+a
where k is an odd integer and a is finite. The coefficient functions sa(x, y) and c(x) are analytic functions of x E Fi and y e Fj, i, j - 1, ..., No. All kernels in the examples in Section 2 are of this type with k - 1 or k - 3, i.e., K: is antisymmetric in z. This will be assumed henceforth and is essential in handling the Cauchy-singular integrals in the numerical quadrature (see Lemma 12). This assumption, the so-called parity condition, holds quite generally for kernels obtained from boundary reduction of elliptic boundary value problems (see [15]). R e m a r k 1: This includes in particular the kernel which occurs in the Neumann problem for the Helmholtz equation - A U - k2U - O. Here the fundamental solution is
eiklx-yl
-
S
-yI '
o (x , y) . Since we and the operator K ~ occurring in (2.7) has the kernel ~-~G have e i k z _ cos(kz) + isin(kz) = fl(z2_____))+i f 2 ( z 2) Z
Z
Z
Z
with entire analytic functions fl, and f2, we can express the kernel function in the form g ( x , y) - Ko(x, y) + 1C(x, y,x - y) with/C as in (3.1.8). The integral in (3.1.7) is in general to be understood in the Cauchy principal value sense, i.e.
p.v. f K(x, y)u(y)dsy = lim f K(x, y)u(y)dsy. ~-~o Jr\B~(~) Jr
(3.1.9)
Here B~(x) - {y E IR3 9Ix - Yl < ~} denotes the open ball of radius s about the point x. We assume that the kernel K(x,y) is such that the limit in (3.1.9) exists (see [19] and Section 5.4 ahead for details). Approximate solutions to (3.1.6) are obtained by the Galerkin method. Given a dense sequence {vL}~=0 of finite dimensional subspaces of L2(F), we solve 21L E V L
(Au L, V> -- ( f , V>
~ V E V L.
(3.1.10)
The Gs inequality (2.9) and the injectivity (2.10) of the operator A ensure the unique solvability of (3.1.6), (3.1.10) and the quasi-optimality
299
Fully Discrete Multiscale B E M
of U L, provided L is sufficiently large [11]. We denote by PL the orthogonal projection
PL L2(r) -~ V L,
((v - PLV), qo) - 0
(3.1.11)
V qOe V L.
Proposition 1. Assume (2.8)-(2.10). Then, for every f E L2(F) and sufficiently large L, the approximate problem (3.1.10) is stable in the sense that IIPLAuLllo >__C8 IluLIIo for all u L E V L. (3.1.12) In particular, there exist unique solutions quasi-optimally to the unique solution u of
UL
Of (3.1.10) which converge
(3.1.6), i.e.,
!1u - uLllo -< C inf Ilu-Vllo.
(3.1.13)
vEV L
For a proof, see e.g. [11]. Our interest is here in so-called multiscale discretizations which are based on special bases for the spaces { V L }~=0, which we define next.
3.2
Multiwavelet basis
To define V l C L2(F) we divide U ~ into 4 t subsquares (resp. congruent subtriangles) {U~ } by successively halving the sides 1 times. Then we define the spaces j = 1,...,No, k-
1,...,41 }
of discontinuous, piecewise polynomials of total degree d in local coordinates (i. e.IId -- span{~Pl ~2 . Pi >_ 0, Pl + P2 _< d}) both for the reference triangle and square (notice that this is possible due to our multiwavelets being allowed to be discontinuous; it does not improve the asymptotic rate of convergence to use, e.g.the full tensor product space on quadrilateral elements here). Throughout the construction of the multiwavelet basis in this section, the notion of orthogonality is understood with respect to the inner product (3.1.5). Let {qb~}, u = 1, ...,0 d with Od : (d + 1)(d + 2)/2, be an Lu(u ~ orthonormal basis of I~0 . - Hd(U0) (such a basis can be generated by the Gram-Schmidt process). The dimension of V t is Nl - OdNo4 I. To unambiguously identify all degrees of freedom, we introduce the multi-index I-
(j,l,k,v),
l <_j <_No, 1E INo, 1_
,
l < u < Od. m
D
300
T. von Petersdorff and Ch. Schwab
Let Tkt denote the affine transformation which maps L/tk to L/~ and define the function ~I" F ~ lR by ,( )_{2tqh~oTk t ~j ~o(j,k,l,,,)[rj 0
inL/~ otherwise,
~(J,k,/,~)[r\r~--0"
For l E IN0 an orthonormal basis of V t is given by the functions { ~I [ I E Zt } where Zt = { ( j , l , k , v ) [ j = 1 , . . . , N o , k = 1 , . . . , 4 t, v = 1,...,Od }. Obviously, the spaces V l form a hierarchy, i.e., V 0 C V 1 C""
C V l C V l+l C " ' ' .
As usual, we define a sequence of spaces W l as orthogonal complement with respect to (., 9) of V t-1 in VI: W l :'- { ~D e V I I (~, ~)) = 0
V ~ e Y l-1 }.
(3.2.1)
Then V TM = V l @ W l+l and we obtain the multilevel splitting V n : - W 0 ~[~W 1 @ . . . ~ W
L
(3.2.2)
where W ~ "- V ~ Hence every function UL E V L admits a unique decomposition u L=w ~
l+...+w
L,
w t EW L,I-O,...,L.
(3.2.3)
Let P-1 - 0 . Then w l--- (Pt - P t - 1 ) uL in (3.2.3). To construct an orthonormal basis for W l we proceed similarly as for Y I. We start with the space ~1 _ { h i u[u ~ E Hd(L/~),k = 1 , . . . , 4 } of discontinuous piecewise polynomials. Then define the orthogonal complement l~ 1 := { u e ~zl [ ( u , v ) - 0 V v e 170 }. Let r denote an orthonormal basis of W 1. Such a basis (which is not unique, of course) can be obtained using the Gram-Schmidt process starting with a basis of ~1 (given by 21~3~ o T1, V = 1,...,0d, k = 1 , . . . , 4 . For an example with d - 1 we refer to [27]). To define the multiwavelet basis of W l, we introduce for l _ 1 the functions r F -~ IR by ,
tcj ( r
{ 21--1~)v 0 T1 0
in H~ otherwise,
r
[r\rj -- 0. (3.2.4)
A basis of W ~ - V ~ is r
~I
for I - (j, 0, k, u).
(3.2.5)
301
Fully Discrete Multiscale B E M
Hence for all 1 _> 0 an orthonormal basis of W l is given by the functions {r E ~ } where ~0 "-- fO
9- { ( j , l , k , u ) [ 1 <_ j <_ No,1 <_ k _< 41,1 <_ u <_ 30d}
for / > 1.
(3.2.6)
R e m a r k 2: The multiwavelets r in (3.2.4) are in fact fully orthonormal with respect to the inner product (3.1.5). They are, moreover, piecewise polynomials in local coordinates. This will be essential in the quadrature error analysis in Section 5. By (3.2.2) an orthonormal basis of V L for L E ]No is given by {r
I E & U . . . U JL }.
The L2-projectors PL onto V L admit the explicit representation ZtL -- PLZt -- Z
(3.2.7)
(Zt, ~ I ) qPI
IEZL
and we have for w I in (3.2.3) w l -- (P1 - P l - 1 ) u
--
Z
(~t,Cj)Cj .
(3.2.8)
JE,TI
Therefore the norm of functions in L 2 (F) can be characterized by the multiwavelet expansion coefficients. Proposition 2. For every u E L2(F), there holds O0
ilul 2
,
~
(3.2.9)
/=0 JEJt
where ~ denotes the equivalence of norms. Moreover, the higher order Sobolev norms of smoother functions can be estimated by properly weighted sums of multiwavelet coefficients. Proposition 3. Let 0 < s < d + 1. Then for every every L E IN, L Z 22ls Z /=0
[(u, r
where u - 0
for 0 < s < d + 1 and u m
E II;__~HS(rj) and
L No < C ~ 22tS[]wl[[2 < C LU Z []u[[~.(rj) , (3.2.10) /=0
JEffl
u
j=0
1 for s - d + 1.
302
T. yon Petersdorff and Ch. Schwab
The proof is obtained exactly as that of [27, Proposition 4.2]. The "energy norm" equivalence (3.2.9) and the one-sided bound (3.2.10) are the only ingredients necessary for the analysis of the multiwavelet scheme. In particular, it is not necessary to have (3.2.9) in a whole scale of Sobolev spaces. We use the multiwavelet basis (3.2.4), (3.2.5) in the Galerkin equations (3.1.10). To this end, we write u L in the form L
Uj~)j,
(~IL , ~)j)
/ - - 0 J E,.T!
and denote by ~7 - (uL)jEjI;I--O ..... L the vector of unknown coefficients of u L. It is determined by the linear system L
I E 3~,,l'- 0,...,L.
(3.2.11)
l=O dE Jr We denote the NL x NL matrix by A L, i.e., ALj "- (r
ACj),
I E ~,,J
E ~,I,I'-O,...,L.
(3.2.12)
Then (3.2.11) corresponds to the linear system AL~ -
where f - ((r f))1eJz,,~'=o ..... L. Note that A L is not symmetric in general. The condition numbers of the sequence {A L } of matrices is bounded:
Proposition 4. There exists K* E lR such that for all L cond2 (A L) <_ K*. Proof: This follows from the stability (3.1.12) and the norm equivalence (3.2.9). 3.3
Consistency analysis for the compressed Galerkin s c h e m e
The wavelet basis {r defined in (3.2.5), (3.2.4) has vanishing moments in local coordinates. More precisely, for all J E ~ , 1 >__1, u (r o
o Xj)(s)s ~ ds - 0
for [a[ _ d.
(3.3.1)
303
Fully Discrete Multiscale B E M
The vanishing moment property (3.3.1) implies the smallness of certain entries of the matrix A L. This is due to (3.3.2)
[ D ~ D ~ K ( x , y ) ] < ix _ y[2+lc,l+l,~l
(here D~, D~ are Cartesian derivatives in lR3 acting on a smooth extension of K to a tubular neighborhood of F), which is an immediate consequence of (3.1.8). By S j we denote the set (3.3.3)
SJ • { X e F [ C j ( x ) r 0 }
and we define d j j, -" dist(Sj, S j , ) . Then there holds L e m m a 1. The entries A Lj,j, in the Galerkin matrix (3.2.12) with d j j, > 0 satisfy _ ~t'7.,4-2(d+ 2) 2 - ( d + 2 ) ( l + l ' ) . IA~j,I < ,,jj, (3.3.4) The proof follows by (3.3.1) and the Taylor expansion of the kernel about the barycenters of S j and Sj, (see, e.g., [27] and [31] for details). We will now show that most of the NL • NL entries of the stiffness matrix { A j j , } can be replaced by zero without affecting the convergence rates of the resulting "compressed Galerkin scheme". To this end we introduce the following truncation strategy: -
A j j, "-
{ A j j, 0
if daj, < St,t, otherwise,
(3.3.5)
where {Sl,t, } is a matrix of truncation parameters at our disposal. To estimate the consistency error thus introduced we define the block matrices At,l' :'- {Ag, g'}geJ~,g'eJt,,
with 5~, 3], as in (3.2.6). Analogously we define At,r, l, l' = 0 , . . . , L. These are submatrices of the respective stiffness matrices. In the following lemma we estimate the effect of the truncation (3.3.5) on each block. L e m m a 2.
ilAl,l' -- Al,l'
oo
< ""'l,l'
1 -- ""Vl'l'
~
max{~?,l,, 2-2l }.
(3.3.6) (3.3.7)
T. von Petersdorff and Ch. Schwab
304 Proof:
We have with Lemma 1
-
I1", '' - ~'," I1~
max
J E.Tz
< --
~
IAj, j,I
jt Ejlt d j jr ~$1,1t
Cmax J EJ~
r/-2(d+2) 2 -(d+2)(l+l') ~.gj, ji E3"ll djjt~--Sl,lt
We estimate the terms from the S j, closest to S j directly and majorize the remaining terms by an integral
llA' '' ~" i1~
The estimate for
<
g'Y.~-2(d+2)9-(d+2)(l+l')
_
VVl,
+
f
<
dist(z,SJ)~_$1,1t C ~Vl, l,['~-2(d+2)o-(d+2)(l+l').
l,
-
.er
dist(x, Sj)-2(d+2)2-(d+2)12 -dl' dsz -~ t~l,l,-2(d+l)2-(d+2)l 2-- dl') 9
ILAt,t,-/kt,t, !1z follows in the same way with J and J'
interchanged. We estimate next the number of nonzero elements in At,l,, denoted by A/'(At,t,). L e m m a 3.
~r(kl,l') ~ Y l - l Y l ' - i
min{C( 2-2l + 2 - 2 r + J~,l,), 1}.
(3.3.8)
Proof: We note that for each C j , J E ~ there are at most 1 + (2 -21 + C~,r)22r values of J' E ~ such that dist(Sg, Sj,) < ~t,l,. The stiffness define finite matrices AL,A L with respect to the multiwavelet basis {r dimensional operators A L, r . V L _.+ (yL)t where (vL) ' denotes the dual space of V L. We have the following consistency estimate for the difference between these operators. T h e o r e m 1. Let s, g E [0, d + 1] and assume that s+d+l a___ 2 ( d + l )
~+d+l ' &-- 2 ( d + l )
"
(3.3.9)
Assume further that the truncation parameters {6t,,' } in (3.3.5) satisfy t~l,l' >_ max{a2-L2a(L-L)2 a(L-r), 2 -t, 2 -~' }
with some number a > 1.
(3.3.10)
305
Fully Discrete Multiscale B E M Then, for u 6 HS(F), fi 6 HS(F) there holds
I<(,A L - AL)pL?2, PLU)I < Ca-2(d+l)LVNL(S+s)/21[u[Isllull~, where u
-
121 4- 122, C
121 122 - - 0
l f 8 < d-~-
-
(3.3.11)
iS independent of a and L and
0 1
-
if 2(d+l) ~+d+~ < O~and ~+d+ < & 2(d+l) otherwise,
1 and g < d + 1,122 - 1 if s - d + 1 and g -
d + 1, and
122 - 1 otherwise.
Proof:
Using Proposition 3 we find
where the matrix E L is now given by ELj,
_
2~(L-I)2~(L-I') (ALj,
_
f~Lg, ) .
We estimate [IEL]I2 using the Schur Lemma (see, for example [18], p. 269) w i t h "),g - 2 - l . Recall that At,l, and Al,l, denote the blocks of 2~ L and A L corresponding to the levels 1 and l'. We estimate with Lemma 2 and (3.3.10)
IE~j,I~J,
J' E,7~, L = Y ~ 2s(L-l)2~(L-l')2-l' /'--0 L
_
IIAt,~, At,/,ll~ -
2"(L-I)2~(L-I')x-2(d+2),,l,,,2- (d+2)12- (d+l)~' max{6~,,, ,2 -21' }
/'=0 L _< C y ~ 2s(L-l)2~(L-l')~-2(d+2)Vl,l, /'=0
2-(d+2)12-(d+l)l'~, t, ,
since hi,t, > 2 -t' by (3.3.10). Furthermore, (3.3.10) gives
Z IEL,I'Y,,
J' EJt, L <_ C ~ 2s(L-l)2~(L-l')2-(a+l)~2-(d+l)l'2-t /'--0 min{a-2(d+l)22(d+l)g2-2(d+l)~(g-l)2-2(d+l)a(g-l') ' 22(d+l)t, 22(d+1)/' } L < Ca-2(d+l)2 -I ~ 2(s+d+l-2(d+l)a)(L-l)2 (~+d§ <_ C L vl a - 2 ( d + l ) , ~ j
l'=O ,
306
T. von Petersdorff and Ch. Schwab
The estimate for the column sum follows completely analogously. The consistency estimate Theorem 1 allows one to show that the compressed Galerkin scheme (3.3.5) has essentially the same asymptotic convergence rate as the original scheme while the number of nonzero entries in ~L is considerably smaller than N~.
Theorem 2. Let u(t) - 0 for 0 <_ t < d + 1 and u(t) - 3/2 for t - d + 1. A s s u m e (3.3.10) with 1 > a > (s + d + 1)/(2d + 2) for s < d + 1 and a = 1 for s - d + 1, and analogously for 5. Let L be sufficiently large. Then there holds 1. /f the constant a in (3.3.10) is sufticiently large, the compressed Galerkin scheme
(3.3.12)
.4n ftL - PLY is stable, i.e.
II v llo
>
I1,~11 o
v,~v
~.
(3.3.13)
2. A s s u m e in addition that f, u E Hs(F), 0 _< s _< d + 1. Then
ilu - z2L [io < C (logNL) u(s) N ; s/2 Ilulls.
(3.3.14)
3. A s s u m e further that for any g E H~(F) the solution qo of the adjoint equation A*qo - g belongs to H ~(F) for some 0 < g < d + 1. Then
II <_c
(logN/) "(8)+v(~) NL (s+~)/2
-
Ilull, Ilgll~. (3.3.15)
4. For ~t,t' as in (3.3.10) and a,& <_ 1, the number of nonvanishing entries A f ( A L) in the compressed stiffness m a t r i x 71L is bounded by
"Af ( fi'L ) --
O(NL(log NL)2) O(NL log NL )
if
a - 5 - 1, otherwise.
(3.3.16)
Proof:
1. It is well known [11] that the Gs inequality (2.9) implies for sufficiently large L the discrete inf-sup condition which can be written as
IivLi[o
< ~ IIAvLil(v~),
V vL ~ VL
(3.3.17)
Using (3.3.11) with v L E V L and s - g - 0 we obtain with (3.3.17)
tI L Lii > iiAvLiir162 >- c:' iivL IIo - C~-~(d§ ') IIvL llo "
307
Fully Discrete Multiscale B E M This gives, for sufficiently large a, V v L E V L.
(3.3.18)
2. We have
Its- ~!1o <-ii~- P~Utlo + tiP, s - ~ltoUsing (3.3.18)
and ( . , 4 i u i , v i I -- ( A s , v i ) for v i E V i we obtain
yielding
I1~- ~LIIo -< The first two terms are estimated using the approximation property and the continuity of A. The estimate for the third term follows from (3.3.11) with ~ = 0 and p L v L = vL:
3. Let ~L :__ PL~. Then
<_
(A(~ - ~L), ~ _ ~L)I + I(A( u _ ~L), ~L)I"
The first term can be estimated by C [ ] u - ~11o I!~- PL~Ilo, which gives the desiredbound using (3.3.14) and the regularity of ~. For the second term we have
The second term on the right hand side can be estimated by (3.3.11). Since (~i _ P i n E V i we have for the first term, using (3.3.11) with s--0
<- C N L g/2Lv(~) II5L -
PL~I[oI1~11~
<- CN~ ~/~L~r (11~L - silo + Ilu - PLUlIo)I1~11~,
T. yon Petersdorff and Ch. Schwab
308
4. We note that for each Cj, J E 3~, there are at most 1 + (2 -2/ + C5~,l, )2 -2/' values of J' E 3~ such that dist(Sg, S j,) < St,t,. Thus, up to logarithms, the compressed Galerkin scheme preserves the optimal convergence rates of the original Galerkin scheme without compression. R e m a r k 3: After the compressed stiffness matrix is computed we solve the linear system by an iterative method like Richardson iteration, starting on the coarsest level and using the results as starting values for the next level. Since the condition number cond2(fi. L) is bounded independently of L by (3.3.13) the iterative solution of the linear system with an error of the order of the Galerkin error requires O(1) matrix-vector products, i.e., at most O(NL(lOg NL) 2) operations. R e m a r k 4: In order to compute the ./V'(2~zL) nonzero entries of the stiffness matrix one first has to locate the matrix entries which satisfy d j j, <_ 6ll' in (3.3.5). A naive search of all matrix elements would require O(N~) operations. However, it is possible to use a tree-oriented search procedure of complexity O(NL(log NL) 2) which produces a set of nonzero elements which contains all elements satisfying dgj, <_ (Sll', but is still of size
O(NL(IOgNL)2).
R e m a r k 5: Since the boundary integral equation (2.17) was obtained from the boundary value problem (2.1)-(2.2), an approximate solution ~rL of (2.1)--(2.2) can be obtained by inserting ?~L into the potential ansatz (2.4), (2.6). For the resulting error U ( x ) - UL(x)] in the solution U(x) of the elliptic boundary value problem at any interior point x 6 gt, the estimate (3.3.15) (with s = g = d + 1 and smooth boundary F) implies i
!
Iv(x) - ~rL(x)] ~_ C ( x ) ( l o g g L )
a
NL(d+~)[Iflld§
(3.3.19)
i.e.twice the convergence rate achieved for the boundary density in L2(F). Analogous convergence estimates hold also for derivatives of the solution for which representation formulas can be obtained by differentiating the representations (2.4), (2.6) with respect to x E ft. For example, if one uses the simplest Haar wavelet (d = 0) for the boundary integral equation (2.6) of the Stokes problem (2.14), the pressure representation formula (2.16) yields approximate pressures ~L which converge pointwise at interior points with the rate h 2 (log h) a where h denotes the boundary meshwidth. Finally we note that in general, the constant C(x) in (3.3.19) blows up as x ~ F. Nevertheless, with appropriate postprocessing the full rate of convergence (log NL) a NL (d+l) can be recovered also at the boundary [35, 36].
Fully Discrete Multiscale B E M
309
R e m a r k 6: Although our construction of multiwavelets works for piecewise smooth surfaces F, we emphasize that the solution u and the auxiliary functions ~ in Theorem 2, item 3, belong only to H ' ( F ) for some s which is (possibly substantially) smaller than d + 1, even if the data f and g are piecewise smooth. This is of course due to the edge and vertex singularities induced by the unsmoothness of F. The corresponding reduced convergence rate can be compensated, however, by employing properly graded subdivisions of F rather than the quasi-uniform ones used here. For an analysis of a multiscale Galerkin scheme on such graded meshes on polygons, we refer to [26]. R e m a r k 7: We will use Theorems 1 and 2 to analyze the effect of the quadrature error. We will choose the quadrature in such a way that the matrix after truncation and quadrature still satisfies (3.3.6), (3.3.7), (3.3.10). Since only these inequalities were needed in the proof of Theorem 2, exactly the same statements will also hold for the resulting, fully discrete method with truncation and quadrature.
w
Quadrature error analysis
So far we assumed that the entries A j j, of the compressed stiffness matrix ~L are computed exactly. Except in very special circumstances, however, only approximate values Ajj, that must be obtained by numerical quadrature are available. In this section we present and analyze a quadrature strategy which a) preserves the asymptotic convergence rates of the compressed scheme in Theorem 2 and b) essentially retains the asymptotic complexity (3.3.16) of the compressed scheme. This will be achieved by tensor product Gaussian quadrature formulas of properly selected orders. The case of triangular elements is treated by conical product rules. Our purpose, then, is to determine a family of Gaussian quadrature rules Q j j , to compute approximations A j j , - Q j j , A j j , of the nonvanishing entries 2~jg, of the compressed stiffness matrix .~L, such that (3.3.13)-(3.3.15) are preserved. We will show that this can be done with O(NL (log NL) 3) kernel evaluations in the far-field, i.e.all off-diagonal entries of the compressed stiffness matrix and with O(NL (log NL) 4) kernel evaluations for the singular integrals. The quadrature error analysis will utilize the consistency analysis introduced for the compression error analysis (see Remark 7). The outline of the section is as follows: in Subsection 4.1 we collect some classical derivative-free error estimates for Gaussian quadrature in one dimension and generalize them by a tensor product argument. In Subsection 4.2 we investigate the analyticity of our integrands. Particular attention is paid to the size of the region of analyticity. Subsection 4.3 then discusses ^
310
T yon Petersdorff and Ch. Schwab
the strategy for the numerical evaluation of the regular integrals and its complexity. Subsection 4.4 contains an analysis of the quadrature of the singular integrals arising on the diagonals of the blocks Al,v. Throughout this section we denote by G n f the n-point Gauss-Legendre quadrature applied to f(x) in [-1, 1]. Further, we denote by Ep C C the closed ellipse with foci at z = • and with semi-axis sum p > 1.
4.1
Quadrature error estimates for analytic functions
In this subsection we collect some known error estimates for Gaussian quadrature formulas for analytic functions. They have been used in the analysis of the discretization error for second kind boundary integral equations in [14]. We begin with a classical derivative-free quadrature error estimate in one dimension.
Proposition 5. Let f (x) be analytic in [-1,1] and admit an analytic continuation f (z) into the dosed ellipse s C C with loci at • and semi-axis sum p > 1. Then
[E~f[ =
If,f - G~f[ < C p -2'~ m a x [f(z)[ --
zEOs
'
(4.1.1)
where I f "- fl_ 1 f(x)dx and E n - I - G n is the quadrature error operator. This estimate goes back to Davis, see, e.g.[7, Eqn. (4.6.1.11)]. Higher dimensional analogs of it can be obtained by a tensor product construction.
Proposition 6. Let f E C(~I X ~-~2). Define
If--IlI2f'-~ and let
f(~l,~2)d~ld~2
/~ 1
2
Ni Qig := Z %(i) gtcj ,.(i) ),
i=1,2
j--1
denote quadratures with w~i) > 0 and ~Ji) E ~i, i - 1, 2. Then IEfl
-
1(I - QIQ2) fl
max < lf~llmax I(/2- Q2)f(~l, ")l" F " [f~2l .. e~= [(11-- Q1)f(. ~2)[, --
~1E~=~1
(4.1.2)
Fully Discrete Multiscale B E M
Proof:
311
Let Q - Q iQ2 denote any tensor product formula. We write
(I-
Q) f
-
(11/2 -- Q1Q2) f - (Ix& - I1Q2 + IIQ2 - Q I Q 2 ) f
--
I1 [(/2 -- Q2) :] + Q2 [(11 - Q1) : ]
and estimate
I(I- Q).fl N2
__ Ileal
I(h - Q2) Y(zx, ")1 + ~
max
j=l
X 1~"~ 1
___ I~11 max 1(/2 - Q2)
w~
(I1 - Q1)
y(.,zj )
f(xx, ")1 + If~21 ma~ I(L - Q1) f(',x2)l,
since [ f t 2 [ - )--~j~i wj(2) by our assumption on the positivity of the weights w~2)." From Propositions 5 and 6 we deduce the basic derivative-free quadrature error estimate which we will use below. P r o p o s i t i o n 7. Let B - [-1, 1], f~l - f~2 - B x B , and f ( u , u ' ) " f~l x f~2 --+ JR. Assume further that for every u E ftl, f (u, u') a d m i t s an analytic continuation as a function of U~l or u~2 to Ep2 C @ for some p2 > 1 with
M2"-
[
m a x ] m a x max u e B x B _ u~ eB u~2eOs
[f(u,u')l+
max
u~ eOs
)
m a x l f ( u , u ' ) [ ~ < o o,
u~eB
J
(4.1.3) and that for every u ~ E f~2, f ( u , u ~) a d m i t s an analytic continuation as a function of Ul or u2 to $pl C @ for some pl > 1 with
M1
-
max
u'EBxB
{ma
max
ulEB u2EOs
,,/u,u'/,+
max
UlEOs
u2EB
(4.1.4) Then, for every n l, n2 E IN,
[(I -- an~an~ Gn!a:~) f[ < C {pl 2n' M 1 + p22n2M2}.
(4.1.5)
Proof: We apply Proposition 6 and must therefore estimate the quadrature error for the double integral
and its counterpart with indices "1" and "2" interchanged. Applying Proposition 6 once more to each of these two quadrature errors, we get
312
T. yon Petersdorff and Ch. Schwab
that [Ell
S
max u~ EB
21f~11 max
uEBxB
+ max
fB
%EB
+
2[f~2[ max
max
u~EBxB
s f (U; Ul, u~2)du~2 -- Gn~ f (u; ul, .)
ulEB
f (u; ul, u2)du 1 - G ul f (u;.,
'
'
'
~
~'~)
s f(Ut, U2; u')du2 - Gn~ f ( u l , "; u')
+ max fB f(Ul, U2; u')dUl - Gn~ f(., u2; u') u2EB We apply Proposition 5 to each of the four one-dimensional quadrature errors in the above bounds. This yields (4.1.5). 4.2
A n a l y t i c i t y of the kernel in local c o o r d i n a t e s
To apply the error estimate (4.1.5) to the compressed Galerkin scheme, we must investigate the analyticity of the kernel K(x, y) in local coordinates, i.e.the analyticity of Kjj, (u, u') - K(aj -l(u), ~;1 (u')),
1 < j, j' < No
(4.2.1)
(the dependence of/~ on j and j' will not be explicitly indicated when it is clear from the context). Lemma 4. i) For every u E l~~ Kjj, (u, v) is a real analytic function of ve
uo\{u}
if ~;1(?~) E
rL,,
otherwise.
There exists a constant ~/ > 0 depending only on the global shape of F and on the domains of analyticity of the charts {~j } and of the functions s~ in (3.1.8) such that for every u' E blo, u' ~ u, Kjj, (u, v) admits, as a function of v, an analytic continuation (for convenience again denoted by
gjj,(u, ~)) ~o~
~ B(u', r)'- {~ e r
whore0
(u'- ~)T ( u ' - ~ ) < r~)
I
(4.2.2)
9
Fully Discrete Multiscale BEM
313
ii) Conversely, for every u E L/~ Kjj, (v, u) is a real analytic function of V
E
[ [ ~-6
if (u) e otherwise
and admits, for every u ~ ~ u E blo, an analytic continuation for v E B(u ~, r) with r as above. Proof: It is sufficient to prove only the first part of the lemma since the second part is completely analogous. We show first that, for given u, u' E H--o, the kernel Kjj, (u, v) is real analytic in B(u', r) n ]R2. By assumption, the charts ~-1 (u) are real analytic functions of u E H0 and bijective. Therefore, with x : ~-1 (u), y : ~;1 (u') and the numerators sa(x, y) in (3.1.8) being analytic on the ranges of the charts ~-1, ~ 1 , the compositions sa(t~;1)(u),~;l(u')) are real analytic for u,u' e Ho. Their domains of analyticity are determined by the domains of analyticity of the charts ~-l(u) and of s~. Next, we consider the analyticity of
. We distinguish several cases. Case i) j : j'. Since the charts (~j)-i . L/--6 __+ Fjj are analytic and bijective, there exists a global constant 7 such that for u,u' E ~ 0 , 1 < j < No. }1/2 Further, for x ~ y the Euclidean distance I x - Yl = ( x - y)T ( x - y) is real analytic in x for fixed y and vice versa. Since the charts (~j)-I are analytic on H -~, for fixed u,u' E H0, u ~ u', the expression i ~ - l ( u ) - ~-l(v)I is an analytic function of v E B(u',r)MIR 2 for r < min{'~, ~,-1 l ~ - l ( u ) - ~-l(u')]} where ~ > 0 depends only on the domains of analyticity of the chart ~-1. The assertion of the lemma follows from the definition (3.1.8) of the kernel after possibly reducing the value of 7. Since Kjj, (u, v) is a composition of real analytic functions with ranges included in the proper domains of analyticity, it admits a complex analytic extension (e.g.via the classic power-series argument) as a function of v E B(u ~, r). Case ii) j ~ j' and dist(Fj,Fj,) >_ ~ > 0. Here the assertion of the theorem is true for all u, u ~ E L/~ since the kernel is nonsingular and real
314
T. yon Petersdorff and Ch. Schwab
analytic in u ~ E/go for every u E U ~ and vice versa. Notice that the value of r in (4.2.2) must possibly be reduced depending on ~. Case iii) j ~ j' and dist(Fj, Fj,) - 0. This is the case when two surface pieces are adjacent, i.e.Fj n Fj, is either a line or a vertex. Fix u, u ~ E/go such that
-
I> o m
Then it follows as in case i) that for fixed u E/go the function Kjj, (u, v) is analytic in v E B(u~,r) where r is as in (4.2.2) and vice versa. The numerical quadrature rules Q j j , are constructed on the reference domain lg ~ i.e. the unit square S - ( - 1 , 1) 2 or the unit triangle T - {(Xl,X2) 9 - 1 < xl < 1 , - 1 < x2 < Xl}. To this end, the kernel K j j , ( u , u ' ) is mapped from/g~ x/g~', to the reference domain/go via the affine transformations ~
T~'U~ ----+U ~ Denote by
K j j, (U, U')"-- g j j , ((T 1 ) - l ( u ) , (T/:)--1 (Ut))
(4.2.3)
the transported kernel and let
djg, "- dist (r(j), r ( j ' ) ) ,
r ( j ) . - (~ o 7-/) -1 (u 0)
(4.2.4)
denote the Euclidean distance of the images F ( J ) , F ( J ~) of/go under the respective coordinate transformations. We consider first the case where U ~ - S, i.e.we have quadrilateral elements. ~
L e m m a 5. Assume that djj, )> 0 and that/go = ( - 1 , 1) 2. Then there exists a constant ~/ > 0 which depends only on the kernel, the boundary F and its parametrization such that (i) t'or every ul, u2, u~ E [-1, 1], Kgg,(u, u') admits an analytic extension to U~l E s with it ~ p' - 1 + 72 d gj, (4.2.5)
and max max K j j , (u, u') < M~ Ul.U2.Ul2E[--1.1] UllEep,
j,
(4.2.6)
Analogously, for every Ul, U2, U~ E [--1, 1], K j j, (~, U') admits an analytic extension to u~2 E E;, with p' as in (4.2.5) and
I-
I
max max K j j, (u, u') < M / ~I.U2.U~E[--1.11 u~EC.,
j,
(4.2.7)
Fully Discrete Multiscale B E M
315
(ii) Conversely, for every U'l, u'2, u2 E [-1, 1], Kgg,(u, u') admits an analytic extension to Ul E s with
pand max
1 + 721djj,
!
max K g g, (u,
Ull,Ul2,U2e[--1,1 ] Ul eE.
!
) < M~
(4.2.8)
j,
(4.2.9)
Analogously, for every u~, u~, ul E [-1, 1], K j j , ( u , u') admits an analytic extension to u2 E s with p as in (4.2.8) and max
max
u'1,u~,ul E[-1,1] u2es
!-g gg, (U, U') I< M~
j,
.
(4.2.10)
Here the constant M is independent of J and J'. Proof: Assume first that 1 - l' - 0, i.e.T~ is affine, but does not change the area. Then the assertion (i) follows from Lemma 4, since for every point u E U ~ and every u' E U ~ there exists B(u', r) such that K j j , ( u , v) is an analytic function for v E B(u',r). Therefore we can select C2, such that U --6 CC s CC U B(u',r(u,u')). u ' E/d ~
The analyticity of K j j , (u, v) and its homogeneity implies also the bound (4.2.6) for l' - O. The proof of (ii) is analogous. The case l, l' > 0 is then obtained by a scaling argument. We reduce the case that U ~ - T of triangular elements to the case where U ~ - S = ( - 1 , 1) 2 via the degenerate mapping (sometimes also called the "Duffytransformation") u - 9 (~), u' - 9 (~'), with ~ given by
O(~) --
~1 ) --1 -~-(~1 -[- 1)(~2 + 1)/2 "
(4.2 11)
We define in this case the transformed kernel by
~,,. (~. ~'/- ~,. ( ( ~ / l o ~(~/. (~:/-~ o ~(~'/)
(4 ~ 1~/
Note that an application of n x n tensor product Gaussian quadrature to K j j , ( ~ , ~l) in the unit square corresponds to using a conical product rule for K j j , in the unit triangle. To estimate the quadrature error on triangles, we need an analog of Lemma 5 for the transformed kernel (4.2.12).
316
T. von Petersdorff and Ch. Schwab
L e m m a 6. For the kernel Kjj,(~, ~') in (4.2.12) statements (i) and (ii) of Lemma 5 remain true/'or Kjj,(~, ~'), with possibly different, but absolute (i.e.independent of J, J') constants 7 and M in (4.2.5)-(4.2.10). Proof: As in the proof of Lemma 5, we first consider l - l' = 0. As before, we obtain from Lemma 4 the analyticity of the kernel Kgg,(u, u I) defined in (4.2.3) on T x T. It therefore remains to show the following: if f(u) is analytic in T then f(r is an analytic function of ~ E S. When f is analytic in T, there exists Ro > 0 such that
f(u) -
~_, ~1 f(a) (~0)(u- Uo)" V u 0 E T , V [ U - U o [ < R o . c~EIN~
(4.2.13)
We show that (f o ~)(~) can be expanded into a convergent power series about any point ~0 E T for which uo - r To this end, elementary estimates show that I= - uol ~ = I ~ ( ~ ) - ~ ( { o ) 1 2
< 3 I{ - {ol ~
for ~,~o E T. This shows that inserting u - &(~) into (4.2.13) yields a power series for f(r which converges for sufficiently small [ ~ - ~olNow the assertion follows for l = l' = 0 as in the proof of Lemma 5. For l, l' > 0, we use once again a scaling argument. Lemmas 5 and 6 will be used when the surface is subdivided either exclusively into quadrilaterals or exclusively into triangles. The arguments in the proof can, however, also be used in the case of mixed partitions of F consisting of both quadrilaterals and triangles. Corollary 1. Statements (i) and (ii) of Lemma 5 remain valid for the ker-
nels and
o
4.3
Q u a d r a t u r e for t h e n o n s i n g u l a r i n t e g r a l s
We analyze the quadrature for the nonsingular integrals, i.e.for those entries
A~j, - fr fr K(x, y)r162
(y)dszdsy
(4.3.1)
of the stiffness matrix for which dist (suppCj, suppCg,) > 0
(4.3.2)
317
Fully Discrete Multiscale B E M
(note that (4.3.2) implies (cOg, Cg, > - 0). According to (3.2.4), every Cg(x) is a scaled and transported copy of some r a piecewise polynomial basis function of 12r 1. As indicated in Remark 7, we will determine the number of Gauss points in such a way that the consistency estimates (3.3.6) and (3.3.7) still hold for each block. This and corresponding estimates for the singular integrals in the next section imply, via Theorems 1 and 2, optimal (up to logarithmic terms) convergence rates of the computed solution (i.e., with compression and quadrature) in the boundary energy norms as well as at interior points of the domain. Finally, we estimate the complexity of evaluating the nonsingular integrals. The numerical evaluation of the singular integrals is the topic of the following section. Basic error estimate. On the reference element, the multiwavelets Cg(x) are piecewise polynomial functions. This allows to derive quadrature error estimates from corresponding results for polynomial density functions. Lemma 7. Let Lt~ - ( - 1 , 1 ) 2. Let A,A' E IN0 and J = ( j , A , k , u ) ~_ fix, J' - (j', A', k', v') E fix' be such that dgg, > O. Let further
E
0_<max{ C~l,oc2} <_d
co vO,
E
ovO
0_<max{ C~l,c~2} _
be polynomials of separate degrees d and d ~, respectively. Denote by l a ~ j ( x ) / a x l -~ the surface d e m e n t at x = n-~l(v) and define
Then the following quadrature error estimate holds:
lds, l -
(4.3.3)
(4.3.4)
Here !!~11- EO<max{c~l,c~2}
318
T. yon Petersdorff and Ch. Schwab
that A = A' = 0. Due to the analyticity of the charts ~j, the surface element Ids'l is analytic in v~ 6 [-1, 1] and thus admits an analytic extension to v~ 6 gp2 for some p2 > 1. Since #(u') is, for fixed u~ 6 [-1,1], a polynomial in u~ of degree _< d, it is sufficient to refer to Lemma 5 to show the analyticity of fjg,(u, u') in (4.3.3) as a function of u~. The analyticity of f j j, in the remaining three variables is seen in the same fashion. This completes the proof for the case A = A' = 0. The case that A > 0 and/or A' > 0 is deduced from the previous one by a scaling argument, keeping in mind the growth of 7r and of # for large complex arguments and the bounds for M1 and for M2 derived in Lemma 5. The following lemma presents a corresponding estimate for triangles. L e m m a 8 . L e t U ~ = T = { ( X l , X 2 ) ' - I < Xl < 1 , - 1 < x2 < Xl}. With the notations as in Lemma 7, det~ne
Lg,(v, v ' ) : = fgj,(O(v), r
+ 1)(v[ + 1).
(4.3.5)
Let
~(u)-
Z
o
~.u". ~(u)-
~
0
~.u~, u eU ~
be polynomials of total degrees d, respectively d' on bl~ Then the following quadrature error estimate holds:
(4.3.6) <_ 02 -~x-~' I1~11ii~11dj~, {p-~.,+d+l, + p~-~.=+~,+,} denote, as before,II=II-~ I~I and II~IIi, defined analogously. The constant C in (4.3.6)is independent of nl, n2, J, and J'.
where we
Proof: We proceed exactly as in the proof of Lemma 7, but use Lemma 6 in addition to obtain the analyticity of fgg, as a function of vi, v i' after applying the transformation ~. It remains to verify the bounds for Mi and pi corresponding to fjg, (v, v'). The bounds for the kernel follow from Lemma 6 whereas the analytic extensions of the surface elements ds, are bounded independently of J, J' by a scaling argument. It remains to consider the growth of 7r(~(v)), #(r for complex v, v' with large absolute values. To this end observe that
.(a(~)) -
~
e.v"
O
for certain coefficients ~a, since 7r(u) is a polynomial of total degree d. Thus 7r(O(v)) is, for fixed Iv1[ _< 1, growing as O(]v2[d) for large Iv2[ and
Fully Discrete Multiscale BEM
319
vice versa. The polynomial #((~(v)) is discussed analogously. Taking into account the Jacobians Idsxl composed with the affine maps T~ and the additional linear term (Vl + 1)(v~ + 1) stemming from the Jacobians of (I)(v) and (I)(v'), the assertion follows. Q u a d r a t u r e strategy. Based on Lemmas 7 and 8 we are now in a position to estimate the quadrature error for those entries Ajj, of the compressed stiffness matrix 2~L for which the integrand is nonsingular, i.e.for which the distance dgg, between the respective boundary elements defined in (4.2.4) is positive. The singular case will be discussed in the next section. To define the quadratures, we recall Remark 2, i.e. that the multiwavelets Cg are, in local coordinates, piecewise polynomial functions. Consequently, we apply on each of the four pieces of supp Cj a tensor product Gaussian quadrature rule, or a conical product rule if the boundary elements are triangular. Throughout, we denote by n l the number of Gauss points used in the quadrature over F(J) and by n2 the number of Gauss points used for the quadrature over F(J~). In either case, 0. Quadrature error we denote the resulting quadrature rule by Q'~IQn estimates will be obtained by applying Lemmas 7 and 8 to each piece. We saw in Lemmas 7 and 8 that Gaussian quadratures exhibit exponential convergence, provided the integrands can be extended analytically to Cp D [-1, 1]. The convergence rate p is, by Lemma 5, basically determined by djg,/max{diam(F(J)), diam(F(J'))}. If p > 1 were independent of J, J~, uniform exponential convergence would result. When all boundary elements F(J) are of approximately the same size (quasi-uniform partitions), this can indeed be shown. For the multiscale discretizations under consideration here, however, the quantity djg,/max{diam(F(J)),diam(F(J'))} can become arbitrarily small, for example in the block ALo, and hence p arbitrarily close to 1. The resulting degeneracy in the convergence rates Pi in (4.3.4), (4.3.6) can be compensated by subdivision of the larger element, as we show in Lemma 11 below. Let us first introduce the notation (d) - ( d if F(J) is quadrilateral, ( d + 1 if r(J) is triangular. This is convenient since the quadrature error bounds (4.3.4) and can both be expressed in the form +(d) .~_ P2-2n2+
(4.3.7)
(4.3.6) (4.3.8)
We consider first the case where the distance between boundary elements r ( J ) and r ( J ' ) is larger than max{diam(r(J)), d i a m ( r ( J ' ) ) } .
T. yon Petersdorff and Ch. Schwab
320
L e m m a 9. Let J : (j, l, k, u) and J' : (j', l', k', u') with l > l' be such that djg, _ "),-12-1' (4.3.9) where dgj,^ is as in (4.2.4), and let fjg, (u, u') be as in (4.3.3). Let approximations A j j , to all nonzero entries A j j , of All, for which (4.3.9) holds be computed by ^ A j j, - Q,u' Qn2gjj, (u, u') (4.3.10) where
g j j, (u, u') =
/jj,(u,u') if r(j),r(j') quadrilateral, /jj,(r if r(J) triangular and r(j') quadrilateral, / j j , (u, r if r(J) quadrilateral and r(J') triangular, r if r(J),r(J') triangular, /jj,(r
and the product Gaussian quadratures are applied to each of the four pieces Cg and (bj, are polynomials in local coordinates. on which the multiwavelets ^ We denote by Au, the block of elements obtained from numerical quadrature. Assume further that the nonzero entries of Au, for which (4.3.9) is violated are computed exactly. Then there hold the block consistency estimates Ilfikl,l' -- ~kl,l'][ ~__C2-(d+l)(2L-l'-l)21'-I (4.3.11) oo
[[fikl,l, - fikl,l, ][1 ~_ C2-(d-J-l)(2L-l'-l)21-1',
(4.3.12)
provided we select the number of Gauss points in (4.3.10) according to nl >__9(d, J, J'),
n2 >_ 9(4, J', J),
(4.3.13)
where V~(d, J, J') - ~
+
2 log 2 "7 + l - l '
+ (d + 2)(2L - l - l ' )
2 log2 (1 + .),2t+lt~jj,)
Proof: We apply Lemmas 7 and 8 with ,k = l, ,V = l', 7r - C j , and # - C j, to each of the four pieces on which Cj and C j, are, in local coordinates, polynomials. Due to the way the Cg are normalized, we have
Ii~]!- 0(2*3,
I!~11--0(2*'3,
Hence we have, for J, J' such that (4.3.9) holds, the error estimate
A j j,
-- 2~jj,
]
{(1
~ )--2n2-~-(d)}
+ (1 + .),21'dj j,
(4.3.14)
Fully Discrete Multiscale B E M
321
Consider first (4.3.11). We have here a trial wavelet ~/)J, and a test wavelet Cg of levels I and l', respectively, which satisfy, according to (3.3.5), dgg, ~ 5ll' <_ 2 -(L-l-t') by (3.3.10) with the "worst case" assumption a - & = 1. Since the size of supp(r is O(2 -2t') and (3.3.10) selects a neighborhood of size 0(22(L-I-t')) of "active" C j, for each C j, we deduce that each row of the compressed block At,l, contains at most 0(22(L-t)) nonzero entries Ajj, to be numerically integrated. To achieve (4.3.11), we require therefore that the quadrature approximations A j j, of each of these A j j , satisfy ^
A j j, - A j j, ! ~ C2-(d+l)(2L-l-l')21'-12-2(L-l)
-- C2-(d+2)(2L-l-l')"
Next, consider (4.3.12). By the same argment as above, a column of -a-l,t, contains asymptotically at most O(2 e(L-t')) nonzero entries A j j,. To guarantee the error bound (4.3.12), we require for the quadrature approximations A j j, of these nonzero A j j, that ^
IA j j ,
- f~jj, I ~- C2-(d+l)(2L-l-l')2l-l'2-2(L-l')
= C2-(d+2)(2L-l-l')"
Now we reason as above. We determine lower bounds on the quadrature orders n l, n2 that are sufficient for (4.3.11) and (4.3.12). For nl, we get from (4.3.14) that 1 + ")'2ldjj, ) -2nl +(d)
~
2-(d+2)(2L-l-l')21+l' ~2jj,
=
2-(d+2)(2L-l-l')21+l'(,),2l~jj,) 2 , . ) , - 2 2 - 2 l "
Taking log 2 on both sides and simplifying, we obtain nl
> .'--_....L__
2
1+
2 log 2 7 + l - l' + (d + 2)(2L - l - l') 2 1 o g 2 ( l + 7 2 1 d j j ,)
This implies the asserted bound for nl. The bound for n2 is obtained completely analogously, with 1 and l' interchanged. Lemma 9 addresses only the case that (4.3.9) holds. As explained above, however, many elements in off-diagonal blocks as e.g.ALo are such that (4.3.9) does not hold. This will be overcome by binary subdivision of the larger of the two panels F(J) and F(J') until (4.3.9) is satisfied for all subelements. We assume w.l.o.g that 1 > l' and that diam(F(J')) >_ diam(F(J)). The following lemma shows how to construct the required subdivision of F(J').
322
T. yon Petersdorff and Ch. Schwab
L e m m a 10. Assume diam(F(J')) _> diam(F(J)), (4.3.2)and that (4.3.9) does not hold, i.e., that 0 < djg, < ~'-12 -t'.
(4.3.15)
Then there exists a binary partition {r(j) 9J E A(J, J')} of r(J') such that d j j >__~-12-~ for all J e A(J, J'). Moreover, the number of sube/ements F(J) C r(j') is bounded by Ih( J, J')l <-- C('~')(1 - l').
(4.3.16)
Proof: The triangulation of F(J') is given by A(J, J') = :7:g(J') where the function $'j is recursively defined as follows:
7j(J) .- { {2}
Uk=l 7,(Jk)
if d j j >__~,-12-[ otherwise,
(4.3.17)
where Jk, k = 1 , . . . , 4 denote the indices (j,[ + 1, k,~) of the four subtriangles of triangle J. This means that triangles are divided until their diameter is smaller than their distance to F(J). One verifies that the above recursion terminates due to the assumption dgj, > O. The bound (4.3.16) is evident in the case that the surface piece Fj is planar, e.g.in the case of a polyhedron where C(~/) is bounded by 12. For general curved boundaries it might be larger, depending on the curvature present, but is always bounded independently of 1 and l ~. The smallest subelements F(J) generated by the above procedure will belong to level l* = 0(-log2(dgg,)), i.e.they result from at most O(l* - / ' ) bisection steps applied to F(J'). R e m a r k 8: Evidently, all subdivisions can be performed in the parameter domain L/~ where each subelement can be easily identified and stored. R e m a r k 9: If 1 = / ' , i.e.for the diagonal blocks, no subdivisions are needed to ensure asymptotically, i.e.as L --+ co, an optimal convergence rate of the fully discrete scheme. In practice, however, the decision whether or not to subdivide the regions of integration should be based on the geometric distance to the singularity versus the element diameter. In this way greater robustness for "irregular geometries is achieved. L e m m a 11. Let J = (j, l, k, u) and J' - (j', l', k', u') with l > l' be such that (4.3.15) holds where dgg, is as in (4.2.4). Let approximations A j j , to the corresponding nonzero entries A j j, o[ Au, be computed by the composite quadrature rule ^
"~JJ' --
Z QuTM Qun~gjj(u, ul), JEA(J,J~)
323
Fully Discrete Multiscale B E M
where the integrand g j J, (I$, U') is as in (4.3.10) and the quadrature orders satisfy
2 log2(/- l') + ((d} + 2 ) ( 2 L - l - l ' ) + l ' - l + ( 1 - l ' - 1)(d)
nl(J,J',D>
2 1 o g 2 ( 1 + 7 2 '-i+l) (4.3.18)
and
n2(J, J' l-) > 2 log2(/-l') + ((d} + 2)(2L - l - l') + l ' - l '
-
2log
2(1+23,)
2 (d) "
(4.3.19)
Let the remaining nonzero entries of Al,l, be computed exactly. Then the consistency estimates (4.3.11) and (4.3.12) hold.
Proof" We apply Lemmas 7 and 8 to the quadrature errors for each of the pairs J, J, J E A(J, J~), which results in the error bound B j j - C2"-'-2~'22i { (1 + ~/21+1-[) -2nl 2 (l-l-D(d} nt- (1 +
27)-2'~2 (d)}.
^
Then the quadrature error for the matrix element A j j, can be estimated by JEA(J,J')
As in the proof of Lemma 9, we require A j j, - f l j j , ] ~ C2 -(d+2)(2L-l-l')
(4.3.20)
For fixed k E [l', l], the number of those j - (j, [, k, u) E h(J, J') which satisfy 1 - k is bounded independently of I and 1~ (see Lemma 10). Therefore (4.3.20) holds if Bjj JEA(J,J') l
2 (l-[-1)(d) + (1 + 2"),)-2'~2 2 (d) 1" l'=l'-t-1
This holds if ,
2(/_i_l)(d ) < ~ 1 2_(d+2)(2L_l_l, ) - (1- b
T. yon Petersdorff and Ch. Schwab
324 and if 2l'-~ (1 + 2-),)-2'~= 2 (d) < -
1 ~2-(d+2)(2L-l-l') (1-
This implies (4.3.18) and (4.3.19). Consistency error of the nonsingular quadrature. The preceding error estimates show that the numerically integrated, compressed stiffness matrix 2~L preserves the asymptotic convergence of the full Galerkin scheme (up to logarithmic factors).
T h e o r e m 3. Assume that all nonsingular integrals in the compressed Galerkin stiffness matrix ,~L are computed by product Gaussian quadratures with orders as described in Lemma 9 and Lemma 11, so that the block error estimates (4.3.11) and (4.3.12) hold. Then the corresponding approximation fiL of u satisfies (3.3.13)-(3.3.15). Proof: Observe that (4.3.11) and (4.3.12) imply (3.3.6) and (3.3.7) in the "worst case" a = & = 1 with equality in (3.3.10). Thus Theorems 1 and 2 hold also for the numerically integrated, compressed stiffness matrix ~L (with the singular integrals evaluated exactly). Complexity of the nonsingular quadrature. We estimate the complexity of the numerical evaluation of the nonsingular integrals in the compressed stiffness matrix .~L using the Gaussian quadratures as described in Lemmas 9 and 11, respectively. T h e o r e m 4. The computational complexity for the numerical integration of the O ((NL(logNL) 2) nonvanishing, nonsingular entries of the compressed stiffness matrix ft L with the quadrature strategies in Lemmas 9 and 11, respectively, is bounded by 0 ((NL(log NL) 3) kernel evaluations in local coordinates. The proof is based on careful summation of the number of quadrature points over all degrees of freedom. The details can be found in the appendix. R e m a r k 10: In our asymptotic work estimates we used only certain simplified bounds for the number of quadrature points. In computational practice, however, the full expressions (4.3.13), (4.3.18), and (4.3.19) should be used. Note also that with a less sophisticated selection of nx, n2, one can still ensure the asymptotic convergence rates for the fully discrete scheme, but at the expense of higher powers of log NL in the work estimates.
325
Fully Discrete Multiscale B E M
4.4
Quadrature of the singular integrals
We turn now to the quadrature of the remaining entries of the compressed stiffness matrix ~L, i.e.those Ajj, for which dist(supp C j, supp 1/)j, ) -- 0.
(4.4.1)
Throughout this section, we assume w.l.o.g, that 1 _ l'. We distinguish three basic cases:
a) r(j) c_ r(j'), b) r(j)
and
r(j,)
share an edge, and c)
r(j)
and
r(j,)
share a vertex. T r e a t m e n t of point functionals. Before discussing the singular integrals, we consider the terms (cCg , C j,) (4.4.2) which arise from the point functional c(x)u(x) in (3.1.7). In cases b) and c) we have obviously that (cCj, ~)g,) -- O. Due to I _ l', we have r(j) c_ F ( J ' ) and hence that f (CCj, C j,) -- I C(X)r162 Jr (J)
(x)dsx .
(4.4.3)
We begin our analysis with the observation that ~ o clF~ is an analytic function of u E U ~ = ( - 1 , 1) 2 (we discuss only the quadrilateral case since the error analysis for triangular U ~ is, after using Duffy coordinates, completely analogous). A change of variables in (4.4.3) together with the definition (3.2.4) of the Cg yields
-
2''-'-2
Juo c o (T~ o ~j)-x G G , Ids, Idu.
(4.4.4)
We approximate the double integral with an n-point Gaussian quadrature rule and use the derivative-free error estimate of Proposition 5. T h e o r e m 5. If the double integral (4.4.4) is approximated by an n x n point tensor product Gaussian quadrature with n - n ( d , L , l , l ' ) >_ d +
(d + 1)(2L - l - l ' ) - 2 ( / - l') 2 log 2 (1 + 72 t)
(4.4.5)
the block consistency estimates (4.3.11) and (4.3.12) are preserved and the total quadrature work for the point functionals is bounded by CNL (log NL) 3 integrand evaluations.
T. von Petersdor]] and Ch. Schwab
326 Proof:
Let Sjj, ( u ) . =
o
Idol du.
o
Since r and r are polynomials on/40 and co ~:1, idol are analytic on ~--6, the integrand in (4.4.4) is, for fixed ul 6 (-1, ]), analytically extendable into ~cp 3 [_ 1, 1] with p = 1 + 72 ~ and likewise for u2. The maximum M of the extended integrand on 0~:p is, as before, bounded by Cp 2d. Hence, using Propositions 5 and 6 we get
l( I - G'~IG~2) f JJ' I <--C2"-' (1 + 72') -2n+2d .
(4.4.6)
Analogous error bounds hold also for/2 ~ a triangle, if we map it to a square with (4.2.11), provided d is replaced with {d) then. As explained in Remark 3.3 we want to ensure that the errors caused by (4.4.6) satisfy the estimates (3.3.6) and (3.3.7). To this end observe that in each row/column of a block Al,t, of the compressed stiffness matrix there are at most a finite, fixed number of entries of the form (4.4.2). Therefore the required block consistencies (4.3.11) and (4.3.12) hold if the error bound (4.4.6) is of this order, i.e.if 2t'-I (1 + ~/21) -2n+2d <_ 2-(d+l)(2L-l-l')21-1'. After elementary manipulations we obtain sian quadrature. The work per each entry uations. In block ~kl,t, the diagonal has quadrature work for the point functionals L
l
)2
Wc<_CE 22'E(n(d'L'l'l') l=0
l'=0
(4.4.5) for the order of the Gausof type (4.4.2) is n 2 kernel eval0(22t) entries, hence the total is bounded by t
L
<_CE22'E l=0
[
(L_/,)2 ] d2 + ( l + l o g 27) 2 "
l'=0
Elementary estimates yield the asserted bound for the work. The case F(J) C_F(J'). This case arises on the diagonals of the blocks At,t,. Discarding the point functionals, we must evaluate
Ajj,
lim f
6--+0 j , z , y ) E r ( J ) x r ( J ~ ) I--yl>e
K(x y)r162
(4.4.7)
From the rule for variable substitution in principal value integrals (see, e.g.[19, 33]), we get
Ajj, - 2t+l'-2
cut
co(u)((b~ o T~)((b~, o Ti~,)du
~ (u, u')(~b~ o T~)(~b,, o Tic,)Ids=l l' +2/+/'-2 ~---~0 lim S u'ul)Ebit Xblkl,, Kjj, Ids'l du'du. I~-~,1>~ (4.4.8)
Fully Discrete Multiscale B E M
327
Here the point functional c0(u) is analytic with domain of analyticity independent of 1,1'. Its quadrature can therefore be treated exactly as in Theorem 5. R e m a r k 11: If the charts nj, nj, are local tangential coordinates to the surface, we have co(u) - O. This is the case considered in [10]. We denote henceforth by A j j , the double integral in (4.4.8). R e m a r k 12: For the evaluation of the singular integrals it is sufficient to consider the case where both regions of integration are of the same size. This is evidently so for 1 - l'. If, however, I > l', the larger patch U~', has to be dyadically refined toward U~ r~ U~', with I - l' levels, as in Lemma 10. This yields a series of near singular integrals which can be integrated in the same fashion as described in Lemma 11. The total complexity of these near singular integrations is estimated as in Theorem 4 and amounts to O(NL(logNL) 4) operations. Therefore only (a bounded number of) singular integrals over F(J) C r ( J ' ) , Y - (j',l, k,u') E A(J, J') for which F ( J ) N F ( J ) r 0 remain to be analyzed. For these integrals, however, the sizes of F(J) and F(J) are (asymptotically) equal, i.e. IF(J)I .~ IF(J)I I
2-2/.
!
We will show now that for the singular double integrals (4.4.8) with equal sized domains of integration the singularity can effectively be removed by coordinate transformations. According to Remark 4.4, we may translate/rotate U~ - L/~ to U0l - {u" 0 _< ul _< h - 2 - l , 0 < u2 < ul} so that it remains to evaluate
Bjj-
H(bu, u ' - u ) d u d u '
2 l+l' lim f
(4.4.9)
where we define with p - u' - u, u' - u + p ~
g ( u , p ) - Kjj,(u, u + p)((bv o T~)(U)(r
i I
o ~-i~,)(U+ p)Idsul Ids~+pl .
~
Note that Kjj, and Ids] in H(u, v) are independent of I. The first transformation consists in introducing the relative coordinates [10] p-
u' - u
q-u
u' - p + q
u=q.
(4.4.10)
Elementary manipulations show that the region U0~ x U0l of integration becomes the union of 6 domains in IR4 i.e. 6
Uo,XUo,- U v,,, b-I
V,-
{ (p, q) l Pj e V,,j, qj e V , , j + 2 , j - 1, 2 }
328
T. von Petersdorff and Ch. Schwab
with Pi E V.,i, i = 1, 2 and qi E V.,i+2, i = 1, 2, and Vl,1 = I - h , O] "121,3 = I-P2, h] "~2,1 --" [-h, O]
V2,3 = [-Pl, h]
])3,1 "-- [-h, O] '~3,3 : ~)2 -- Pl, h] V4,1 = [0, hi V4,3 -- [--P2, h - Pl] '~)5,1 - - [0, h]
V5,3 = [0, h - Px]
V6,1 - - [0, h] V6,3 -- [P2 -- Pl, h - Pl]
])1,2 = ])1,4 = ])2,2 -])2,4 = ])3,2 = ])3,4 = ])4,2 = ~)4,4 -"1)5,2 = '])5.4 = V6,2 - - " ]26,4 :
[ - h , Pl] [-P2, ql] [Pl, 0] [-P2, ql 4- Pl - P2] [0, Pl 4- h] [0, ql 4- Pl - P2] [Pl -- h, 0] [--P2, ql] [0, Pl] [0, ql] [Pl, h] [0, ql 4- Pl - P2]
(4.4.11)
where h = 2 -l. Therefore 6
(4.4.12)
B j j - 2t+t' ~-~olimE / . H(q,p)dpdq. .--1 ~>
Next we map { PIP1 E ])tt,l,P2 E ])tt,2 } to L/ol and transform q via p+(h,h) x -p P+ (h,O) r - p + (h, O)T
#=2, #=3, #=4, #=5, P - p + (h,h) T # = 6 ,
~_
~ - (h, h)T + q # - 1 , q-~ #-2,
lS-(h, 0) T + q q q q
#=3, #=4, #=5, #=6.
(4.4.13)
This shows that BJJ
-
2t+l' e~olim/ u~ (H1 (13) -4- H2(/3) 4- H3(/3))dis I~I>~
(4.4.14)
where
HI (p) --
f h-pl ~oql (H(q~R) 4- H(q 4-p~ -p)) dq~ Jo
-/.,~1 -/.o1-,1+,2 :H:~, (h,h) T -/~) + H(~ + (h,h) T -/~,/~-(h, h)T))d~,
H=(~)
1 --P2
-
J0
H3(/~) =
Jp
2 JP2
(,-,
(,,, o)T_
+ (,,,
(,,, (4.4.15)
Fully Discrete Multiscale B E M
329
The functions Hi(/3) have singularities in the corners of//0~. The purpose of the transformations was to render these singularities weakly singular, i.e.the leading singularity is cancelled out. This is a consequence of the following lemma. L e m m a 12. There exists kjj, (u, v), analytic in u E ld ~ for every fixed v ~ O, and homogeneous of degree - 2 in Ivl such that
H (u, v) - kjj, (u, v) + njj, (u, v).
(4.4.16)
Here kjj, is antisymmetric with respect to v, i.e. -
-kx(u,-v),
v y~ 0
(4.4.17)
and the remainder Rjj, (u, v) corresponds to a weakly singular boundary integral operator. In particular, Rjj, (u, v) iS analytic in u E bl ~ for every fixed v ys O, and analytic in v ys 0 for every u E ld ~ and it satisfies the
estimate IRjj, ('l~, v)] ~ C Iv] -1 .
Proof: The decomposition (4.4.16) is a consequence of Taylor expansions of the smooth parts of H(u, v) about v - 0 and of the pseudo-homogeneity of the kernel K ( x , y ) in local coordinates (see [33]). The antisymmetry (4.4.17) of kjj, follows from g ( x , y ) = Ko(x,y) + E ( x , y , x - y) with K: as in (3.1.8). 13: The antisymmetry (4.4.17) implies in particular the TricomiGiraud-Mikhlin condition
Remark
I=l
kjj, (u, v) - 0.
Notice however that (4.4.17) is actually a stronger condition which is, nevertheless, always satisfied for zero order boundary integral operators arising in the boundary reduction of second order elliptic PDEs, as was shown in [15]. The idea is now that due to the antisymmetry (4.4.17) the integrand H(u, v) in the definition (4.4.15) can be replaced by Rjj, (u, v). Hence the Hi (/3) are only weakly singular at the vertices of U0l . After the transformations (4.4.10) and (4.4.13) the integral is ready for numerical quadrature. For convenience of exposition, we will state the detailed result only for H1 (/3) in (4.4.14); the other two cases are treated in exactly the same fashion.
T. yon Petersdorff and Ch. Schwab
330
T h e o r e m 6. Let (I)(~) - (~1,~1~2) n- "H0~ --+ (0, h) • (0,1) denote the Duffy transformation. Then
B jl
_ 21+t'
f (~1, ~2; (~1,(~2)d02d~lld(2d(l
~ HI (i5)dis -
where the integrand f -- ~1 {H(r
(~(~)) + H(r + q)(~),-(~(~))}
is independent of l and l' and can be analytically extended in each variable past the region of integration. Let G~: G~: Gq: Gq~ f denote the quadrature approximation to B j1j by properly scaled Gaussian quadrature formulas with (d + 1)(2L - l - l ' ) l' nl >_ d + 2 ( / + 1) I l + 1' (4.4.18) respectively n2 _ d +
(d + 1 ) ( 2 L - l - l ' ) 2 log 2 5
l' { log 2 5'
(4.4.19)
nodes where 5 E (1, 4.62) depends only on the regularity of the chart aj (for regular charts it is uniformly bounded away from 1; a precise value was obtained for genera/surfaces in [34]). Then the block consistency estimates (4.3.11) and (4.3.12) are preserved and the total quadrature work for the singular integrals (4.4.9) is bounded by CNL (log NL)2 kernel evaluations. Proof: The function H({,i5) + H({ + i5,-i5) is analytic for i5 ~ 0. It remains therefore to investigate the situation at i 5 - 0. We write H({,iS) + H({ + i5,-i5) - H({,iS) + H(O,-i5) + i5" (01H)(r where r
~),-i5),
~) is analytic. By Lemma 12, this yields
H({,/5) + H(Ct + ~,-~) - Rjj,(?t,)) + Rjj,(?I,-~) + ~. (01H)(~(i5, ~),-i5). Thus the integrand is weakly singular at i5 - 0. The analyticity of the integrand f with respect to { follows from that of the kernel Rjj, of Lemma 12 and the charts, whereas the analyticity with respect to ~ can be shown as in [34], Theorem 1. Next we estimate the quadrature error using a tensor product argument as in the proof of Proposition 7. We use, however, for the nl-point quadrature error the estimate
oh g(x)Zrp(x/h)dx - G nl gZrp( . /h) <_ Ch 2hi+l-p,
(4.4.20)
Fully Discrete Multiscale BEM
331
where 7rp is a polynomial of degree at most p independent of h and g E C 2nl ([0, h]) independent of h. For the n2-point quadrature we use, as before, Proposition 5. This yields the error bound
Here C = C(nl, d) is independent of 1,1', and n2, and 5 > 1 depends only on the domains of analyticity of nj and of sa in (3.1.8). From the analysis for the consistency error due to the matrix compression, we must have (see the proof of Theorem 5) E < C2-(d+l)(2L-l-l')21-1'. m
This gives the asserted quadrature orders (4.4.18), (4.4.19). The work estimate is obtained as follows.
<_
L
l
/=0
/'=0
<_ cZ2
lz
l(
1+ l + l
(l+(2L-1))
l~=O
/=0 L
<
C~22'
{2L1 + (2L-1)4/(1 + 1) 2 }
/=0 L
<
CL~22'l+Cn
L
4 ~ 2 2 ' ( 1 + 1 ) -2 ,
/=0
/--0
where we used (6.1) with g - 1 and k - 2. F(J) and F(J') share an edge. By Remark 12 it is once again sufficient to consider r(j) and F(J) C_ r(j') belonging to the same level l (i.e.1 - D. By translation and rotation of L/lk and U~ this case can be further reduced to
Bjj
-
2 t+l'
ul
H(u, u' - u)dudu',
(4.4.21)
where h - 2 -l and the integrand function H(u, u ' - u ) defined in (4.4.9) is pseudo-homogeneous of degree - 2 with respect to the second argument. The key to the regularization is again an appropriate variable transformation which we present in the following lemma.
T. yon Petersdorff and Ch. Schwab
332
Lemma 13. There holds 0
H(ul, u2; Ul, - Ul, u2, - u2 )dul2 duldu2dul ' _
{~
~01 ~01 ~0h ~2
h [H (r - ~r](1 - 8), ~r]8; ~r](1 - 8), - ~ ( 1 + 770)) + H (r - ~(1 - r/), ~r/; ~(1 - r / ) , - ( ( 1 + r / - O))] d~"
+
[H (r ~(1 - 0 ) ; - ~ ( 1 - r / ) , - ~ ( 1 + r / - 0)) + H ((2, ~;-~r](1 - 0 ) , - ~ ( 1 + r/0))] dr
)
(4.4.22)
Proof: [30, Chap. 3.3] We m a p U ~ - { u ' ' 0 < u i < h , - u i < u~ < 0} onto L/~ by setting u' = (Wl,-w2)7-. Hence
u'-u-
Wl -- Ul / -(w2+u2)
and
SJJ -
~
H(u, u' - u)dudu' = B j3 + B:3
where
B j1j _
/oh/o 1/o 1/o wl H(ul, u2; Wl -
U l , - ( w 2 + u2))dw2dwldu2dul
and
2 /oh/o l/h/oWl
Bjj
-
H(ul, u2; Wl - u l , - ( w 2 + u2))dw2dwldu2dul.
1
2 yields Setting t~l = Ul - W l in B j1j and ~1 - Wl - Ul in B j3
H(ul, u 2 ; - t ~ l , - ( w 2 B j2j _
+ u2))dw2d~ldu2dul,
h/0 1/o 1/oWl ol H (Wl -- Ul, U2; Ul, --(W2 -t-u2))dw2dwld~ldu2.
In both terms the singular arguments are now independent of Ul (resp. Wl) and the transformed integrand is analytic in these variables. We exchange the order of integration to move these integrations to the innermost
Fully Discrete Multiscale BEM
333
position. Since t h e t r e a t m e n t for each of t h e t e r m s is a n a l o g o u s , we conc e n t r a t e now only on B je j . T h e region D of i n t e g r a t i o n c a n be split i n t o t h r e e s u b d o m a i n s , r e s u l t i n g in
B j2j __ B j2,1 j + B j2,2 j + B j2,3 j with
fob/i~=fo~=-~1K1
B j2,1 j -
B jj2'2
__
"
W2--Ul
(W2~ ?~1~U2; ?~1~--(W2 -~" u2))du2duldw2,
K2 (w2, ill, U2; ill,--(W2 "b
Bjj2'3--~oh//~oh--~1 K2(w2, ill, u2; i l l , - ( w 2 2
~-
u2))du2df~ldw2, u2))du2dfildw2
w h e r e we set
K1 (w2, Ul, u2; ?~1,-(w2 + u2)) -
2
H (Wl -- ?~1, U2; ?~1,--(W2 "+- U2))
dwl
and
fih K2(w2,1~1, u2; ?~1,- (w2-~-?-$2)) -
H (W 1 -- Ul, U2; 1~1,--(W2 -~- U2))
1"~-U2
dwl.
We s u b s t i t u t e variables as follows: In B 2'1"
(100)(w2) (w2) (lO JJ
fix ~2
=
0 0
1 0
in B2'2- 9
JJ
(.)(ol ~21 ~2
a n d in
~1 ~22
=
-1 0
1 0
1 1
fix u2
~
'UX
"--
u2
0 0
1 0
1)(w2) (w2)(1 1
1 1
fix u2
r
fil u2
1 0
=
0 0
--1 1
'I~X
,
~22
-1 1
~x ~22
0 0
72x '52
,
B j2,3. j =
(0 l l)(w2) (w2) (1 0 0 -1
0 1
1 1
fil u2
r
fix u2
=
1 0
-1 1
T h i s yields
Bjy ~,~-
/o~ioO~/o ~ K1 ((v2, ~x -
22 /0~/002/:~
Bjj
=
~2, ~22;~2x - ~2,-(tb2 + ~2))d~2d~xd(v2,
K2 (~b2- ~2x,~b2- ~2, ~22;t~2- ~22,-(~b2-~x
+~2))d~xd~2d(v2
.
T. yon Petersdorff and Ch. Schwab
334 and
B j2,3 j -- ~oh fO~2 ~0 ~2 K2(w2 -~2~ ~)2-~1~ ~tl; ~)2 -~tl, -(~)2 +Ul -u2 ))d~l d~2d~2. Swapping the innermost integrations in B j2,3 j and exchanging the variables j , we can combine B j2,2 j and B j2,3 j into a single term, re?~1 and fi2 in B j2,2 sulting in
2
~oh~olV2~O~1
Applying the triangular coordinates
W2 = ~, ~,I -- ~ , U2 -- ~T]O in the first integral and
in the second integral yields, after backsubstituting K1 and K2, the first two terms in (4.4.22). The substitutions for B~j are exactly the same, with u and w interchanged, and yield the last two terms in (4.4.22). The main point of the transformation (4.4.22) is that for H as in (4.4.9) and kernels as in (3.1.8), the integrand on the right-hand side of (4.4.22) is analytic in all variables of integration. Thus, standard Gaussian quadratures will yield the consistency required by the Galerkin scheme.
Let Theorem 7. Denote by /(wl , ~, r/, 0) the integrand in (4.4.22). G~' G~ 1G~2G~ 2f denote the quadrature approximation to S j j by properly scaled Gaussian quadrature formulas with nl >_ 2d +
(d + 1)(2L - l - l ' ) 2(/+ 1)
l' ~ l + 1'
(4.4.23)
n2 >_ 2d +
(d + 1 ) ( 2 L - l - l ' ) 2 log 2 ~
l' ~ log2 ~'
(4.4.24)
respectively
nodes where ~ is as in Theorem 6. Then the block consistency estimates (4.3.11) and (4.3.12) are preserved and the total quadrature work for the singular integrals (4.4.21) is bounded by CNL (log NL) 3 kernel evaluations.
Fully Discrete Multiscale BEM
335
Proof: From the definition of H in (4.4.9) we see that [dsu[, [dsu,[ are, in the (r ~, 77,O) coordinates, (piecewise) analytic functions with domains of analyticity independent of I and l ~. The product r162 is, in triangular coordinates, a polynomial of the form E C0~0~1~2r~~ cti_~4d
We claim that the integrand is (piecewise) analytic in the triangular coordinates and independent of the level 1. The independence of 1 follows directly from its definition. The analyticity as a function of (77,0) for ~ ~ 0 follows from H(u; v) being analytic in u and also in v ~ 0, since ~ ( 1 - 0) and 1 + ~0 (resp. 1 - ~ and 1 + 77- 0) do not vanish simultaneously for any (77, 0) E [0, 1]2. This also shows the analyticity with respect to ~ provided ~ 0. The analyticity at ~ = 0 (and thus the uniform analyticity in ~) follows from H(u; v) being pseudo-homogeneous of degree - 2 in v. Using the pseudo-homogeneous expansion of H(u; v) with respect to local polar coordinates in v (see, e.g., [33]), we see that the factor ~2 introduced by the triangular coordinates cancels the singularity of H. This shows that the integrand is regular at ~ = 0. Analyticity at ~ = 0 is then shown as in [34]. To estimate the quadrature error, we use a tensor product argument. For the double integration in (~,~) E (0, h) 2, we use the error estimate (4.4.6) and for the integration in (77,0) E (0, 1) 2, we use Proposition 5. This is analogous to what was done in the proof of Theorem 6. We get the error estimate
S - [Sj~ - G~le~le~2e~2 fl ~ C2 l-~l' (2 -(/-[-1)(2n1-[-1-4d)
-~-(~-2(n2-2d)) .
Matching this error bound with the block consistency estimates yields the lower bounds for quadrature orders n l and n2. The work estimate is obtained as in the proof of Theorem 6, bearing in mind that we now have to sum up 22/(nl)2(rt2) 2 over all blocks (thus the power of 1 + 1 in the denominator is reduced by one resulting in one additional power of L).
r(J) and F(J') share a vertex. The final case where supports contain a common vertex is also treated using a special coordinate transformation similar to [10, 30]. With F(u, u') = H(u, u ' - u ) , the integral takes the form 21-[-l'
_
1=0
21+r
2--0
i "--0
ltl F ( - u l , u2; ul, u~2)du2duldu~2du~1 ~=0
fU h ~sl ~uh ~s"1 F(-ul,UlS;Ul,t ~tiS)UldSdul UldS t ' du '1, 1--0
--0
~----0
~--0
T. von Petersdorff and Ch. Schwab
336
using u2 = UlS, u'2 - u2s'. We move the integrals over Ul, Ui outside and split the integral over (ul, u~) E [0, h] 2 into two triangular parts uh
1=0
fu h
~=0
A(ul,u2)du~dul
A(ul, Ul t)ul dt dul
-
+
1=0
0
~=0
0
d(U'lt, Ul)U'ldtdu'1
using u~ = u l t and Ul - u'lt, respectively. The first term in the sum is
2l+l' ~ h
/~
Ul --0
the
second
0
term
~sl ~sl F(-Ul, Ul s; ult, ul ts)u 3 t ds'ds dt dul, --0 ' --0 is
analogous.
Now
note
that
the
function
u ~ F ( - u l , UlS; ult, ults) is analytic, hence standard Gaussian quadrature can be used. We can now estimate the error as in Theorems 6, 7 using the tensor product argument of Proposition 6. This will yield again lower bounds for n l, the number of nodes for the Ul integration, and for n2, the number of nodes for the u2 integration of the form (4.4.23), (4.4.24), with 2d replaced by 3d. The quadrature work is estimated as follows:
L
l
/=0
/'=0
<_ cZ2 ' /=0
<--
L
CL4E
/'=0
1+ l + l
(1 + ( 2 L - / ) ) 3
22/ -< CL422L"
/=0 Thus we have shown T h e o r e m 8. The singular integral where F(J) and F(J') share a vertex can, after reduction to integration domains of equal size as described in Remark 12, be treated with the coordinate transformations described in [10, Section 2.3]. The product Gaussian integration of the regularized integrands needs, to ensure the block consistency estimates (4.3.11), (4.3.12) and thus the asymptotic convergence rates, O(NL (log NL )4 ) kernel evaluations.
w
Conclusion
A multiwavelet basis for L 2 on curved, piecewise smooth surfaces r C IR3 has been constructed. The multiwavelet families are fully orthogonal and
Fully Discrete Multiscale BEM
337
the construction applies for an arbitrary degree of approximation d E ]No. We have shown that stiffness matrices for corresponding multiscale Galerkin discretizations for a class of strongly elliptic boundary integral operators of order zero on piecewise smooth surfaces in ]R3 can be compressed to O(NL(log NL) 2) nonvanishing entries while retaining essentially (up to logarithmic terms) the full asymptotic convergence rates of the uncompressed scheme, even in negative norms. The location and required accuracy of the nonzero entries in the compressed stiffness matrix can be determined a priori and explicitly, thus bypassing the need to generate the full, dense stiffness matrix prior to compression. Due to our multiwavelets being piecewise polynomial in local coordinates, standard tensor product Gaussian quadrature can be used for the computation of the entries in the compressed stiffness matrix. We have given explicit estimates for the required quadrature orders and shown that the approach of [10, 30] for the quadrature of the singular integrals can be used in a multiscale context as well. The total work for all quadratures was estimated to be O(NL(logNL) 4) kernel evaluations. The condition number of the compressed stiffness matrix was shown to be bounded, so the iterative solution with standard methods, as e.g.Richardson iteration or generalized conjugate gradient methods, can be obtained with an accuracy comparable to the discretization error in O(NL(logNL) 4) operations, provided we start on level 0 and use the approximation on level 1 as an initial guess on level 1 + 1 for l = 0 , . . . , L - 1. The consistency and quadrature error analysis presented here is quite flexible and applies also to Galerkin schemes for strongly elliptic operators of nonzero order provided piecewise polynomial, biorthogonal multiwavelet bases for H+I/2(F) with the appropriate number of vanishing moments are available. For operators of positive order, an analytical regularization of the double integrals ("integration by parts") must be available, yielding weakly singular integrals [21]. We mention further that using families of biorthogonal multiwavelets with the same approximation order, but a higher number of vanishing moments than our family will allow us to remove some more logarithms in our complexity estimates [31]. As long as these families are piecewise polynomial, our quadrature error analysis will still apply. We emphasize in closing that no computational experience with the method presented here has yet been obtained. Although the work estimates are slightly better than corresponding ones for, say, the panel clustering method [9], it must be borne in mind that these estimates are purely asymptotic. They have little predictive value for the performance of an actual implementation of the method which depends on many other factors as well. The implementation of the multiscale scheme analyzed here is under way at present.
T. von Petersdorff and Ch. Schwab
338 w
Appendix
In this appendix, we supply the proof of Theorem 4. Proof: We will utilize the following estimate which follows immediately from integration by parts and asymptotic expansions of the exponential integral function (see, e.g., [23])-
f
x 1
tl
Xtq-I
(logt) k dt <_ C(k, e) (log x)* ~
k,e e IN0
~
x > 1 ~
9
(6.1)
Since the asserted work estimate is asymptotic and since O(.) may depend on the geometry of the domain and on the value of the constants 7 and a in (4.3.13), we assume for our estimation that 7 - a - 1 (this only affects the constants in the work estimate). We will also assume without loss of generality that 1 _ l'. Throughout, C will denote a generic constant which is independent of l , l ' , L but which may depend on other parameters, as e.g., d, g~, a, 7, etc. We consider first the work W1 corresponding to the case where no element is subdivided, i.e.the situation of Lemma 9. Here (4.3.9) and the truncation criteria (3.3.5) and (3.3.10) (with a = & - 1) imply with our assumption that 2 - 1 ' <_ dgj, <_ 2 L - l - l ' . (6.2) The total work for the integrals in block )kt,t, satisfying (6.2) is given by (n2)2
_
Z
~
J,J'
(~(d, J, j,))2 (~(d, J', j))2
2 -I~ < d j j I <_2L - t - I t
kernel evaluations. Now, for fixed J', dgg, - 2-lr where r(i,j) Estimating the sum by an integral we get
<
,-,'
[
2 +
- dist((i,j),{i
- j}).
2(1 + log 2 r)
[d l' - l + (d + 2)(21- l - l') ] 2 x ~ + 2(1 + l' - 1 + log 2 r) rdr. Changing variables ~ = 2~'-l+lr, r - 2l-l'-l~ yields 1/171'l' " 1
< --
C221' f2
<e<2c-~+~
[d + l - l' + (d + 2)(2L - l - l') ] 2 l - it +log 2
x d+
l'-1 + (d+ 2)(2/-l-l')] log 2
J
2
22(l-/' + 1)~dr.
339
Fully Discrete Multiscale B E M
To apply the Cauchy-Schwarz inequality we estimate
w,'," 1 .-
<~<2L--t+I
[,t+
l-l'
+ (d+ 2)(2L-l-/,)]4
J
1 - I' + l o g 2 ~:
~d~.
+ q)4 _< C1(p4 A-q4), we get
Using (p
1
< -
C d 4 2 2 L A- C 2 2 L
a b + log 2
=2
~d~
with a - 1 - l' + (d + 2)(2L - l - l') >_ 0 and b - l - l' >_ 0, independent of ~. Substituting ~ - 2b~, we estimate
2L--/+l(
=2
)4
a
fdf
b + log 2 f
f2b+L-t+l()4a
-
=2~,+~
2_2b~d ~
log 2
_2b+L --I+1
a42 -2b [
~ d~. (log 2 p) 4
J. = i
Using (6.1) with g 2 L-l+l
1, k - 4 and substituting back, we get 4
=2
b + log 2 ~
-
l - l' + (d + 2)(2L - l - l')] 4 22(L-l+1) l-l'
+ L-1
+ 1
]
and using p4 + q4 _< C2(p + q)4 for p, q > 0, we arrive at Wit, l'
<
=
C22t [ d + C 2 2L
2Ll-l'
1 - l' + 1 - l'] +L-l+
rd + L - l ' + l
1
4
J
22(L-l+1)
2(L - l')
l
Further, using again (6.1) we obtain analogously
Wl2,l '
22l/2<e<2~,-'+x [4l ' -+l + ( d + 2 ) d( 2 L -log l - l '2) ~]
C2 2L d + L _ l + l
~d~
"
The total q u a d r a t u r e work for entries which are computed according to L e m m a 9 is therefore bounded by L
l
w, <_cEEwt"' l=O l' =0
T. yon Petersdorff and Ch. Schwab
340
Since W~,l' = W1l' ,l and W~"' _< C ( )W~ 1 / 2t'( ) 1 / 2 W~ t' that
W~'l' ~ C22L
,we obtain first
k,L-l+l]; 2 ( L - !! ~2 (k ,2(L-/-t)L - l ' + l ) 2]
d4+
(L - /)a + (L - /')a
]
Summing over all blocks, the first d4-term results in an O(L2NL) bound. It remains to estimate the second term: L
_<
l
(L -/)4
.jr
(L-/,)4
c22L Z E (L -1 ~+ 1) 2 ( L - l + 1) 2 l=0 V=0
c22L ~-~ (L -1 + 1) 2 /--0
+ (L + 1 - / ) 2 L
--
1':0
(L_/)4
(L + 1 - l ' ) 2 (L + 1 - / , ) 2
L3 + (n+l-/)2
c22LZ(L+l_/)3 1--0 c22LL 3.
Consider next the work W2 for those entries ~Lj, which are computed as specified in Lemma 11, i.e.by subdividing F(J') as described in Lemma 10. We assume that l > 1' (see Remark 9). The quadrature orders are given by Lemma 11 and we assume that ni (k) = ni (J, J', k) are equal to the lower bounds in (4.3.18), (4.3.19). Since by assumption 3' = 1 and 1 > 1~, we estimate <
(d + 2)(2L - l - l ' ) + ( l - / r - 1)d + 2 log2(/- ~:) 2 log2(1 + 2t-~+l) (d + 2)(2L - 1 - l ' ) + ( 1 - k - 1)d + 2 log2(/- k)
<
-d +
-
2
z
2(/-
2(t- t + 1)
k + 1)
where we defined z - (d + 2)(2L - 1 - l') + 2 log2(/- k). Analogously z
n2 (/r _ 2 log 2 3" To estimate the work W2, we consider first T'T"I'I',,2for one block Al,l, with I > l'. There are O(22t') "large" elements r(J') with diameter O(2 -t')
341
Fully Discrete Multiscale B E M
which contribute to this block. The near singular case of Lemma 11 occurs when a "small" element F ( J ) of level / satisfies 2 -4 < d-j j, < 2 -t', i.e.F(J) belongs to the zone M(J',l)
- {x e F" 0 < d i s t ( x , F ( J ' ) ) < 3 . 2
-l'
}.
We partition A 4 ( J ' , l ) into 1 - l' concentric rings A4k -- {x E F" 2 -(k+l) <_ d i s t ( x , F ( J ' ) )
<_ 2-k},
k - l' + 1, ..., 1.
The area of A4k is O(2 - l ' - k ) so that the number of elements F ( J ) of level l and of size 2 -2l belonging to A4k is 0 ( 2 - 1 ' - k / 2 -2t) -- O(22t-t'-k). For each r(j) e Mk, quadrature over r(j') requires subdivision of r(j') as described in Lemma 10 up to level k by Lemma 11. Therefore the quadrature work for one pair F ( J ) C A4k, k - l' + 1, ...,/, and F ( J ' ) is k
k----l' + 1
kernel evaluations. Multiplying by the number of elements F ( J ) C A4k, summing over k - l' + 1, ...,1 and multiplying the sum by the number of F ( J ' ) of level l' yields l
W 2 l'l' _<
C221'
k
E
E
k=l' + l
kml' + l
Setting k - l' + j , , k* - k - l' and using the above simplified bounds for h i ( k ) gives the bound
_j.
<_
j,----0
2
z k*----1
d+ l-l'
+ 1-k*
where z - log 2 j , + (d + 2)(2L - 1 - l'). Changing the order of summation
342
T. von Petersdorff and Ch. Schwab
gives l-l'
/,l
<
,+
l-l'
C22t
z z2-k*( , +
Z)z z), 1 -- l' + 1 -- k* 2
l - l' + 1 - k*
k* =1 j . =k* l-l'
l-l' -k*
k*--i
j--0
C221
2-J
___ c 2 2 ~ ~ 2-~" d + t _ t , _ ~ l _ k ,
2
s2
k*----1 !-ff
52
= C 2 2 ~ 2 -(~-t'+l-k) d+ g k=l
where 5 - log2(/- l') + (d + 2)(2/- 1 - l') is independent of k. Hence we get the bound l-l I
2
k=l
Expanding the square, passing from sums to integrals and using (6.1) with k - 1 and e - 0 yields "2
c22,[
<
52
d+ 1-l'
< C22t [1 + log2(/- l') + (2L - l - l ' ) 1-1'
(log2(/- l') + (2L - l - / , ) ) 2 .
Summing over all blocks gives L
E
-
/,/'=0
L
.,'," _< c E
l=O l'=O
L
/-1
/=0
/'=0
L
l-1
l--O L
/'=0 1-1
l
_< C E 22` ~
(
1+
..2
2L_l_l,)2
l-l'
(l~
(log2(/-l')+(2L-l-/'))2
+ (2L - l -/,)2 +
(2L_l_l,)4
(l - l') 2
<_ C E 22` E (2(L - l) + (l - l')) 4 (l - / , ) - 2 + (2(L - l) + (1 -/,))2 /=0
/'=0
_ C E 22` (L - 1 ) ' E (1 - / , ) - 2 + E (l-/,)2 + ( l - 1)(2(L - l) +/)2 /=0
l'=O
l'=O
.
343
Fully Discrete Multiscale BEM A careful examination of all terms finally gives W2 ~_ CL322L. The proof of Theorem 4 is thus completed. References
[1]
Alpert, B., A class of bases in L 2 for the sparse representation of integral operators, SIAM J. Math. Anal. 24 (1993), 246-262.
[2]
Alpert, B., G. Beylkin, R. Coifman, and V. Rokhlin, Waveletlike bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comp. 14 (1993), 159-184.
[3]
Beylkin, B., R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141-183.
[4]
Costabel, M. and W. L. Wendland, Strongly elliptic boundary integral equations, J. Reine Angew. Math. 372 (1986), 34-63.
[5]
Dahmen, W., S. PrSt~dorf, and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993), 259-335.
[6]
Dahmen, W., S. PrSt~dorf, and R. Schneider, Multiscale Methods for pseudo-differential equations on smooth manifolds, in Wavelet Analysis and its Applications, C. K. Chui, L. Montefusco, L. Puccio (eds.), Academic Press, New York, 1995, pp. 385-424.
[7]
Davis, P. J. and P. Rabinowitz, Methods of Numerical Integration, Academic Press, Boston, 1975.
IS]
Elschner, J., The double layer potential operator over polyhedral domains II: Spline Galerkin methods, Math. Methods Appl. Sci. 15 (1992), 23-37.
[9]
Hackbusch, W. and Z. P. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math. 54 (1989), 463-491.
[10]
Hackbusch, W. and S. A. Sauter, On the efficient use of the Galerkin method to solve Fredholm integral equations, Appl. Math. 38 (1993), 301-322.
[11]
Hildebrandt, S. and E. Wienholtz, Constructive proofs of representation theorems in seperable Hilbert spaces, Comm. Pure Appl. Math. 17 (1964), 369-373.
344
T. yon Petersdorff and Ch. Schwab
[12] HSrmander, L., The Analysis of Linear Partial Differential Operators III, Springer, New York, 1985. [13] Hsiao, G. C. and W. L. Wendland, On a boundary integral method for some exterior problems in elasticity, Proc. Tbiliss. Univ. Math. Mekh. Astronom. 257 (1985), 31-60. [14] Johnson, C. and L. R. Scott, An analysis of quadrature errors in second-kind boundary integral methods, SIAM J. Numer. Anal. 26 (1989), 1356-1382. [15] Kieser, R., []ber einseitige Sprungrelationen und hypersinguliire Operatoren in der Methode der Randelemente, Dissertation, Stuttgart University, 1991. [16] Kumano-go, H., Pseudodifferential Operators, MIT Press, Boston, 1981. [17] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, 2nd edition, Gordon and Breach, New York, 1969. [18] Meyer, Y., Ondelettes et Opdrateurs, vol. 1-2, Hermann, Paris, 1990. [19] Michlin, S. G., MuItidimensional Singular Integral Equations, Pergamon Press, Oxford, 1965. [20] Moritz, H., Advanced Physical Geodesy, Abacus Press, Tunbridge Wells, UK, 1980. [21] Nedelec, J. C., Integral equations with non-integrable kernels, Integral Equations Operator Theory 5 (1982), 562-572. [22] Oswald, P., On function spaces related to finite element approximation theory, Z. Anal. Anwendungen 9 (1990), 43-64. [23] Olver, F. W. J., Asymptotics and Special Fhnctions, Academic Press, New York, 1975. [24] von Petersdorff, T. and C. Schwab, Wavelet approximation for first kind boundary integral equations on polygons, Numer. Math. (1996), to appear. [25] von Petersdorff, T. and C. Schwab, Fully discrete multiscale Galerkin BEM, Report 95-08, Seminar fiir Angewandte Mathematik, ETH Ziirich, Switzerland, September 1995.
Fully Discrete Multiscale BEM
345
[26] von Petersdorff, T. and C. Schwab, Wavelet approximation with mesh refinement for integral equations on polygons, Report 95-10, Seminar fiir Angewandte Mathematik, ETH Ziirich, Switzerland, October 1995, to appear in Z. Angew. Math. Mech. (Proc. ICIAM 95, Hamburg, Germany). [27] von Petersdorff, T., R. Schneider, and C. Schwab, Multiwavelets for second kind integral equations, Technical Note BN-1153, IPST, Univ. Maryland, College Park, August 1994, SIAM J. Numer. Anal., to appear. [28] Rathsfeld, A., A wavelet algorithm for the solution of the double layer potential equation over polygonal boundaries, J. Integral Equations Appl. (1995), to appear. [29] Rokhlin, V., Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60 (1985), 187-207. ..
[30] Sauter, S., Uber die emziente Verwendung des Galerkinverfahrens zur L6sung Fredholmscher Integralgleichungen, Dissertation, University Kiel, 1992. [31] Schneider, R., Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten L6sung grosset vollbesetezter Gleichungssysteme, Habilitation, Technical University Darmstadt, 1995. [32] Schwab, C., Variable Order Composite Quadrature of Singular and Nearly Singular Integrals, Computing 53 (1994), 173-194. [33] Schwab, C. and W. L. Wendland, Kernel properties and representations of boundary integral operators, Math. Nachr. 156 (1992), 187218.
[34] Schwab, C. and W. L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62 (1992), 343- 369.
[35] Schwab, C. and W. L. Wendland, On the extraction technique for boundary integral equations, Technical Report 96-03, Math. Institut A, Stuttgart University, February 1996.
[36] Tran, T., The K-operator and the Galerkin method for strongly elliptic equations on smooth curves: Local estimates, Math. Comp. 64 (1995), 501-513. [37] Wendland, W. L., Strongly elliptic boundary integral equations, in The State of the Art in Numerical Analysis, A. Iserles, M. Powell (eds.), Clarendon Press, Oxford, 1987, pp. 511-561.
346
T. yon Petersdorff and Ch. Schwab
Tobias yon Petersdor~
Department of Mathematics University of Maryland College Park College Park, MD 20742, USA [email protected] Christoph Schwab
Seminar fiir Angewandte Mathematik EidgenSssische Technische Hochschule Ziirich R~mistrasse 101 CH-8092 Ziirich, Switzerland [email protected]
W a v e l e t M u l t i l e v e l Solvers for L i n e a r I l l - P o s e d P r o b l e m s Stabilized by Tikhonov Regularization
Andreas Rieder
Abstract. Additive and multiplicative Schwarz algorithms are described for the efficient iterative solution of a linear system arising from the discretization and Tikhonov regularization of an ill-posed problem. The algorithms and their convergence analyses are presented in an abstract framework which applies, for instance, to integral equations of the first kind discretized either by spline functions or Daubechies wavelets. Numerical experiments are supplied to illustrate the quality of the theoretical results and to compare both Schwarz algorithms and both discretization schemes.
w
Introduction
The mathematical modeling of many technical and physical problems leads to operator equations (1.1) of the first kind, K f -
g,
(1.1)
for the unknown object f with observed data g. We only mention two typical examples: acoustic scattering problems for recovering the shape of a scatterer (see, e.g., Kirsch, Kress, Monk and Zinn [21]) and hyperthermia as an aid to cancer therapy (see, e.g., Kremer and Louis [22]). To fix the mathematical setup, let K be a compact nondegenerate linear operator acting between the (real) Hilbert spaces X and Y. Then it is well known that the problem (1.1) is ill-posed, that is, its minimum norm solution f* does not depend continuously on the right-hand side g. Small perturbations in g cause dramatic changes in f*. Solution techniques for (1.1) M u l t i s c a l e W a v e l e t M e t h o d s for P D E s W o l f g a n g D a h m e n , A n d r e w J. K u r d i l a , a n d P e t e r O s w a l d ( e d s . ) , pp. 3 4 7 - 3 8 0 . C o p y r i g h t (~)1997 by A c a d e m i c P r e s s , Inc. All r i g h t s of r e p r o d u c t i o n i n a n y f o r m r e s e r v e d . ISBN 0-12-200675-5
347
348
A. Rieder
have to take this instability into account (see, e.g., Engl [14], Groetsch [16], and Louis [24]). One of the theoretically best understood and most commonly used techniques for the stable solution of (1.1) is a Tikhonov regularization combined with the method of least squares (see King and Neubauer [20] and Plato and Vainikko [27]). In this, (1.1) is approximated by the finite dimensional normal equation (g;Kt
+ c~I)f: '~ = K?g ~
(1.2)
with a positive regularization parameter a. (Throughout the paper I denotes either the identity operator or the identity matrix of appropriate size.) Here, Kl = KPt where/~ 9X --+ ~ is the orthogonal projection onto a finite dimensional subspace ~ C X. In (1.2), ge is a perturbation of the (exact but unknown) data g caused by noise which can not be avoided in real-life applications due to the specific experiment and due to the limitations of the measuring instrument. The perturbed data g6 are assumed to satisfy IIg-g611Y ~_ -c with an a priori known noise level ~ > 0. In this paper we present additive and multiplicative iterations for the efficient solution of (1.2) based on a multilevel splitting of the underlying approximation space Y~. To this end we will split V~ into orthogonal subspaces of increasing dimension. The number of subspaces involved is called the splitting level and the subspace with the smallest dimension is referred to as the coarsest space. Our approach not only offers all the advantages of multilevel splittings but also yields an asymptotic orthogonality of the splitting spaces with respect to an inner product related to problem (1.2). The latter fact will be essential for the presented convergence analyses. In the case of the additive algorithm we will rely on well-known convergence results for general additive Schwarz type methods (see, e.g., Hackbusch [17], Oswald [26], Xu [34], and Yserentant [35]). To obtain our convergence result for the multiplicative algorithm we can not apply Xu's Fhndamental Theorem II (see [34]) which yields too rough an estimate for the convergence speed. Even the refined analysis of the multiplicative Schwarz iteration presented by Griebel and Oswald [15] will not give our result. We will comment on this in further detail at the end of Subsection 3.3. The outline of this paper is as follows. In the next section we give more details on the regularization of problem (1.1) by the normal equation (1.2). Also in the next section we introduce the multilevel splitting of the approximation space and prove some of its properties. Section 3 is devoted to the additive and multiplicative Schwarz iterations. After a motivation we define and analyze both iterations in an abstract framework. The convergence theorem for the multiplicative algorithm will be proved by a connection between the iteration matrices of the
Wavelet Multilevel Solvers for Ill-Posed Problems
349
additive and the multiplicative iteration. Both Schwarz iterations enjoy the following two qualitatively different convergence results: 1) For a fixed splitting depth, the convergence improves as the discretization step-size decreases (or, what is the same, as the dimension of the approximation space increases). 2) In case the coarsest space is fixed, the convergence rate is independent of the discretization step-size and of the splitting level. Section 3 includes also a representation of the algorithms with respect to wavelet or pre-wavelet splittings of the approximation space. In the remainder of the paper we apply the proposed iterative schemes to integral equations on L2(0, 1). Here, we present two families of test function spaces which satisfy the hypotheses of our abstract theory. These spaces are spline spaces and the spaces of the Daubechies scaling functions on the interval (see Cohen, Daubechies, and Vial [6]). The numerical realization of the methods in this setting is considered next. We show that approximate integration, which will be necessary in a general application of the algorithms, does not deteriorate their convergence behavior. By an analysis of the computational complexities of both methods we find an implementation which has for one iteration step the same order of complexity as a matrix-vector product and which reproduces the increasing convergence speed when the discretization step-size decreases. In the final Subsection 4.4 we supply various numerical experiments. Amongst other things, those experiments on one hand support our theoretical findings and on the other hand demonstrate clearly the limitations of our convergence theory. Wavelets have already been used for the treatment of inverse problems. For instance, we refer to Dicken and Maafl [12], Donoho [13], Liu [23], and to Xia and Nashed [33]. A first study of multilevel algorithms in connection with ill-posed problems was done by King in [19]. A comparison of King's method with the methods presented here can be found in some detail in [29]. Therefore we will not comment on this matter any further in the present paper.
w 2.1
Preliminary considerations
General assumptions and parameter selection
In the sequel we will assume that the sequence {~}t of finite dimensional approximation spaces, which underlies the normal equation (1.2), is expanding, i.e., Vt C ~+1, and that the union t31~ is dense in X. Under
A. Rieder
350 these assumptions the quantity "r, := I l K - K,I[ = [ I K ( I - P,)ll,
(2.1.1)
which will be crucial for the further analysis, satisfies 71+1 _ 71
and
7t -+ 0 as l --+ oc
iff
K is compact,
(2.1.2)
(see, e.g., Groetsch [16]). The operator norm in (2.1.1) is defined by []KI] sup{[[KulJY]U e X, [[u[[x = 1}. An a priori (a = a(l,c)) as well as an a posteriori (a = a(l,e, ge)) choice for the regularization parameter a in (1.2) is established by Plato and Vainikko [27] leading to the convergence of f['a to the minimum norm solution f* as r tends to zero and 1 goes to infinity. Moreover, the resulting convergence rate is optimal in c. For instance, applying the a posteriori parameter selection yields the error estimate
[[f~'~ - f*[[x <_ Cp (v~ + 7~)
(2.1.3)
which holds true under suitable smoothness assumptions on f* [27, Theorem 3.3]. It is our goal to provide efficient multilevel solvers for equation (1.2) combined with one of the above mentioned parameter selection strategies under the general assumption of a fixed noise level c. In this framework a is bounded below by a positive constant a0(r uniformly in the discretization level l, which gives that ~/l <_ V/ao(r
< V~
for l sufficiently large.
(2.1.4)
The above inequality guarantees a high performance of our multilevel solvers as will be discussed in Subsections 3.3 and 4.3. Remark. In general, the algorithms we present can be applied to solve operator equations of the second kind (#I
-
A)f
= g
where A : X --+ X is a compact and symmetric operator and # E lR satisfies # > sup{A [ A is eigenvalue of A}.
2.2
Multilevel splitting of the approximation spaces
The basis of all multilevel algorithms is the decomposition of the approximation space into subspaces. Therefore we define the space Wt as the X-orthogonal complement of V} with respect to the larger space k}+l:
Wavelet Multilevel Solvers for Ill-Posed Problems
351
~/1+1 -- V/ 0 W! where @ denotes the X-orthogonal sum. Consequently, we have the orthogonal multilevel splitting
l--1 V/ -- V/min 0
~
Wj,
/min ~ l - -
(2.2.1)
1.
J ~/min
By Qj we denote the orthogonal projection from X onto Wj. In the next lemma we show that compact operators vanish asymptotically on the complement spaces Wl. L e m m a 1. Let V~ and Wl be the spaces defined above and let K 9X --+ Y be a compact linear operator. Then,
IlK Qt[I
_~ 71 --~ o
as
l - + cxD
where 74 is defined in (2.1.1). Proof: The orthogonality of Vl and Wl gives PIQl = 0. I[g Q~]I - IlK (I - Pt)Ql[[ <_ IlK (I - Pl)l[ = ~/l.
Therefore,
The regularized normal equation (1.2) can be reformulated as a variational problem find f~'~ C Vl:
a(f~ '~, vt) = (K~g ~, vl)x
The bilinear form a : X
x X -+ IR, defined by
for all vt E V~.
a(u, v) := (Ku, K v ) y + a (u, v ) x
(2.2.2)
(2.2.3)
is symmetric and positive definite. The operators .41 = K~KI + aPt and Bl = Q1K*KQI + aQl are associated to a via a(ul, vt) = (Alul, Vt)x for all ul,vt E V1 and a(wl,zl) = (]31wt,z~)x for all wt,zz E Wl, respectively. Later, we will rely on the following strong Cauchy inequality which basically says that the spaces V~ and Wl are not only X-orthogonal but also asymptotically orthogonal with respect to the inner product on X induced by the bilinear form a (2.2.3). The corresponding norm [[ 9[]2 _ a(., .) is called the energy norm on X. T h e o r e m 1. Let Vl and Wm be defined as above and let m >_ 1. The strong Cauchy inequality
l a(vl, wm)[ _< rain{l, 7 m / v / a } [] vl [[a [[ wm [[a
(2.2.4)
holds true for a11 Vl E Vt and for ali Wm E Wm. Further, let j ~ l. Then, ]a(wj,wl)] <_ min{X,Tj/vf-a} m i n { 1 , T l / v ~ } for all wj E Wj and for all wt E W1.
[! wj [[, [[ wt ][,
(2.2.5)
A. Rieder
352
Since v t and wm are orthogonal in X we have that a(vt, Wm) (Kvt, Kwm)y. Further,
Proof:
la(v ,wm)i
=
I(KtA -1/2 At1/2
=
T4"O l~-l/2R1/2Wm)Y I
<_ IIKtA~I/211 ]lA~/2vtllx ligQrnBml/21] ]lB~2wmlIX IIKt.all/21[ Ilvl Ila llgQmll [IBml/2ll IlWm Ila" Using arguments from spectral theory it is easy to verify that ilKt A~ 1/2 [J _ 1 and libra1~211 < a-l~ 2. Thus, (2.2.4) is proved by IIgQm]l <_ 7m (Lemma 1). The second inequality (2.2.5) can be proved analogously. w 3.1
T h e a d d i t i v e and m u l t i p l i c a t i v e Schwarz m e t h o d s
A b s t r a c t f o r m u l a t i o n of t h e i t e r a t i o n s
The philosophy of multilevel methods is to approximate the original large scale problem (in our context: (1.2) resp. (2.2.2)) by auxiliary problems defined on suitable subspaces of the original space (in our context: (2.2.1)). Of course, the auxiliary problems should be solvable at lower costs than the original problem. We refer, e.g., to Oswald [26], Xu [34], and Yserentant [35] for a detailed motivation and consideration of the multilevel concept. Now, we introduce the concept of subspace corrections (see, e.g., Xu [34]). Suppose that we have a given approximation u~,ld to the solution rJt of (1.2). If the residue r~)ld = At u~)ld - K~g ~ is small we are done. Otherwise, we consider the equation .Al el - r~)ld (3.1.1) for the error et " - u ~ l d - f:'a. Instead of the large scale problem (3.1.1) we solve restricted equations with respect to each of the subspaces of the splitting (2.2.1):
]3j ej "Almin elmin
= Qj r~)ld,
=
for /min <_ j _< l -- 1,
(3.1.2)
P/min r~)ld"
We observe that
II jll.
< x- a(wj,wj)< (1 +'/2/oz)oL Ilwjll
(3.1.3)
for all wj E Wj, which is an immediate consequence of Lemma 1. Hence, 13j can be approximated well by a I on Wj and ~ = a-1 Q~ r~,ld
for lmin _< j _< l-- 1
Wavelet Multilevel Solvers for Ill-Posed Problems
353
may be viewed as reasonable approximations to the ej's defined in (3.1.2). Finally we are in a position to define the subspace correction of u~)ld relative to Wj by u~4e7
._
u~ld _ ~ j
_
~ld
Qj ( , A / u ~ l d -
_ O~-1
g;ge)
and relative to V/min by /,new "-- Zt~ ld Vimin "--- e/min
- - Zt~ ld -- ,A - 1 (,Al u~ ld /min / ~ m i n --
Kt*
ge
)"
Starting with an approximation u~ E ~ , the (Jacobi-like) additive Schwarz iteration produces a new iterate by performing the subspace corrections simultaneously, that is,
tt~ +1 ---~ U~
(Atu~ - K[g ~)
p~dd
-- ~/,/min
#-
'
0,1,2,
(3.1.4)
....
Here, u~ E lit is an arbitrary starting guess and C~dd l,lmin =
A~linPtmin ~m - -
l--1
-1
q- a
QJ-
J=/min
The (Gaug-Seidel-like) multiplicative Schwarz iteration is somewhat more involved. Here the subspace corrections are used sequentially to update the actual iterate v~" For # - 0, 1 , 2 , . . .
(3.1.5)
(.Alv~ - K~g e)
V~ +lmin/l
--
V~ -- .AL 1. P / m i n mln
V~ +(j+l)/l
=
V~ -t-ill -- o~-lQj(c41v~ +ill -
g~gS), j
=/min,-..,/-
1.
The multiplicative iteration can also be written in the form V~'~-i -- V~t
['lmult
-- ~"/,/min
(Atv~ - K;g 6)
'
# = 0, 1 2 '
~''"
(3.1.6)
where the action of v/,/minPmUlton an element dt E lit is given recursively as follows: Let /~mult Cmult
~ / m i n ,/min ,
-
-
-
.A-1
/min
and/min < j < 1. For dj E Vj define
J,*mn Vj- Vj by Zj
-
~r,,,.,, j_l,/min
J,lmindj
--
zj - O~
Cmult
--1
Pj-ldj, Q j - 1 (r
-- dj).
354
A. Rieder
For the proof of the equivalence of both formulations (3.1.5) and (3.1.6) we refer to Xu [34, Section 3.4]. As an initial guess for the additive as well as the multiplicative iteration we suggest to take u~ - v~ - - fe'~ /min, the solution of (2.2.2) with respect t o ~/lmin The effort to calculate fe'a is less than 9 /min the computational work for one iteration step. L e m m a 2 Let u~ = f~'~ "
/min"
Then,
II- ? - S;'"llo < V liKii: +
inf
Ill[ '~ - vllx.
vEl/}mi n
Proof: The proof is analogous to the proof of C6a's lemma (see, e.g., Ciarlet [5]). 3.2
Matrix versions of the iterations
In this section we present matrix versions of the iterations (3.1.4) and (3.1.5) given suitable Riesz bases on V~ and Wt. Therefore, we assume that X is a function space over the compact interval [a, b] C lR; X = L 2 (a, b) for example. The results obtained can easily be generalized to multiple tensor products of X with itself. Let ~ E X be a compactly supported function satisfying the refinement or scaling equation M~ - 1
~(z) = ,fff ~
h~ ~(2z - k)
(3.2.1)
k=0
with coefficients hk E IR. In the wavelet terminology, ~ is called scaling function (see, e.g., Chui [2], Daubechies [9] and Louis, MaaB, and Rieder [25]). For the ease of presentation we neglect, at the present time, necessary boundary modifications and suppose that V} = span{~t,k]k = 0 , . . . , n t - 1} C X
(3.2.2)
for all 1 k l* > 0 where ~t,k(x) "- 2 t / 2 ~ ( 2 x - k ) . Further, let there exist an]k = other function r which is called a wavelet, such that Wt = span{r 0 , . . . , m l - 1} and M,r - 1
r
= vff
Z k--0
with coefficients gk E JR.
g~ ~(2z - k)
(3.2.3)
355
Wavelet Multilevel Solvers for Ill-Posed Problems
Since the sum oftwo functions ft - ~-,k dkCPl,k 6 Vl and q, - ~-,k dtkCt,k 6
+I 99/+1,k. Applying Wl is in Vl+l it can be expressed by fl + ql - ~-~kclk
both refinement equations (3.2.1) and (3.2.3) we get the relation clk+1 -- Z h k _ 2 i e i i
l
+ Egk-2jdj j
l
which we write in matrix notation as el + l _ g:+ l el -f" Gt/ + 1 dt
(3.2.4)
9
Clearly, Ht+l : IRn'+l --+ ]R n' and G I + I : I R nl+l --+ IR mr. The solution f['~ of the variational problem (2.2.2) (resp. of the normal equation (1.2)) can be expanded in the basis of I~ as f [ ' a = ~ k ~l,k~OZ,k. The vector ~l 6 IRn~ of the expansion coefficients is the unique solution of the linear system At (t = ~t (3.2.5) where the entries of the positive definite matrix At and of the right-hand side/~l are given by (A1)i,j
=
(K~pt,i,Kcpt,j)y
(/3t)y
-
(g~, g c p t , j ) y .
(3.2.6)
+ a (r
(3.2.7)
The following lemma enables matrix representations of the operators Qj.At, /min < j < -1 _ l - 1 , and A /minP/minA/, which are the building blocks of the Schwarz iterations (3.1.4) and (3.1.5). For a proof see [29]. _
L e m m a 3. Define the restrictions
forj
~'~l,j
"=
g j + l g j ... g l - l g l
61,j
"-
Gj+I Hj
9]R n' --+ IR nj,
... Hi_ 1 HI 9IR TM --~ IR mj
1 -- 1. For vt -- ~ k Ck~l,k 6 Vt we have that Qj,AlVl -
nt -1 E ( ~[,J B ; 1 ~l,j Al cl ) ~l,k, k k--O
where By is the Gramian m a t r i x (Bj)r,s - (r
lmin<_j
nt --1 A-1
lminPlminAlvl -- Z k=O
t A-1 ct (7"~/,/min lminqf~l'lminAl )k
~l,k,
356
A. Rieder
Now, the abstract additive iteration (3.1.4) translated into an iteration acting on (3.2.5) reads Z~ +1 -- Z~
(Atz~-/3t)
t~'dd
--
~/,lmin
# = 0,1 2
~
~
~''"
(3.2.8)
with an arbitrary starting guess z~ E IR'~t and where
Cl'lmin -" ~t~/'/min add
t
A-1 /min qf~l,lmin
1-1 q- Or- 1
E
t B j -1 ~l,j. ~l,j
(3.2.9)
J=/min
The matrix version of the multiplicative algorithm is Z~"1t-1
--
where C m~
[~mult
-- ~/,/min
(Atz~ - /3t)
~
# - 0, 1 2
~ ~'" "~
(3.2.10)
is defined as follows"
/,/min
Let
Z~
/~.mult
= A -1 and/min < j < l For dj E leo j define /min -- " IRnj --+ IRnj by
V/min,/min
mult
,/min
"
wj
--
Ht
,/min dj
__
wj - OL- - 1 G jt B-)-._llGj
Cjmult
Cmult
j_l,lmin Hj dj
(Ajwj
- dj).
In the remark at the end of Section 4.2 we will see that the application of B~-1 to a vector can be realized in a very efficient way. The starting guess due to Lemma 2 is Z~ -- qf~,/min~/min
3.3
-- "]'~/,/mint
A-/miniqfit/,/min]~l.
(3.2.11)
Convergence analysis
The errors with respect to the energy norm of the # - t h iterates u~' and v~ of the iterations (3.1.4) and (3.1.5), respectively, can be estimated by
and
ilu~,_ re,a
_< p llu?
llv~- ~,~
I1o <_ P"rnllv --fl'
with the convergence rates Pa "-[IMadd I l,lminll a and tim "-- l] Mmult /,/mini[ a. Here, Mt da 9X ~ X and M ~"'' 9X ~ X are the iteration operators (er,/min /,/min rot propagation operators) of the Schwarz relaxations (3.1.4) and (3.1.5), respectively, (see, e.g., Hackbusch [17]). Hence, the sequences {u~'}~ and {v~}, converge to g ' ~ if the convergence rates Pa and Pm are smaller than 1, respectively. Furthermore, the convergence is faster the smaller the corresponding convergence rate is.
Wavelet Multilevel Solvers for Ill-Posed Problems
357
Explicit representations of the iteration matrices in question are (see the above cited literature) l--1
I - padd ~'l,lmin .,el - I -- T)/min -- ~
add ,/min ---
(3.3.1)
7j
j=lmin
for the additive variant (3.1.4) and 71-1) (I -- 71-2) . . . . . .
M l m~lt ,/min = ( I -
(I-
"~min) ( I -
T)/min )
(3.3.2)
for the multiplicative one (3.1.5). We set Tj - c~-1 QjAI as well as ~)/min : "A12inPlminfl[l"
Provided a mild decay assumption on 7j (2.1.1), we have the following convergence result for the additive algorithm. T h e o r e m 2. Let rll be an upper bound of 7L (71 <_ rll) satisfying VII __< 771-1 a n d
l-1 ~
7]j < C o lllmin
(3.3.3)
J=lmin
with a positive constant C n which does not depend on 1 or on lmin. Then l[ A/f'add '~'~1,1minli~ < 2 C~ aLmin (1 + fflmin(1 -I- max{Cr/, Crlmin}) )
(3.3.4)
where 6r/min -" T]/min / ~"~. /f a/mi n _< 1 (i.e., /min iS sufliciently large) the inequality (3.3.4) can be reduced to !1-tW ~ l,lmin !1a < _ 2 c o ( c o + 2) O'/min Proof: We roughly sketch the proof. For the details we refer to [29]. It is well known (see, e.g., Griebel and Oswald [15]) that p~ _< m a x { l l r~l, I1 - r=l} where r l and F2 are positive constants such that
rx III vt III2 _< II vt II2 _< r2 III v~ !112
(3.3.5)
holds true for all vt E Vt. Here, the norm II1" LII is the so-called additive Schwarz norm on Vt and in our situation it is given by III vt ill 2 "= a 1-1 II P/rainY/I[ 2 + c~ ~"~J-/min II Qjvt II}. Using the Cauchy inequalities (2.2.4), (2.2.5), and the estimate (3.1.3) one can show that (3.3.5) is satisfied with
and
(Cri-t-l) tTlmin)O'lmin)
F1
-
1/(1 + 2C~(1 +
1"2
--
(1 + 0 .2/min ) (1 -4- 2 C~ O'/min ).
The above theorem has to be interpreted in the following ways:
(3.3.6)
(3.3.7)
358
A. Rieder
1. For a fixed splitting level L = 1 -/min _> 1 and a fixed noise level e > 0 the convergence rate Pa tends to zero. Moreover, p,,
-
p,,(O
-
as l
oo,
(3.3.8)
where the constant in the above O-expression does not depend on 1 and L or on a. 2. In view of our general assumption that the noise level ~ > 0 is fixed, the additive iteration admits a convergence rate Pa which is bounded smaller than 1 uniformly in 1 provided that Z]lmin < V/-~/(2C~7 (Cr/-[- 2)) or ~lmin ( ( ~/a0(r The quantity a0(e) was introduced in (2.1.4). In this case/min depends on r and it increases when r decreases. 3. If/min is fixed independently of ~ and 1 then the additive iteration may even diverge for r too small. Still, we can use the additive iteration as a preconditioner for the conjugate gradient method applied to (1.2). For more details on preconditioning the conjugate gradient method we refer, e.g., to Deuflhard and Hohmann [11] and Hackbusch [17]. R e m a r k . Readers familiar with multilevel preconditioners in the context of elliptic PDEs may wonder why we do not replace A -1 as well as B~-1 /min /min <_ j _< I - 1, in (3.2.9) by the spectrally equivalent identity matrix I. Thus ~ dd would gain a simpler structure but we would lose property W/,lmin (3.3.8). As a consequence the additive scheme (3.2.8) would not be attractive any more as an iteration in its own right. For large scale problems (1.2) the additive iteration is meaningful only if (3.3.8) holds (see also Section 4.3). In other words: the X-orthogonality of the splitting (2.2.1) is a crucial ingredient for our multilevel algorithm and must not be relaxed. In the remainder of this section we will use the estimate (3.3.4) to establish an upper bound for the energy norm of M l,lmin mu't which is of the same type as the right-hand side of (3.3.4). In a first step we show a relation between the iteration operators M l,lmin "dd and M l,lmin" mu't To this end we define Ej
-
(I-
7j-1)(I-
7j-2) . . . . . . ( I -
"~min ) (I -- ~/min ) for /min + 1 < j < l,
E/min
--
(I - ~)/min)'
and
E/mi n-1 - I.
Note that Et = ~/fmult The identity "v~/,lmin 9 I -- ~jfmult
l-1 __ ~lmin "~" ~ j=lmin
~
Ej
(3.3.9)
359
Wavelet Multilevel Solvers for Ill-Posed Problems
follows immediately from the trivial equations E j - 1 - E j - 7 j E j - 1 , / m i n "41 < j < l, and E/min_ 1 - E/min - ~)/min W/min-1 (see Xu [34, Lemma 4.3]). Simple algebraic rearrangements verify Tj E j
-
lmin + 1 <_ j _ l-- 1,
"fjj - 7"j T-j - I E j _ I ,
71min Elmi n
71min -- "~min ~/min"
Substituting the above equations into (3.3.9) we end up with
mult ,/min
I-
--
l-1 ~)/min -- ~ 73j J=/min l-1 "1- % i n ~)lmin "4-
(3.3.1)
_~ ~j "~--1 E j _ l j=lmin+l l--1
Ml add ,/rain -}- 71rain ~)/min -f-
7"j ~ j - l S j - 1 ,
J=/min+l
which implies IMmult a II /,/min I1
_ <
I I ~,f add
I-L"-t/,/min [[a +
II~min'~/min Ila
(3.3.10)
1--1
+
~
IIT~7j-ill,, liE~-llla.
J=/min + 1 An upper bound for the energy norm of M'aa/,/min is already known (3.3.4) Consequently we concentrate on estimates for the remaining terms in (3.3.10). Lemma
4.
The following estimates hold true for/min
1171min ~:)/min Ila < IITj Tj-llla
(1 + 6r/min2) min{ 1, O'/min },
nt"
1 _ j _< 1 - 1:
(3.3.11)
2 2 _< (1 + aj)(1 + aj_l) min{1,aj) min{1,aj_l}, (3.3.12) j-2
[IEj-l[ia
<_
max{1,a~}, H k--/mi n
(3.3.13)
where we set aj = rlj/V~ with vii as in Theorem 2. (The empty product is considered to be 1.)
A. Rieder
360 Proof:
Let vt E ~ be arbitrary. Then,
l177min T)'minV' II2
Q/min,Al T)/mi n Vl)
--
O~-2 a (Q/min ,Al ~)/min Vl,
<
(1 + if2/min )
----
(1 + {Y/min2) a (~)/min Vl ' 71min ~)/min Vl),
-
-
where we used (3.1.3). The strong Cauchy inequality (2.2.4) gives 117~minD,minV, ll~ _<
(1+ O'/mi 2 n ) min{1, O'/mi n } II~'m~nV'lla IlTlmi n ~)lminVl I1~.
Since ilTPlm~nvlll= <_ IIv, ll= (~)lmin is an orthogonal projection in V} with respect to a(., .)), the estimate (3.3.11) is proved. The verification of (3.3.12) follows the same pattern. Again we will use (3.1.3) and (2.2.5)"
IITj ~-lV, II2
a -2 a(QjA, ~-lVl, QjA, ~-lv,)
-
<_ (1 + a~) a(7~_~v,, Ta"~ - l V l ) 2 min{1, aj} min{1, aj_l} _< (1 + aj)
• 115-1v, llo li7~ 7~-1v,11o. We proceed to estimate IlTj-lil~ where we will need that
a -1 <wj,wj}x <_ (1 + a2)
<]~;lwj, Wj>X for all wj
e Wj,
which is a reformulation of (3.1.3). So, we have that la(Tj-lV/,
Vl)l
(Q~-lAlvt, Qy-lAlvl)x
--
OL-1
<
2 (1 -t- ffj-1) (B-fllVj-lAlVt, Qj-lAlVl)X
___ (1+
2 o'j_l)IIB~-J~Qj-~.A, IIo
llv, ll 2a
which together with IIBj_ -1 1Qj-IA,II= - 1 implies that
I17~-111 a <_ 1+~ j=- l "
(3.3.14)
Therefore we have established the desired bound for ]lTj 7j-1 ]la. For the estimate of liEj_lila we claim that
III - 7k]]~ _< max{ 1, a~ }, /min~ k _ l - I.
(3.3.15)
Wavelet Multilevel Solvers for Ill-Posed Problems
361
Because I - 7~ is symmetric in ~ with respect to a(., .) our claim (3.3.15) will follow from
-a~ a(vl, vt) <_ a ( ( I - 7-k)Vt, vt) <_ a(vt, vt)
for all vt E lit.
(3.3.16)
The upper bound is a trivial consequence of a(Tkvt, vt) >_ O. Further,
a(7-kVt, Vt) < 117 vtlla IIv/II
117 11 Ilvtll 2
(3.3.14)
<
(1 + a~) a(vt, vt),
which gives the lower bound in (3.3.16). Taking into account that [ l I ~)/min ]la -- 1, the definition of Ej-1 and (3.3.15) complete the proof of (3.3.13). We summarize our findings in the following convergence theorem. T h e o r e m 3. Let M mu't be the iteration operator (3.3.2) of the multiplica/,/min rive Sclawarz algorithm (3.1.5). Further, adopt the assumptions (3.3.3) of
Theorem 2. Then,
IIMl,~=tin
O'/mi n (2 Cr/ (1 + O'/min (1 + max{ Cn, O'/min )) )
+ 1+ a2
/min "~" Cr/ ff/min
(1 4- cr2
/min
)2
1--3
II
max{l, a ~ ) ) ,
k=/mi n
where ak -- rik / V/-d" h c O'/mi n < 1 (i.e., /min is sufficiently large) the above estimate can be reduced to II~v~/,/mi ]~/~multn IIa (~ 2 ((C o "4- 1) 2 -4- 2C~) ~7/min 9 (3.3 917) Proof: We only have to put the pieces together. The estimates provided in Lemma 4 and Theorem 2 applied to (3.3.10) establish the statements of Theorem 3. The statements 1 and 2 after the proof of Theorem 2 hold correspondingly true for the multiplicative iteration. The asymptotic relation (3.3.8) remains correct when Pa is replaced by Pm, that is, Pm -- Pm(/) -- O ( rlt- L / V/-d )
as I -+ c~o.
(3.3.18)
A few additional comments are in order. By (3.3.10), the quality of our estimate for the convergence rate IIMmu~tt,tmin Ila of (3.1.5) was limited from the very beginning by the convergence rate II"*/,/mi I a/r,dd n Ila of the additive iteration (3.1.4). However, our numerical experiments presented in Section 4.4 (see Table 2 and Figure 8) indicate that the multiplicative algorithm (3.1.5)
362
A. Rieder
converges much faster than its additive counterpart (3.1.5). This is the typical result often encountered in the numerical comparison of the convergence speeds of additive and multiplicative algorithms with respect to nonoverlapping subspace splittings. A thorough explanation for this observation is still missing. Only in special cases one can prove that the multiplicative version is twice as fast as the additive one (see, e.g., Hackbusch [17, Section 11.3]). Even though (3.3.17) does not reflect the correct convergence rate quantitatively, it describes the real situation at least qualitatively correctly in a~/fmult decreases with the the following sense: for a fixed noise level e, Itl...l,lminil same order as r]/min when lmin increases. From this point of view (3.3.17) is more precise than the corresponding estimate which we would get by applying the abstract results of Griebel and Oswald [15]. Their Theorem 4 combined with the related remark yields (with lmin large enough that a/min < 1) 1-
Pm <
-
F1 (1 - a 2 /min
)
)1/2
(1 + log2(2n ) (F2 - F1)/2) 2
with F1 and F2 from (3.3.6) and (3.3.7), respectively. Now, fix the splitting level L. Since F1 - 1 - O(~t_L/V/-~) as l --+ c~, the above estimate implies the asymptotic decay
(r . . . . . . )
which only has half the order of decay of our result (3.3.18). w
A p p l i c a t i o n to integral equations
We demonstrate the performance of our proposed iterative schemes for the solution of integral equations. For simplicity we limit ourselves to problems on the unit interval, but the algorithm can be carried over to problems on Cartesian products of intervals as well. 4.1
Preliminaries
Let K : L2(0, 1) ~ L2(0, 1) be an integral operator with nondegenerate kernel k which is square-integrable over the unit square [0, 1]2. Then, the integral equation of the first kind, 1
K f(.) -
~0
k(.,y) f ( y ) dy - g(.),
(4.1.1)
363
Wavelet Multilevel Solvers for Ill-Posed Problems
is ill-posed (K is a compact operator). To estimate the decay rate of 7t as 1 goes to infinity, we consider scaling function spaces (3.2.2) of order N. This means that the error of the best approximation of f E L2(0, 1) by an element in Vl satisfies (4.1.2)
inf II : - v~llL: < c~ liD ":IIL~ a:
vt C Vt
as long as : e Hr(0,1) for r e { 1 , . . . , N } , where Hr(0,1) stands for the L2-Sobolev space and D r for the (generalized) differential operator of order r. In (4.1.2), 51 = 2 -l is the discretization step size associated with V/ and the positive constant Cr depends only on r. In the next section we give two examples of families of scaling function spaces of arbitrary order. The following lemma is a straightforward consequence of (4.1.2). L e m m a 5. Let {Vl }l be scaling function spaces of order N . Further, let K be the integral operator (4.1.1) and suppose that the composition D r K* is a bounded operator on L2(0, 1) for some r E { 1 , . . . ,N}. Then, (4.1.3)
7l = I1(1- P~)K*II < C~ IID~ K*II a:, where (Jr is as in (4.1.2).
lmin for both multilevel iteraWe proposed the starting guess u ~ - vp - - f~'<~ tions (see Subsection 3.1). Here we can establish a more detailed estimate for the corresponding initial error. Numerical examples demonstrating the quality of the initial guess are presented in Section 4.4 in Table 1. L e m m a 6. We adopt the assumptions of Lemma 5. AdditionMly we assume that D S f * e Hs(O, 1) for some s e IN. Let u~ - f['~. . Then mln
I1~? - K'<~II.
___
CE (VG
+
max{(~[,(~/Lin})
with a positive constant CE independent of e, 51 a n d ~/min"
Proof: []~0
1
Taking v
__ f e , a
l
-" P/min f*
a
II ~ C K [ill
e,a
in the abstract estimate of Lemma 2 we obtain
-- Y/min
f.
IlL 2
proceed with II~P - Y'~ l !1~
_< c/+ (il Y': ~
-
f*
with CK
__
x/IIKII 2 + a
.
We may
IIL: + II(S- P~min)f*llL:)
<_ CK ( Cp (x/7 + "y~) + C. IID~Y*IIL~~/Lin )" For the last inequality we used (2.1.3) and (4.1.2). By (4.1.3) Lemma 6 is proved with CE = CK max{Cp, (Jr Cp IID~K*I! + c~ llD~/*IIL~}.
364
A. R i e d e r
In the framework of scaling function spaces of order N _ 1, both Theorems 2 and 3 apply because the requirements (3.3.3) are met with ~ t - C r liD r g*ll 5~ for 1 _ r _ g . The decay rates (3.3.8) and (3.3.18) read now Pa/m(1) : O(~lr_._LIVf'~) as 1 --+ 00. (4.1.4) The constant in (4.1.4) is independent of l, L, and a. Let ~ , 1 >__ 0, be the space that contains all functions being constant on the intervals [2-Zk, 2 - t ( k + 1)[ (k - 0 , . . . , 2 z - 1). This is a simple but admissible choice: ~ is of the form (3.2.2) when we set ~ - X[o,l[, the characteristic function of the interval [0, 1[. The complement spaces Wt have also the dimension 2 t and they are spanned by the Haar wavelet r = X[o,1/2[-X[1/2,1[ (see Daubechies [8]). The refinement equations (3.2.1) and (3.2.3) hold true with ho = hi = l/v/'2 and go - - g l = 1/s/~, respectively (M~ - M e - 2). It is not hard to see that {~}l>o is a family of scaling function spaces of order 1. The asymptotic behavior (4.1.4) is illustrated in Figure 1 where approximations to the convergence rate Pa(/) of the additive algorithm (3.1.4) are plotted for L - 4, a - 0.001, and for several 1. The underlying kernel is k l ( x , y ) - x - y if x >_ y, and k l ( x , y ) - 0 otherwise. Besides the convergence speed we plotted the rate Pa(/) / p~(1 - 1).
qa(l) -
(4.1.5)
Since (4.1.4) is valid with r = 1 and L - 4 we expect qa to be bounded: qa(1) <_ 0.5 for 1 large. To improve the poor decay rate of pa obtained 0.8 0.6 -4 0.4
\*-
-
0.2 0.0
-
~
7
;
0.5 ..-.= -_--~ - 4
....
9
q~
-
................
Pa
;
1'0
~
11
t
12
1
F i g u r e 1. Solid curve: convergence rates pa of the additive iteration (3.1.4), /min --/-4, with respect'to the Haax wavelet and with a - 0.001. The underlying kernel is kl (x, y) = x - y if x > y, and kl (x, y) = 0 otherwise. Dashed curve: the rate qa (4.1.5). The theoretical bound 0.5 for qa is drawn as a dashed straight line. On level 1 - 6 the iteration diverges. in the Haar wavelet case we introduce scaling function spaces containing polynomials of higher degree. We present two different families of functions which both can be considered as further developments of the introductory example above.
Wavelet Multilevel Solvers for Ill-Posed Problems
365
D a u b e c h i e s w a v e l e t s and spline wavelets on the i n t e r v a l
4.2
We will briefly recall the wavelet systems on the interval [0, 1] constructed by Cohen, Daubechies, and Vial in [6]. This construction is a modification o f the Daubechies wavelet family on the real line (see Daubechies [8]). Let ~N and r - ~)N be the Daubechies scaling and wavelet function of order N _ 2 which both have compact support in [1 - N, N]. Define l* to be the smallest integer such that 2 t* _> 2N. For l _> l* there exist 2N edge r such that each of scaling functions ~o, ~v~, and 2N edge wavelets r the sets -
-
X1
_
k ]O<_k<_N
1}
U {~l,k[N<_k_<2 t - N - l } Y~
-
t2 {~,kl 2 t - N < - k - < 2 4 - 1 } ,
{r176 U {r
I N <_ k <_ 2z - N -
1} U {r k 12 l - N < k < 2 l - 1},
is a family of orthonormal functions. (For convenience we use the notao l x) and fl,k(X) 1 1 tion f0l,k(X) _ 2l/2 fk(2 - 21/2 fk(2/(1 -- x)) only for edge functions.) The spaces Vld and W d of dimensions nt - ml - 2l, V/d "- span Xl
and
W d : - s p a n Yt,
(4.2.1)
have all the properties required in the previous sections. For instance, {vtd}t>l. are scaling function spaces of order N (see Lemma 7 below). The edge functions satisfy, as the interior functions, a kind of refinement equation. The corresponding coefficients are tabulated in [6]. The Daubechies scaling function and wavelet of order 2 are shown in Figure 2 as well as the necessary modifications for the left boundary. R e m a r k . Since the wavelet basis in Wld is an orthonormal basis, the corresponding Gramian matrix Bt coincides with the identity matrix, which simplifies the matrix representations of both iterations (3.1.4) and (3.1.6) (see (3.2.9) for the additive iteration). L e m m a 7". Let Vld be the spaces (4.2.1) with underlying scaling functions ~lOk ~91l,k, and ~z,k of order N Further, assume that f 6 H r(O, 1) for some r 6 { 1 , . . . , N } . Then,
IIf - PlflIL 2 <_ CN,r IIDr fllL 2 ~lr
(4.2.2)
where P1 " L 2 (0, 1) --+ Vld is the orthogonal projection and Cg,r is a positive constant. In particular, the spaces Yl d have order N.
A. Rieder
366
Proof: We give a rather elementary proof using Taylor expansion, the orthonormality as well as the vanishing moments of the wavelets. The intersection of C~176 1), the space of infinitely differentiable functions in (0, 1), with Hr(0, 1) lies dense in Hr(0,1). Thus, it suffices to verify (4.2.2) only for f E Coo(0, 1) n H~(0, 1). The decomposition Vtd = Vtdl @ wtd_l is orthogonal and the basis in W d is orthonormal. Hence, Oo
II/
N-1
2~ - N - 1
E (E j=l
~fll~2
+ kE= N I
k=O
2j - 1
+
~ k=2J -N
I(f,r
(4.2.3)
We proceed by estimating I(S,r -- ]fb.~ f(x)Cj,k(X)dx] for Y _ k <_ 2J - N - 1 where Ij,k = [2-J(1 -- N + k),2-J(N + k)] is the support of Cj,k. The function f E Coo(0, 1) n H~(0, 1) admits the Taylor expansion
f(x) - Er--1f(")(Xj,k)#W /~=0
( X - Xj,k) v + (r --1 1) w
"
"
(x - t) r-1 f(r)(t)dt J,~
with Xj,k -- 2-J (1 -- N + k). The moments of Cj,k up to order N - 1 vanish identically (see [6]), which results in 1 (r - 1)v9
<
(x - t) r-1 ,k
If(~)(t)i dt ICj,k(X)l dx
j,k
( 2 N - 1) r
-< (r- 1)! r
1)
(4.2.4)
The latter estimate comes from the Cauchy-Schwarz inequality applied to the integral with respect to t at first and then to the x integral. We also used the normality []r = 1. In the very same way the reader may convince himself that the inner products of f with the edge wavelets satisfy I
_< C~v,r ]lf(r)[lL 2 ~jr,
i - 0, 1.
(4.2.5)
The assertion (4.2.2) follows now by (4.2.3), (4.2.4), and (4.2.5) when taking ~-~2 j - N - 1
into account that Z-~k--N
I[f(~)]2]L2(b~) ,
< ( 2 N - 1)[If(r)[] 2L 2 " ~
We now consider the spline wavelets. The N - t h order B-spline SN is defined as the N-fold convolution of X[o,1]. The B-splines have compact support in [0, N] and they satisfy the refinement relation N
k--0
367
Wavelet Multilevel Solvers for Ill-Posed Problems
1
1 v
-1
-2 Figure 2. Top: Daubechies scaling function and wavelet of order 2. Middle: edge scaling functions for the left boundary. Bottom: edge wavelets for left boundary.
The corresponding B-spline wavelet of order N is well known (see, e.g., Chui [2] for a comprehensive introduction to spline wavelets). As in the case of the Daubechies wavelets we will need edge splines and edge spline wavelets to yield scaling function spaces of order N on [0, 1]. For N = 2 we will give explicit expressions. A general description of B splines on the interval can be found, e.g., in the book of Schumaker [30]. The general case of spline wavelets on the interval is considered by Chui and Quak in [3] (see also Quak and Weyrich [28] for the corresponding fast algorithms). The graph of the linear B-Spline $2 is shown in Figure 3. We have that S2(x) "- S(x) -
1
~(S(2x)+
2S(2x-1)+
S(2x-2)
).
(4.2.6)
To span the linear functions on [0, 1], we need two edge splines S O and S 1 given by S~ - 1-x if 0 _ x _ 1, So(x) - 0 otherwise, and
A. Rieder
368 S 1 (x) = S~
- x). The modified scaling equation is 1
-
The space Vl8 of dimension nl = 21 + 1, Vts "- span{ S ~ S], Sl,k l k - 0 , . . . , 21 - 2 },
(4.2.7)
S~ and S](x):= S~)(1- x) (1 > 1), coincides with the where S~ space of continuous functions which are affine-linear on [2-*k, 2-*(k + 1)[. The order of V~" is 2 (see, e.g., Strang and Fix [31, Theorem 1.3]).
0 0
1
0
2 -0.6 -I
Figure 3. The linear B-Spline $2 (4.2.6), its corresponding (pre-)wavelet Cs (4.2.8) and the edge spline wavelet r (4.2.9) for the left boundary (from left to right). The interior wavelet r r
=
given by
1 S(2x)1--0
~3 S ( 2 x - 1) + S ( 2 x - 2)
_ 35 s(2
- 3) +
s(2
(4.2.8)
- 4)
(see Chui and Wang [4]), has compact support in [0, 3] (see Figure 3). The edge wavelets r and r can be calculated directly and they fulfill r176
-
S~
el(x)
-
r176
11 1 - ~ S(2x) + ~ S(2x - 1) -
(see Figure 3). Again, we set r and define W~ "- span{ r
r176 r
r
S ( 2 x - 2), (4.2.9)
:= r 1 7 6
x) (l > 1),
r k I k = 0,..., 2 t - 3 }
for 1:2_ l* - 2. The dimension ofWt s is ml = 21 and the reader may convince himself that Vl~_1 = Vls @ W~, l _> l*. Since the constructed bases
Wavelet Multilevel Solvers for Ill-Posed Problems
369
in Vls and W[ are only Riesz bases, the functions r r C s are usually called semi-orthogonal wavelets or pre-wavelets in the wavelet literature. Remark. The Gramian matrix Bt with respect to the basis in Wls is a band matrix with band width 3, and Bl differs from a Toeplitz matrix only at both corners of its diagonal. Utilizing these special features together with the Cholesky decomposition (see, e.g., Deuflhard and Hohmann [11]), it is possible to compute the action of B~ 1 on a vector with CO(ml) operations. The additional storage is independent of the dimension. 4.3
N u m e r i c a l r e a l i z a t i o n , c o m p u t a t i o n a l c o m p l e x i t y , a n d imp l e m e n t a t i o n issues
The sensitive point in using the proposed methods lies in the computation of the entries of the matrix Al (3.2.6). Since the integrals (f, ~l,j)L 2 c a n be calculated only approximately we choose a quadrature rule of the form m
Ql,j(f) -- (~l1/2 ~
Wt f(Xl,t,j)
(4.3.1)
t=0
with weights wt E IR and abscissae xl,t,j E [0, 1] (see Sweldens and Piessens [32]). If Ql,j is exact for polynomials up to degree m and if f E Cm+l, then I(f, :t,j)L= --Qt,y(f)l ~ CQ m a x
xE[0,1]
If (m+l)(x)l ~tm+3/2
(4.3.2)
where the positive constant CQ does not depend on j. The error estimate (4.3.2) follows from Peano's theorem (see, e.g., Davis and Rabinowitz [10]). Approximating the action of K on ~l,j by the integration rule (4.3.1), we have to deal with a perturbation Kl -- KPI of Kl where m
.- Q
,j(k(x, .)) - : - l / : Z
w
t=0
Provided the degree of accuracy m (4.3.2) of the quadrature rule is sufficiently high, we will show that the convergence analysis of Section 3.3 carries over to the perturbed linear system At ~t - ~l with
(Al)i,j
-
(K~I,i,K~I,j)L: + a (~l,i,~t,j)L2
(4.3.3)
370
A. Rieder m
1
m
(4.3.4)
s=0 t=0
0 + a (~ol,i, ~t,j)L ~-
and (/~l)j - 6t1/2 Y~t wt f f ( x ) k ( x , xt,t,j)dx (see (3.2.6) and (3.2.7)). Lemma 8. Let VI be either VId (4.2.1) or Vis (4.2.7) and let B2I be defined as above with the quadrature rule (4.3.1) of accuracy m. Further, suppose that the kernel k of the operator K (4.1.1) is (m + 1)-times continuously differentiable with respect to y. Then,
7~ " - I l K -
g r i t - 0(6tm+').
Proof." Obviously, IlK- Ktll< ~'t + IlKt - Ktll. Using Cauchy's inequality for sums and the fact that we have a Riesz basis on VI, it is straightforward to verify that I!(gt - Kt)f[] 2
<__
(4.3.2)
C
[lfll~
n,
max
O<_j<_nt--1 fo max
~,,ve[o,1]
[(K
-
om+l
K)~l,j (x)12 dx
Oym+ 1 k(x, y)
]2
5/2m+3.
The assumption on the kernel gives I [ K l - Kli] - O((~/re+l) as well as "[l -- (-0(5/m+l ) by Lemma 5. Lemma 1 holds also true for K qualitatively: llKQtll - IIKPI+I(IPI)QtI[ _< IIKl+l - KII + I l K - KZll- 7l+1 + 7~. Altogether we found: The convergence results stated in the Theorems 2 and 3 apply also to the iterations (3.2.8) and (3.2.10), respectively, when At is replaced by At. Moreover, the optimal order r of the decay rate (4.1.4) is not affected by numerical integration as long as the quadrature rule has the degree of accuracy r - 1.We demonstrate the efficiency of the multilevel algorithms for the solution of (4.3.3) by giving estimates of their computational complexities. We make thefollowing general assumptions. We will not include the computation of At (4.3.4) into the operation count of the algorithm because this effort has to be raised independent of the specific solver we use for (4.3.3). The application of B~-1 in (3.2.9) is performed according to the remark at the end of Section 4.2. Further, the auxiliary linear system on
Wavelet Multilevel Solvers for Ill-Posed Problems
371
the coarsest level/min with matrix Almin is solved using Cholesky decomposition, where the decomposition is computed once before the iteration starts. Finally, the underlying scaling function spaces ld of order N have dimension nl = 21 + (9(1) and the degree of accuracy of the used quadrature rule is N - 1. Note that the application of Ht and Gt (3.2.4) can be performed with (9(nl) operations. We begin with the additive iteration (3.2.8). The different parts of the algorithm have the following operation count:
1. Computation of the matrix Atmin (4.3.4)" N 2 n /min 2 + O(nlmi n) Here, we did not take into account the evaluation of the kernel at the quadrature points and we supposed that the integral and the inner products are calculated exactly. 2. Cholesky decomposition of A/min and computation of the starting guess (3.2.11)" n 3 /6 + O 2 /min (n/min)" 3. Evaluation of the residue on level l" 2 n~ + O(nt). The matrix At is a dense matrix. dd 9 r~ 2 4. Application of C a/,/min /min + O(nl). In (3.2.9), the sum over j can be realized in O(nt) operations. Thus, s steps of the additive Schwarz iteration (3.2.8) take essentially
Oadd(s)
" - -
s(2n~ +
~t/min2) -t
n3 Imin 2 6 I- O(n/min ) -t-
O(S~tl)
(4.3.5)
operations.
Now, we count the operations of the multiplicative algorithm (3.2.10):
1. Computation of the matrices Aj, /min ~ j <_ l - 1" 8 CH(N)n~/3, where CH(N) is the band width of Ht, that is, CH(N) = 2N for the Daubechies system and CH(N) = N -t- 1 for the spline system. The matrix J.j can be computed either by numerical integration (4.3.4) or by the recurrence relation Aj_I - Hj Aj H t. Both methods are of the same order of complexity. If we use the latter one we have to add, over all levels/min + 1 _ j _ l, the operation count CH(N)nj (nj + nj-1) for computing H j A j H t. mu't 9 8n~/3 + O(nl) Let Ej be the effort to apply 2. Application of C /,/min C jmu~ Then we have that Ej - 2 nj2 + Ej-1 + O(nj ) for/min -t- 1 _< j _< l 9 ,lmi n " With E/min -- n /min 2 we get inductively the above upper bound for El 3. The points 2 and 3 of the operation count of the additive algorithm apply also to the multiplicative algorithm.
A. Rieder
372
Thus, s steps of the multiplicative Schwarz iteration (3.2.10) take essentially Omult(8) "-- (14 S "4- 8 C H ( N ) )
_•_ -4-
n3 /min + C0(s nl).
6
(4.3.6)
operations. If we fix the splitting level L = l -/min then we have on one side that the convergence rates pc(l) and Pm (l) tend to zero for large l and only a few steps of the iterations yield the solution (see Table 1). On the other side O add and O mu~t are dominated by n3/min --" n3-L which gives the unfavorable complexities O add - O(n~) and O mu~t - O(n 3) as I --+ cr However, with a sophisticated implementation we can achieve both, decreasing convergence rate and optimal complexity (.9(n~). The idea is as follows: we allow the splitting level L to grow but not too fast. We therefore define l-l*
L(1) "-
Lo + L//3J
9 l* < l _< 3 (L0 + l* - 1)/2 9 otherwise,
(4.3.7)
where L0 E ]No is a parameter which may be chosen according to the application under consideration. In (4.3.7) L'J denotes the 'greatest integer' and l* is the smallest possible approximation level (see Section 4.2). Now the asymptotic relation (4.1.4) reads
Pc~m(1) -- O ( r l l _ L ( l ) / ~
) -- (.~(t~lr/3/V/~)
as 1 -4 oo.
(4.3.s)
So, we sacrificed one third of the optimal decay rate of Pa/m to have the following operation counts
_< (2s +
+
)
and Omult(s) _____(148 "11-8CH(N) + 81-L~
-~- + O(snt) + O(n~/3).
(4.3.9)
Both estimates above follow by (4.3.5) and (4.3.6), respectively, when em- n~_L(l) < 8 l-L~ n~ + C0(n~/3) 1 --+ cr ploying that n a /min -In the latter implementation, both multilevel iterations may be viewed as direct solvers with complexity orders (.9(n~) as 1 -+ ~ which outperform any direct solver. R e m a r k . a) Since ~dd (3.2.9) is a sum of matrices, its action on a vector ~"/,/min can be performed in parallel. This leads to a speed-up when the additive
Wavelet Multilevel Solvers for Ill-Posed Problems
373
iteration is implemented on a parallel machine. b) Both iterations can be accelerated with the help of matrix compression techniques presented by Harten and Yad-Shalom [18] (see also Beylkin, Coifman, and Rokhlin [1] and Dahmen, PrSgdorf, and Schneider [7]). The matrix-vector product Alvl can be calculated with O(nt) or O(nl log nl) operations if the kernel k satisfies some additional requirements. c) At a first glance, an implementation of the multiplicative algorithm according to its representation (3.1.5) seems to be prohibited because the residue relative to the finest level 1 has to be computed on each intermediate level. With (4.3.7) we get the following operation count for s iteration steps Omult(8)
__< (2 (L(1) + 1) s
+
8 l-L0/6)
n~
+ 0(8 ~ )"
(4.3.10)
So we have that O mu~t - O(n 2 log 2 hi) as 1 -+ co. However, considering only the leading terms of the complexities (4.3.9) and (4.3.10) we realize that an implementation of the multiplicative iteration via (3.1.5) outperforms (3.2.10) as long as
4 L(/) < ~ ( 1 +
CH(N) s )"
When comparing both implementations one also has to take storage requirements into account. An implementation based on (3.2.10) needs additional storage for the matrices A/min+l,..., Al-1. 4.4
Numerical experiments
We shall provide numerical approximations to the convergence rates to demonstrate the theoretical results as well as the performance of the Schwarz relaxations (3.2.8) and (3.2.10). For convenience, we limit ourselves to the Daubechies wavelet system of order N - 2 and to the linear spline system of the same order. In both cases we computed the weights of the quadrature rules such that affinelinear functions are integrated exactly. The quadrature points are of the form xt,t,j - 2 -z (j + t). In all presented experiments the integral in (4.3.4) is evaluated exactly and all kernels are smooth enough to yield 7t - O(~12). The first experiments consider only the additive iteration (3.2.8). The asymptotic behavior (4.1.4) is illustrated in Figure 4 where approximations to Pa with fixed splitting L - 5 are plotted for several 1 and the regularization parameter a - 0.001. The underlying kernel is kl (x, y) - x - y, x _> y, kl (x, y) - 0 otherwise. Next, we compute approximations to Pa where the coarsest level/min -" 2 is fixed and where the kernels are kl and k2(x,y) - cos(Trxy). We
A. Rieder
374 0.5 0.4 0.3
~---==~
0.2
.....
O.25 ,= . . . .
:g .....
~. ql
0.1 v l 7 8 9 10 11 12 F i g u r e 4. Solid curves: convergence rates pa of the iteration (3.2.8), lmin -- l--5, with respect to the kernel kl and with c~ - 0.001 (.: Daubechies wavelet, N - 2; o: linear spline). Dashed curves: the rate qa (4.1.5). The theoretical bound 0.25 for qa is drawn as a dashed straight line. 0.0
know that Pa < 1 uniformly in l if a is not too small. Here, a >_ 0.001 is numerically sufficient (see Figure 5). In both examples above the convergence rates with respect to the linear splines are clearly smaller than those with respect to the Daubechies wavelet. The main reason for this observation is the higher regularity of the linear B-spline: The Daubechies wavelet of order 2 is Hhlder continuous with exponent 0.55 (see, e.g., Daubechies [9]) whereas the linear B-spline is Lipschitz continuous. The following example illustrates the performance of the iteration (3.2.8) by an approximate solution of (4.1.1) with kernel kl and exact righthand side g(x) x 2 ( x 2 - 4x + 9)/12 + (1 - cos(31rx))/lr2/18. Thus, f*(x) - (1 - x) 2 + cos2(37rx/2). In our computations we supposed that g~ is known only at the discrete points j 51 with g~(j tit) = g(j 5l)+ ej where the random errors {ej} are distributed uniformly in [-e,s]. The integrals in the (/31)j's are evaluated by the trapezoidal rule. Table 1 contains the number s of iteration steps to yield an Euclidean norm of the residue smaller than 10 - 4 . a . ]lz~ll. Here, {z~}~ denotes the sequence of iterates generated by (3.2.8) with starting guess either z~ - 0 or -
Z0 -- U/min'~ "-- qf~/,/mint A-1/min~'~l'lmin~l (3.2.11) (see Lemma 6). Our stopping criterion guarantees a relative accuracy IIz~ - ~tll/liz~ll bounded by 10 -4. The approximate solutions with respect to the discretization levels I -- 6 and 1 - 11 together with the minimum norm solution f* are displayed in Figure 6. We considered an absolute error e - 0.04 and got the optimal regularization parameter by trial and error. It took 14 (1 - 6) and 1 (1 - 11) iteration steps (L = 4) to bound the relative accuracy by 0.01. In the remainder of this section we compare the additive and the multiplicative iterations numerically. All computations presented are based on the approximation spaces spanned by the linear B-spline.
375
Wavelet Multilevel Solvers for Ill-Posed Problems 0.6 0.4 ~
0.2 0.0
<* . . . .
6
--0--
~
^
. . . . . . . . . . . . . .
7
9. . . . ~' . . . .
-0- . . . . --0.. . . . .
41 . . . . . .~ . . . . .
9
1
11
9 r
12
0.6~ 0.40.2
1
. . . .
-O. . . . . . . .
9 v . . . .
-^
9
. . . .
-O-
v
0.0
.....i .....................
!
;
~
;
. . . .
~
. . . . .
9
A
,
,
,
6 7 10 11 12 F i g u r e 5. Convergence rates pa with respect to the fixed coarsest level/min 2 (-: Daubechies wavelet, N - 2; o: linear spline). Top: k2(x, y) - cos(it x y). Bottom: kl(x, y) -- x - y, x > y, kl (x, y) - 0 otherwise. Solid curves: a - 0.001, dashed curves: a = 0.005. -
-
Table 2 lists approximations to Pa and #m with /min : l -- 5 and a 0.001. The rates Pa are those plotted in Figure 4. To check the quality of our convergence result for the multiplicative iteration (Theorem 3) we computed the quotient qm which is defined analogously to qa (4.1.5). The values of qm with respect to both kernels kl and k2 are shown in Figure 7. Since (4.1.4) is valid with r = 2 we expect qm to be bounded by 0.25. This is true indeed. Moreover it seems t h a t qm stays bounded by 0.16, indicating a decay rate in (4.1.4) for Pm of the order of about 2.6 rather than 2 which is the order supported by the theory. Hence, the estimates in Theorem 3 are too pessimistic. In contrast to that, qa relative to the additive algorithm is very close to 0.25 (see Figure 4). Thus, the order 2 of the decay rate of pa seems to be optimal and it looks like Theorem 2 describes the real situation exactly. In the last experiment we compare the convergence rates of the additive and multiplicative iterations when the coarsest level is fixed. The numerical approximations are displayed in Figure 8 where/min -- 2 and a - 0.001. In this setting we have convergence of both iterations uniformly in I. As in the former example the multiplicative iteration converges much faster than its additive twin. This was already mentioned in the discussion after equation (3.3.18). However we do not have a theoretical proof for this phenomenon.
A. Rieder
376
Table 1. Necessary iterations of the additive algorithm to guarantee a relative accuracy smaller than 10 -a (e = 0.04, a = 0.001, Daubechies wavelet). The starting guess is either z~ = 0 or z~ = -Ulmin := 7/l,lmin /min n , ,/min "fi (3.2.11).
/min -- 1 -
5
/min -- 3
7
8
9
10
11
12
z~ = o
49
7
4
3
3
2
Z~ "-- Ulmin
12
4
2
1
1
1
z~ = 0
7
7
7
7
7
7
z? - Ulmin
4
4
2
4
4
4
2 1.5
1.5 "% ~
0.5
0.2
Figure (dashed 1 - 6, a that the
%
l
0.4
-,,,,Q, 0.6
0.8
1
....... _~'.'.s . . ,,,,,, ,, .........,~,. . 0.2 0.4 0.6 0.8
1
6. Approximate solutions (solid lines) and minimum norm solution lines) of (4.1.1) with kernel kl (r = 0.04, Daubechies wavelet). Left: - 0.001, right: I --- 11, a - 1.5- 10 -4. Each a is optimal in the sense L2-error Ill* - J~'~IIL2 is minimized.
Thus far we have not brought the issue of effort into the discussion of the comparison of both multilevel methods. Relying on the numerical experiments (see Figure 8), the multiplicative algorithm is at most twice as fast as the additive one, t h a t is, we may assume Pm ~ p2. Two steps of the additive iteration are necessary to yield roughly the same error reduction as one step of the multiplicative iteration. Our investigations on the numerical effort of b o t h methods (see Section 4.3) show however t h a t two steps of the additive algorithm are cheaper than one step of the multiplicative algorithm. So, the additive Schwarz method is the more efficient variant. R e m a r k . In the former sections we have seen t h a t the proposed multilevel solvers work most efficiently if
~t-i(O << ~/ao~(e),
(4.4.1)
where ao(r is as in (2.1.4). For more quantitative statements see Theorems 2 and 3.
Wavelet Multilevel Solvers for Ill-Posed Problems
377
Table 2. Approximations to pa and pm ( l m i n - - 1 - 5, a - " 0.001, linear spline). The underlying kernel is kl. The values of pa are those from Figure 4.
P&
1=7 1.3.10 -1
/=8 2.6.10 -2
I=9 5.4.10 -3
1=10 1.2.10 -3
1=11 2.6.10 -4
/=12 6.0.10 -5
pm
2.5.10 -2
3.9.10 -3
6.1.10 -4
8.9.10 -5
1.3.10 -~
1.9.10 -6
0.16 0.15 v
0.14
!
!
9
10
1'1
A v
1'2
F i g u r e 7. The rate qm , see (4.1.5), corresponding to approximations of pm (/min : l5, O~ - - 0.001, linear spline). Solid curve: kernel kl, dashed curve: kernel k2. Readers familiar with inverse problems might see a serious disadvantage of the multilevel schemes in condition (4.4.1) which requires large discretization levels 1. This does not seem appropriate for first kind problems due to possible instabilities which might amplify the noise. However, in our approach the stability of (1.2) is taken care of by the Tikhonov t e r m c~ I. Here it pays to refine the discretization. The solution f~'a is contaminated by a discretization error and by the unavoidable noise error. Increasing 1 reduces the discretization error without amplifying the noise. The remaining error in f['~ is caused only by the noise (provided c~ is chosen adequately as mentioned in Section 2.1). This can bee seen either by the error estimate (2.1.3) or by Figure 6. The approximate solution on the right is closer to the minimum norm solution than the one on the left because its discretization error is negligible. Its error comes solely from the noise in the data.
A. Rieder
378 0.4 0.3 0.2
" ~ "r
-" "-~- --'-O
0.1 0.0
A
A
9
10
,
6
7
8
T
~
11
12
0.15 0.10
~ " ~" "- -- -~ -_ --<>
0.05 0.00
,
T
,
1
6 7 8 9 10 11 12 F i g u r e 8. Convergence rates with respect to the fixed coarsest level / m i n - - 2 (a -- 0.001, linear spline). Dashed curves: approximations to pa, solid curves: approximations to pm. Top: kernel k2, bottom" kernel kl. References
[1]
Beylkin, G., R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141-183.
[2]
Chui, C. K., An Introduction to Wavelets, Academic Press, Boston, MA, 1992.
[3]
Chui, C. K. and E. Quak, Wavelets on a bounded interval, in Numerical Methods in Approximation Theory, vol. 9, D. Braess, L. L. Schumaker (eds.), Birkh~iuser, Basel, 1992, pp. 53-75.
[4] Chui, C. K. and J. Z. Wang, On compactly supported spline wavelets, Trans. Amer. Math. Soc. 330 (1992), 903-915. [5] Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [6] Cohen, A., I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1993), 5481. [7] Dahmen, W., S. PrSgdorf, and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993), 259-335. [8] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 906-966.
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[9] Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [10] Davis, P. J. and P. Rabinowitz, Methods of Numerical Integration, Academic Press, New York, 1975. [11] Deuflhard, P. and A. Hohmann, Numerical Analysis: A First Course in Scientific Computation, de Gruyter, New York, 1994.
[12] Dicken, V. and P. Maat3, Wavelet-Galerkin methods for ill-posed problems, J. Inverse Ill-Posed Probl. (1996), to appear.
[13] Donoho, D. L., Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition, Appl. Comput. Harmon. Anal. 2 (1995), 101-126.
[14] Engl, H. W., Regularization methods for the stable solution of inverse problems, Surveys Math. Indust. 3 (1993), 71-143.
[15] Griebel, M. and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), 163-180.
[16] Groetsch, C. W.,
The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
[17] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations, Springer, New York, 1994. [18] Harten, A. and I. Yad-Shalom, Fast multiresolution algorithms for matrix-vector multiplication, SIAM J. Numer. Anal. 31 (1994), 11911218. [19] King, J. T., Multilevel algorithms for ill-posed problems, Numer. Math. 61 (1992), 311-334. [20] King, J. T. and A. Neubauer, A variant of finite-dimensional Tikhonov regularization with a- posteriori parameter choice, Computing 40 (1988), 91-109. [21] Kirsch, A., R. Kress, P. Monk, and A. Zinn, Two methods for solving the inverse acoustic scattering problem, Inverse Problems 4 (1988), 749-770. [22] Kremer, J. and A. K. Louis, On the mathematical foundations of hyperthermia therapy, Math. Methods Appl. Sci. 13 (1990), 467-479. [23] Liu, J., A multiresolution method for distributed parameter estimation, SIAM J. Sci. Comput. 14 (1993), 389-405.
380
A. Rieder
[24]
Louis, A. K., Inverse und schlecht gestellte Probleme, Teubner, Stuttgart, 1989. English translation in preparation.
[25]
Louis, A. K., P. Maa~, and A. Rieder, Wavelets: Theorie und Anwendungen, Teubner, Stuttgart, 1994. English version: Wiley, Chichester 1997, to appear.
[26]
Oswald, P., Multilevel Finite Element Approximation: Theory and Applications, Teubner Skr. Numer., Teubner, Stuttgart, 1994.
[27] Plato, R. and G. Vainikko, On the regularization of projection methods for solving ill-posed problems, Numer. Math. 57 (1990), 63-79. [28] Quak, E. and N. Weyrich, Decomposition and reconstruction algorithms for spline wavelets on a bounded interval, Appl. Comput. Harmon. Anal. 1 (1994), 217-231. [29] Rieder, A., A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization, Numer. Math. (1996), to appear. [30] Schumaker, L. L., Spline Functions : Basic Theory, Wiley, New York, 1981. [31] Strang, G. and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, 1973. [32] Sweldens, W. and R. Piessens, Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Numer. Anal. 31 (1994), 1240-1264. [33] Xia, X. G. and M. Z. Nashed, The Backus-Gilbert method for signals in reproducing Hilbert spaces and wavelet subspaces, Inverse Problems 10 (1994), 785-804. [34] Xu, J., Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), 581-613. [35] Yserentant, H., Old and new convergence proofs for multigrid methods, in Acta Numerica, Cambridge Univ. Press, New York, 1993, pp. 285- -326. Andreas Rieder Fachbereich Mathematik Universit~t des Saarlandes Postfach 15 11 50 66041 Saarbriicken, Germany [email protected]
IV. Software Tools and N u m e r i c a l Experiments
This Page Intentionally Left Blank
T o w a r d s O b j e c t O r i e n t e d S o f t w a r e Tools for N u m e r i c a l M u l t i s c a l e M e t h o d s for P D E s u s i n g W a v e l e t s
Titus Barsch, Angela Kunoth, and Karsten Urban
A b s t r a c t . During the past few years, we have started developing
a software package that realizes numerical multiscale methods using wavelets for a class of PDEs. In particular, it includes many wavelet typical routines e.g.for handling refinement masks, computing refinable integrals, multiscale transformations, and multilevel preconditioners. The program is independent of the spatial dimension and follows modern software technology principles. In this paper we describe some of our software tools and provide examples for the use of the package. {}1
Introduction
During the past years, multiscale methods have been proven to be asymptotically optimal efficient numerical schemes for elliptic partial differential equations. This means that the resulting linear systems of equations can be solved with an overall amount of work which is of the order of the number of unknowns. In particular, this implies that the number of iterations for an iterative scheme is independent of the scale. Discretizations allowing such a multilevel structure include uniformly refined finite elements and bases of multiresolution spaces. The latter may be used as a nested sequence of trial spaces where bases, called wavelets, of the complement between two succeeding spaces are available. This availability of direct sum decompositions together with additional analytic properties provide a powerful tool to prove the theoretical asymptotical optimality of the resulting numerical schemes for a whole range of operator equations, including elliptic partial differential and singular integral equations. Multiscale Wolfgang
383
W a v e l e t M e t h o d s for P D E s Dahmen,
Andrew
J. K u r d i l a ,
and Peter
C o p y r i g h t (~)1997 by A c a d e m i c P r e s s , I n c . All rights of reproduction ISBN 0-12-200675-5
in any f o r m reserved.
Oswald
(eds.), pp. 383-412.
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Therefore, we aimed at designing a software package that realizes all promising features of wavelet-based multiscale methods for this class of problems. In this paper we describe some modules that we have implemented during the last years and which might be helpful as a first step to a library for this kind of schemes. Naturally, at the beginning we had to limit the scope of problems we wanted to solve and decided to restrict ourselves to stationary problems which are Galerkin formulations of (1) scalar elliptic problems with variable coefficients and (2) systems of elliptic problems with variable coefficients. Also, as already mentioned, a class of elliptic operators stemming from (3) boundary integral equations can be efficiently treated by wavelet-based multiscale methods. Here the additional problem of compressing the corresponding matrices which are in general not sparse, arises. This can be done by using wavelet expansions [20]. We mention (3) here since it is intended that the work just started in [6] based on [21] is adjusted according to the common features of the problems (1)-(3) such as data structures, preconditioning and iterative methods. The Galerkin discretization of (1), (2) leads to the problem of solving linear systems of equations involving (i) positive definite matrices and (ii) indefinite matrices stemming from saddle point formulations. Both systems may arise in single- and vector-valued form. For example, for a scalar elliptic partial differential equation, including the boundary conditions into the approximation spaces leads to a single-valued problem of type (i) whereas appending them by Lagrange multipliers as proposed in [26] gives a single-valued system of type (ii). Furthermore, the divergencefree formulation of the Stokes problem (see for example [22]) results in a system (i) for vector-valued equations whereas the mixed formulation of the Stokes problem leads to a vector-valued system (ii). As a last example, from the mixed formulation of a second order problem one obtains a singlevalued system of type (ii) [3, 5]. It is important to implement these linear systems in such a way that the solution method has optimal complexity as predicted by the theoretical results and also to use appropriate data structures in order to be able to handle realistic problems in three and more dimensions. The promising features of wavelet-based multiscale methods may be summarized in the following list:
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(a) no geometric decomposition of the underlying domain, (b) independence of the spatial dimension, (c) easy realization of discretizations of higher order, (d) high efficiency for multivariate problems by using tensor products, (e) adaptation of the bases to the underlying problem. Some comments on these topics are in order. Although a geometric decomposition of the domain as in the finite element case allows one to handle more complex geometries, this may lead to difficulties in three and more dimensions. The discretization of multivariate problems using wavelet type methods need not be done by tensor products. However, this approach increases efficiency enormously. This fact is also exploited in spectral element methods [7]. Sometimes the discretization has to fulfill additional requirements due to the structure of the underlying problem. As an example, consider the Stokes problem [22]. For the divergence-free formulation one needs trial spaces of divergence-free vector fields, while for the mixed formulation the trial spaces for velocity and pressure have to satisfy the Lady~enskaja-Babu~ka-Brezzi (LBB) condition. The realization of these conditions in terms of wavelets can be found in [18, 27, 33]. We feel that these advantages deserve attention and make it worthwhile to attempt an implementation. However, we are well aware of an essential drawback wavelet-based multilevel schemes have had until the present, namely, the adaptation to domains different from rectangular ones. The investigations in [8] where multiscale decompositions for Lipschitz domains were introduced may also lead to efficient software for wavelet-type methods on general domains. In view of subsequent investigations in this direction, we have designed our data structures such that they also fulfill the requirements of the construction in [8]. However, for the present we restrict ourselves to the case where the domain is the d-dimensional cube [0, 1]d. 1.1
R e q u i r e m e n t s for software caused by the s c o p e of p r o b l e m s
The above described scope of problems requires the following routines for a wavelet-based multilevel scheme: 9 construction of trial functions of arbitrary regularity, 9 construction of trial spaces that satisfy additional conditions, such as being divergence-free or fulfilling the LBB condition, 9 procedures for setting up the corresponding stiffness matrices and right-hand sides,
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9 iterative schemes for solving linear systems of equations and saddle point problems, 9 interfaces for graphic tools to visualize the solution data. These tools must be built in a way such that 9 data structures do not destroy the optimal complexity, 9 they are independent of the spatial dimension and the single- or vector-valuedness of the problem, 9 two-dimensional single-valued problems should run on a personal computer and three-dimensional problems on a workstation. 1.2
G e n e r a l r e q u i r e m e n t s for software
In addition to the items collected in the past section, there are general requirements that modern software packages should satisfy. These include 9 modularity, i.e., the package consists of independent modules, 9 use of multipurpose tools like data structures independent of the problem, 9 clearly defined interfaces between the different tools, 9 easy extension of the existing collection of subroutines, 9 type safe programming, 9 independence with respect to the underlying hardware and compiler. A well-known possibility to guarantee the satisfaction of these features is provided by object-oriented programming. In particular, the possible independence of the spatial dimension and the single- or vector-valuedness are in favour of an object-oriented language like C + + . In addition, the complex algorithms and software can be arranged and validated more clearly. Finally, these codes are more readable for the user. Many users may prefer packages in FORTRAN with the justification of it being widespread with a perhaps easier parallelization of the corresponding codes. But the above mentioned arguments lead us to use C + + . The remainder of this paper is structured as follows. Section 2 we briefly describe the basic linear algebra structures and tools that are independent of any multilevel background. Section 3 collects the necessary theoretical multiscale concepts and introduces the corresponding data structures that are frequently used in the sequel. Here Subsection 3.1 deals with the
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masks of generators and wavelets and Subsection 3.2 with the setup of the stiffness matrix and the right-hand side as well as the fast computation of their entries. The multiscale transformations described in Subsection 3.3 provide an efficient tool which is the main ingredient of the fast multilevel preconditioners in Subsection 3.4. In Section 4 we point out future problems that we started including in the software package. Finally, the two- and three-dimensional examples in Section 5 give an impression of the range of problems we are able to solve numerically at this point. It should be understood that this paper is more a rough description of the present stage of the developed scientific software tool box than an easyto-use manual of a commercial package. However, a detailed description of these software tools is in preparation [2]. 1.3
A c c e s s to the software
To receive this software package with the corresponding commenting reports, the interested reader may write e-mail to urban 9
rwth-aachen, de
or look at the homepage h t t p : / / w w w , i g p m . r w t h - a a c h e n , d e / ~ u r b a n under the topic Software.
w
Basic linear algebra tools
The scope of problems as described in Section 1 requires in particular the solution of linear systems of equations and saddle point problems. Hence we need some basic tools for handling the algebraic structuies, which will be briefly described in this section. Although there is a whole variety of software packages in C + + available for vector-matrix operations, we found it advantageous to develop our own package that allows us to adapt the data structures to the special requirements of multiscale methods. The class T v e c t o r < t y p e > is a t e m p l a t e c l a s s , where t y p e may be any kind of structure for which the standard arithmetic operators are defined. For convenience, Tvectors with double or integer entries were called v e c t o r and i v e c t o r , respectively. The memory needed to store the vectors is allocated dynamically with respect to the length of the particular vector. Naturally, all the standard manipulations like arithmetic operations, norms and scalar products are provided. To handle different kinds of matrices, we constructed the classes m a t r i x , s p a r s e , symspa and blband. The purpose of the class m a t r i x is to handle all kinds of operations in connection with dense matrices. It is also generated by a template class Tmatrix. Moreover, for the problems described in Section 1 we usually have sparse matrices so that routines are
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required that make use of this fact in order to optimize memory. This is done by the class s p a r s e , which stores the entries of a sparse matrix in the frequently used compressed row .format. In addition, many of the matrices arising from the above described problems are symmetric. Consequently, we have introduced a corresponding class symspa that reduces the amount of memory to store a symmetric sparse matrix by a factor of two. It turns out that many of the system matrices have a block band structure. In spatial dimensions higher than one these matrices become block banded with a 'depth' equal to the dimension. Our implementation of such block band matrices allows an arbitrary depth and is thus automatically independent of the spatial dimension. The basic idea for the class blband that handles such block band matrices is a control flag that checks whether the elements of the matrix are numbers or banded matrices themselves. Additionally, blband contains two pointers called LittleBlocks and GreatBlocks. Both pointers share the same memory, because only one is used. In C + + this is realized with a union. During the construction of an object of blband, the user specifies the number of elements whether the elements are numbers or matrices. In the first case the control flag is set t r u e and memory is allocated for L i t t l e B l o c k s , while in the latter case memory is allocated for GreatBlocks and the flag is set f a l s e . This makes sure that no memory is wasted for storing unneeded objects. If the elements are banded matrices, the same question recursively arises. We emphasize that there is no theoretical limit concerning the depth of this structure. Since the allocation and the assembly of such a matrix is completely dynamical, this means that the depth may even be arbitrary at compilation time. For every line in the matrix we store the borders of the horizontal band. The same recursive strategy can therefore also be used for the implementation of multilevel preconditioners or multiscale transformations, see Section 3 below. A special feature which drastically reduces the amount of necessary memory is linked to the observation that in the above described applications many of the involved blocks may be equal. And again, in every block many of the entries which can be blocks or numbers may coincide. Thus, the entries of these blocks need only be stored once, while their position within the matrix is stored' individually. Using these data structures we implemented solvers for different kinds of algebraic problems. These include in particular the conjugate gradient method to solve linear systems of equations. The preconditioners that guarantee asymptotically optimal efficiency of the cg method are based on multilevel theory and are described in Subsection 3.4. Moreover, several versions of the Uzawa algorithm for solving the symmetric saddle point problems (ii) mentioned in Section 1 are implemented, including the mod-
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ified ones described in [30]. All the iterative routines make essential use of matrix-vector multiplications. Nevertheless, they are independent of the special structure of the matrix. As long as the matrix-vector multiplication is programmed with respect to the structure, the iterative routines may be written without knowledge of the type of matrix. Here the program language C + + reveals its full power: as for classes the use of templates makes it possible to use the type of the matrix only as an argument, so that all routines have to be implemented only once. The compiler automatically generates routines for the different types of matrices. It should be mentioned that this programstyle preserves the main advantage of C + + over C: it is still a type-safe way of programming, i.e., the compiler checks all arguments.
w
Multiscale methods
In this section we review the basic concepts of multiscale methods and provide examples of the data structures as they are implemented in our code. Contrary to the setup for multilevel schemes based on finite elements, wavelet-oriented methods do not make use of a decomposition of the underlying geometric domain. The advantage of this approach is apparent, for instance, in higher dimensions subdivision of a domain into e.g. tetrahedrons may be costly and difficult. In fact, here finer discretization levels are introduced automatically in terms of refinement equations like (3.1.6) and (3.1.7) below. 3.1
M a s k s of refinable functions and wavelets
Multiscale methods can be described by means of a sequence S - {Sj }~~ o of closed nested subspaces of a real Hilbert space ~" depending on the underlying problem. The very general situation on Hilbert spaces in [12, 13, 14] (see also the survey [16]) is adapted here to the case of the underlying domain Ft = [0, 1]d and ~ - L2(f~) yet to be implemented. Of course, by a simple transformation general rectangular domains can also be treated. Thus, we assume
~jo C Sjo+I
C""
C ~j
C Sj+I
C ' ' - C.~",
closy( 0
Sj)-~
(3.1.1)
j=jo
and that these spaces are given in terms of their bases,
Sj = closm-span @j,
~j := {~oj,} : k E Ij},
(3.1.2)
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where ~j,k E ~" are compactly supported functions and Ij C Z d is a finite set of indices. The lower index j0 E IN is assumed to be sufficiently large, guaranteeing for technical reasons that the support of basis functions overlapping opposite faces of the domain do not intersect. It is important to assume that r is stable, i.e., it forms a Riesz basis for Sj, lckl 2 ~
keb
Ilz
ck
keg
II
,
~"
where a ~ b means that a can be estimated by a constant multiple of b and vice versa with constants independent of j. Given an approximate solution in some Sj, one wants to obtain a more accurate approximation in Sj+I by updating the coarser solution. This detail information are modeled by considering an appropriate (not necessarily orthogonal) complement Wj of Sj in Sj+I,
Sj+I = Sj @ Wj, j >__jo.
(3.1.3)
Thus, to construct the spaces Wj, one must find a compactly supported : k E Jj } for Wj where Jj is another (finite) set of indices basis ~j := {r such that ~ J j = ( ~ / j + l ) - (~Ij). Setting Wjo_l := Sjo , Jjo-1 := Ijo, equation (3.1.3) gives rise to a multiscale decomposition of Sj, j--1
sj=
Win.
(3.1.4)
m=jo--1
In addition to the sequence S we assume that we have another sequence with the same properties (3.1.1) and complement spaces 17Vj such that l~j_l_Sj, Wj_LSj, Wj_I_IVj,, j C j ' , (3.1.5)
,~ - (Sj }~~
and correspondingly for their bases (~j, ~j where orthogonality is to be understood with respect to the inner product (., .)y. This framework, called biorthogonal multiresolution analysis, was developed in [9] has been adapted to the situation at hand on the interval and thus on unit cubes in [17]. A simpler construction for lower order functions can be found in [21]. The nestedness of the spaces Sj implies that (~j is refinable, i.e., there exists a mask a j " - - {aJk,m}keb,meij+l such that
~j,k --
~
a j~,m ~j+l,m, k e Ij.
(3.1.6)
mE/j+1
Typically we assume that the refinement equation (3.1.6) is stationary, i.e. the coefficients do not depend on the level j. Hence, the mask coefficients only have to be computed on the coarsest level j0.
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The inclusion Wj C Sj+I implies the representation
r
--
~_,
b~,m qo:)+X,m, k E Jj
(3.1.7)
mEIi+l
for the complement functions with some mask b j "- {bJk,m}kEJ~,mEb+l, where the values of its coefficients should also not depend on j. We assume : j >_ jo, k E Jj } forms a Riesz basis for Y and then we call that 9 : - {r a wavelet system for ~. Note that in the cases of interest the functions ~j,k, l/)j,k are uniquely determined by their masks aJ and b y so that only these coefficients need to be stored. Here we describe this for a simpler case of masks, namely, in the wellknown shift-invariant setting on all of ]Rd where 9v - L2(]R d) and
j E Z,
(flj,k(') "-- 2dj/2tfl( 2j " - k ) ,
k E Z d.
(3.1.8)
In this case, the refinement equation (3.1.6) can be written as U
~9(X) -- Z ak ~:~(2x-k), k--l
x e IRd
l-(/1,
,ld) ' ~ t - (Ul,
,Ud) e Z d
(3.1.9) where supp a = [l, u], i.e., ak = 0 for k r [l, u]. Often the function ~ is also called the generator of 8. Since properties like the order of approximation and adaptation to the underlying problem can be realized by taking different basis systems ~j as trial functions, the software should be independent of the particular choice of qDj,k. To this end, we designed three classes, namely, MaskBorder, Mask and B a s i s as follows. Observe that we can write
d Ij C_ i X_ l { l i , . . . , u i } -" Zj,
li <_ ui,
li, ui E Z,
l < i < d.
(3.1.10)
The class MaskBorder contains the two vectors 1 and u of lower and upper border of the mask indices, respectively. Since the implementation should not depend on the spatial dimension d, the allocation of the integer vectors representing l (LowBorder) and u (UpBorder) is done dynamically with respect to d. The class MaskBorder is used to declare an object of the class Mask which contains the mask coefficients ak, 1 <_ k <_ u. Using an object of MaskBorder as argument for the constructor of an object of Mask, a vector of length d
M kDim "-- II(u i--1
-
+ 1)
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is allocated. Now one can define the mask coefficients as in the following simple example for the cardinal B-spline of order 2 for d - 1 where a0 1 a 2 - ~ , a l -- 1" MaskBorder
N2Bord (I) ;
/ / c o n s t r u c t o r of MaskBorder / / d e f i n e s object N2Bord in 1D
N2Bord. putLowBorder (i) = 0 ; N2Bord.putUpBorder(1) = 2;
/ / s e t s the borders
Mask N2(N2Bord) ;
/ / c o n s t r u c t o r for Mask
N2.put(0) = 0.5; N 2 . p u t ( 1 ) = 1; N2.put(2) = 0.5;
/ / s e t s mask coefficients
For multivariate masks we have to index the multi-indices in l j which have the form of integer vectors. This is done in the canonical way, i.e., for k - ( k l , . . . , kd) 6 I j we define for some object MB of the class MaskBorder d Index(k, MB) "-- E ( k i i=1
d - l i + 1) H (uj - l j + 1). j=i+l
The function Inc increases the multi-index within the borders of MB by one. The multi-indices are ordered lexicographically such that k < l is equivalent to kl < ll
or
kl - / 1 , k 2 < 12,
or
kl - ll,k2 - / 2 , k 3 < 13 and so on.
By this, we obtain k < 1 if and only if Index(k, MB) < Index(1, MB). One often has to realize a loop of the form "k 6 I j , " and this can now be done as follows: ivector k(d); k = MB. getLowBords () ; for
{
(int i=O;
II...
//i.e.k=l
i<MB.getMaskDim();
i++)
Inc(k,MB) ; }
To illustrate its usefulness in dimension independent programming, compare the following two equivalent examples, where in the right column the space dimension only plays the role of a parameter.
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int S p a c e D i m = 2;
MaskBorder TestBord(SpaceDim) ; TestBord. p u t L o w B o r d e r (i) = 0 ; TestBord.putLowBorder(2) = 1;
TestBord. putUpBorder (i) = 2 ; TestBord. p u t U p B o r d e r (2) = 3 ;
Mask Test (TestBord) ;
Test. put(0,1) Test. put(0,2) Test. put(0,3) Test.put(I,1) Test. put(l,2) Test. put(l,3) Test.put(2,1) Test. put(2,2) Test.put(2,3)
= = = = = = = = =
1; 2; 3; 4; 5; 6; 7; 8; 9;
int i, index; ivector k(2); k = T e s t B o r d . g e t L o w B o r d s () ; for (i=l; i<=TestBord.getMaskDim(); { index = Index(k,TestBord) ; Test. putI (index) = i ; Inc (k, TestBord) ;
i++)
}
In order to collect a finite number of refinable functions and to handle linear combinations of refinable functions as in (3.1.7), we designed the class B a s i s which is an array of Mask with some additional information flags, see [2, 32]. This class is also important for handling wavelet systems adapted to bounded domains as constructed in [1, 10, 17, 18]. Their common idea is to take as many as possible of the shifts (3.1.8) whose support is completely contained in the interior of the domain and additionally to build linear combinations of the shifts (3.1.8) near the boundary, such that the approximation order is preserved. All this information is contained in the class B a s i s . Perhaps the easiest way to form multivariate functions is to build tensor products. Hence, we implemented the corresponding operator ".": Mask N2xN2 ; N2xN2 = N 2 ( )
* N2();
With this definition the mask coefficients of N2xN2 have the expected values: a[ a[ a[ a[ a[
0, 0, 1, 2, 2,
0] 2] 1] O] 2]
= = = = =
2.5000000000e-01 2.5000000000e-01 1.0000000000e+00 2.5000000000e-01 2.5000000000e-01
a[ a[ a[ a[
0, 1, 1, 2,
1] 0] 2] 1]
= = = =
5.0000000000e-01 5.0000000000e-01 5.0000000000e-01 5.0000000000e-01
.
It is one of the promising features of multiscale schemes that their basis functions can be adapted to the problem at hand. As an example we consider here the construction of divergence-free wavelets [27, 33] and of
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corresponding trial spaces that fulfill the LBB condition [18] in any space dimension. Both constructions can be used for the numerical solution of the Stokes problem. Their main ingredient is that the differentiation operations of the involved basis functions can be formulated in easier terms of algebraic manipulations of the symbol of the refinable function in the shift invariant setting, a(z) "- E
ak z k, z 6 T "d,
(3.1.11)
kEIj
with refinement coefficients ak as in (3.1.9). Here T ~d denotes the d-dimensional torus. Upon introducing the backward and forward difference operator Vvf(.) "- f(.) - f ( . - e ~) and A . f ( . ) "- f(. + e ~) - f(.), respectively, where 1 ___ v ___ d, f 9IRd --~ IR and e ~ the v-th canonical vector in IRd, we can formulate a basic result which can be found in [27] for the univariate and in [33] for the multivariate case. P r o p o s i t i o n 1. Let 1 <_ v <_ d be some fixed integer. (a) Let ~ 6 L2(]R d) be a refinable function with mask a such that a*(z) := (2a(z))/(1 + z.) is the symbol of another refinable function ~* 6 L2(IR~). Then the difference equation -
Oxv holds. (b) Define ~*(z) "- (1/2) (1 + 2~)a(z). function r with mask ~* such that - - r
Then there exists a refinable
-
Oxv is vafid. The above described' modifications are realized by the functions Mask MaskDivision(int nu, Mask a) ; Mask MaskMultiplication(int nu, Mask a); Mask MaskBackwardDifference(Mask c, int nu, int alpha); Mask MaskForwardDifference(Mask c, int nu, int alpha); The first two generate 12a(z) + z~ and 1 +2 Zva(z) and the last two a Vvc and a A~c, respectively.
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395
Refinable integrals and stiffness matrices
The Galerkin formulation of problems like (1), (2) in Section 1 and its discretization in terms of the basis ~j leads to the linear system of equations
Ar
- f j,
(3.2.1)
where Acj = (a(Wj,k,Wj,L))k,leb denotes the single scale stiffness matrix according to the bilinear form a(., .), fj - ((f, Wj,k))keb is the right-hand side and cj = (Cj,k)keb is the vector of unknown (single scale) coefficients. In the solution process, the first task is to set up the linear system (3.2.1), i.e., to compute the entries of the stiffness matrix and right-hand side. Here the entries of the stiffness matrix are integrals (of derivatives) of refinable functions over all of the domain. Note that this is different from the setup in finite element codes where first an element stiffness matrix on a reference element is computed and then the global stiffness matrix is assembled by using geometric transformations. Here the energy inner products can be reduced to integrals like
;_~ r176 I-I D~'cP'(x - k')dx, #',k' E Z a. J lts
(3.2.2)
i=1
Note that the domain f~ can be treated in terms of the indicator function ~0 which in our case is also refinable. We call the terms (3.2.2) refinable integrals because they are refinable as functions of (kl,..., k s) if all ~i are [19]. It is also proven there that the evaluation of terms like (3.2.2) reduces to the solution of an eigenvector problem which is uniquely solvable, provided that certain multivariate discrete moment conditions are added. A first implementation for the evaluation of (3.2.2) is documented in [25]. This was used here to implement an extended version using the above described data structures. To handle the factors in the integral, we designed the class I n t e g r a l F a c t o r . To explain its use, let us consider the following example of evaluating
/R2(g~ |
D(l'~174
1) D(~174
where D i" = 8ol~, 7 , i E Z d . This is handled as follows: int S p a c e D i m
= 2;
Mask FI, F2, F3; Mask IntegralValues ; F1 = N I ( )
9N I ( ) ;
2) dx
(3.2.3)
T. Barsch et al.
396 F2 = N3() F3 = N2()
9 N2(); 9 N3();
// For the construction only the spatial dimension // is needed. The particular factors are put into // this object of the class. IntegralFactor Functions (SpaceDim) ; Funct ions. AddFactor (FI) ; Functions. AddFactor (F2) ; Functions. AddFactor (F3) ; // // // //
Now the user determines the derivatives" the first parameter is the number of the function, the second one the index of the partial derivative, default value is zero.
Functions.putDer(2,1) Functions.putDer(3,2) integrals(Functions,
= I; = 1; IntegralValues) ;
The class I n t e g r a l F a c t o r contains the masks and the derivatives for the factors arising in the integral. The spatial dimension is the parameter used for the constructor so that this function is independent of the dimension. The masks of the refinable functions are added to the integral by the function Functions.AddFactor. The derivatives can be defined by the user by F u n c t i o n s . putDer ( F u n c t i o n , D i r e c t i o n ) as described above. Finally, i n t e g r a l s ( F u n c t i o n s , Values) evaluates the corresponding integral for all shifts k i 6 Z d, i - 1 , . . . , s, for which (3.2.2) does not vanish. The first parameter serves as input, the second as output. The first one is of type I n t e g r a l F a c t o r and the second one of the type Mask. This means that the result is also an object of the class Mask, which makes it easy to handle in further calculations. Obviously, we can rewrite (3.2.3) as
(fR N1 (~)N~(~-k~)N2(~-k 2) d ~ ) ( / R
N1 (7;)N2(r]-k~)N~(r]-k~)dr]),
(3.2.4) so that the refinable integral of tensor product functions is a (slightly modified componentwise) tensor product of the integral of the components. This is realized by the function ComponentTensorproduct in the following way:
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397
const int SpaceDim = 2; const int NumOfFac = 3; Mask Function_ID[NumOfFac] [SpaceDim] ; int Derivatives[NumOfFac] [SpaceDim] ; IntegralFactor Factor_ID [SpaceDim] ; Mask IntegralValues_ID[SpaceDim], IntegralValues,
Help;
Function_lD[0] [0] = NI(); Funct ion_ lD [0] [1] = NI();
Derivatives [0] [0] = O; Derivatives[O] [1] = O;
Function_ID[l] [0] = N3(); Funct ion_ ID [l] [l] = N2();
Derivatives[l] [0] = I; Derivatives[l] [I] = O;
Function_ID[2] [0] = N2(); Funct ion_ ID [2] [l] = N3();
Derivatives [2] [0] = O; Derivatives [2] [I] = I;
for (int i=O; i<SpaceDim; i++) { Redimension(Factor_iD [i], i) ; // set spatial dimension to i
f o r ( i n t j =0; j
}
integrals (Factor_ID [i], IntegralValues_ID [i] ) ; if (i==O) IntegralValues else IntegralValues
= IntegralValues_ID [0] ; =
ComponentTensorproduct(IntegralValues, IntegralValues_ID [i] ) ;
It turns out that the second tensor product version is much faster than the first one which computes (3.2.3). For the above example, the first algorithm took 36.122 seconds on a SILICONGRAPHICS R4400 workstation while only 0.063 seconds were needed for the second one. This difference gets even bigger when dealing with more factors in the integral or higher spatial dimensions. As already mentioned, the same behavior has also been observed in spectral methods [7]. Since the values of such an integral is also an object of the class mask,
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we have a straightforward way for the setup of the stiffness matrix A% in (3.2.1). As a simple example, let Cj be a basis such that (3.2.5)
a(99j,k, 99j,1) : Ogj a(~Oo,o, ~O0,l-k )
is valid where aj is some constant depending on j. This assumption is always satisfied in the shift-invariant case with bases consisting of derivatives of refinable functions satisfying (3.1.8) and (3.1.9) and constant coefficients in the energy inner product. Assume that the values (3.2.5) are stored in B i l i n e a r V a l u e s and the borders for multi-indices according to the entries of the stiffness matrix in MatrixBorders. T h e n the following procedure realizes the setup: int d; MaskBorder MatrixBorders ;
// spatial dimension
int MatrixDim = MatrixBorders.getMaskDim() ; matrix A(MatrixDim,MatrixDim) ; double alphaj ; ivector k(d), l(d), difference(d); int RowIndex, ColIndex, ValueIndex; k = MatrixBorders.getLowBords()
;
for (int i=0; i<MatrixBorders.getMaskDim(); { RowIndex = Index(k,MatrixBorders); 1 = MatrixBorders.getLowBords() ; for (int j=0; j<MatrixBorders.getMaskDim(); { difference = 1 - k;
i++)
j++)
ColIndex = Index(l,MatrixBorders) ; ValueIndex = Index(difference,BilinearValues); A(Rowlndex,Collndex)
= alphaj * BilinearValues. getl (Valuelndex) ;
Inc (I, MatrixBorders) ;
}
Inc (k, MatrixBorders) ;
}
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For the computation of the solution at a given point we have to evaluate refinable functions, which is done as a special case of i n t e g r a l s for the case of only one factor. 3.3
Multiscale transformations
One of the most important advantages of multiscale methods is the fast realization of asymptotically optimal preconditioners [4, 15, 20], which is based on the efficiency of multiscale transformations, also called the Fast Wavelet Transforms. Let us briefly recall the main ingredient for the above described classes of problems. In view of (3.1.4), any vj E Sj can either be written in single scale representation as
vj - -
E
ek ~ j , k
k~I~ or in multiscale form as j--1
vj - m--jo--1 kE Jm
The transformation Tj 9 ( d m , k ) m = j o - 1 ..... j - l , k e J m ~ (Ck)keb, which takes the multilevel coefficients to the single scale ones is known as multiscale transformation, see [12, 13, 14]. Although the matrix associated with Tj is in general not sparse, it can be written as a product of sparse matrices,
Tj w h e r e ~ m . _ (Tm 0
-
Tj-I
" "Tjo,
(3.3.1)
0 ) , m - j o , . . . , j - 1, and the matrices T,~ are sparse Id
and banded with bandwidth independent of m, since the functions ~m,k are compactly supported [12]. The size of T,~ is equal to #Ira so that the multiplication with Tm can be executed in O ( # I m ) arithmetic operations. Since #Ira ~ 2 m, the multiplication with Tj as a subsequent multiplication with the matrices Tin, m - j 0 , . . . , j - 1, requires only O ( # I j ) operations, i.e., it is of the order of unknowns and thus optimal. Moreover, the application of these transformations is stable since the underlying bases are stable themselves [13]. Solving (3.2.1) is equivalent to treat the system corresponding to the multiscale basis A % d j -- T ~ f j, (3.3.2) where dj - (dm,k)m=jo-1 .... ,j-l,kegm and
A,rj "- T ] A ~ j T j .
(3.3.3)
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T. Barsch et hi.
Here Tj* denotes the adjoint of Tj. Since Tj is not sparse, the matrix A % , if it would be explicitly calculated, would in general not be sparse either. Instead, whenever applying the matrix A% we always realize it in form of the product Tjo " " "Tj-I^" A c j T j - 1 ""'Tjo (3.3.4) that is applied in O ( # I m ) arithmetic operations, m - j 0 , . . . , j - 1. Note that we do not need to determine the inverse of Tj in this context because we never need to convert data from the single into the multiscale basis, only the other way round. Indeed, this can be seen by A % d j = Tj*A c j Tj T~- I c - Tj*A % c. We will now discuss the problem of assembling a matrix Tm first in the univariate case adapted to an interval. Following [17], the index set Im may be divided into the disjoint union Im - I L U I ~ U IRm where # I L and #IraR are independent of m. The matrix Tm can be structured into Tm - (~o, ~lm) where the blocks are given by f,
(r163
_
9
,
9
9
,
9
9
, ,
, i -- O, I.
~176
*
*
(3.3.5)
*2
In this example the first and last two rows correspond to I L and Im R, respectively. Since the entries for every row are the same in the interior part, the matrix can be stored very efficiently using the class blband, see Section 2. The function that sets up the matrix consists of three nested loops: (a) The first loop distinguishes between the mask coeffcients a m for the refinable functions and b m for the wavelet. (b) The second loop consists of the different parts according to the partition of the ~et of indices Im (see loop4rows below). (c) The third and innermost loop is of the length of the mask size and sets up the particular entry of the matrix (for (cols(DIR) . . . ) ). In higher dimensions using tensor products, the matrix Tm will be a tensor product of matrices of the above kind: now we first have to loop over the corners of the unit box, i.e., over {0, 1} d, where in each direction a "0"
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401
stands for taking the generator and "1" for taking the wavelet. This loop again uses the above described mechanism of MaskBorder and i v e c t o r . For each of these functions we recursively start the other loops as follows. Note that rows and c o l s now denote blocks of rows and columns. The procedure loop4rows is called initially with DIR=0. The parameter DIR is the one used for the recursion. The two members of blband, L i t t l e B l o c k s , and GreatBlocks, contain the data of this level (see Section 2)" void loop4rows(blband ~A, ivector ~rows, ivector acols, int ~DIR, int SpaceDim)
{
DIR++; // DIR == SpaceDim => reached end of recursion // DIR != SpaceDim => start next loop in recursion if (DIR==SpaceDim) // ... allocate memory for double-entries in A else // ... allocate memory for blband-entries in A int Counter ;
// represents here different counters
int masklength; // contains different values, // depending on rows(DIR) // first we treat l_m'L : for (/, rows(DIR) kin I_m'L ,/) { if (DIR==SpaceDim) { // setup information about masklength, location within // the matrix A etc. ; then start the inner loop over // the length of the mask:
}
for (cols(DIR)=O; cols(DIR)<masklength; cols(DIR)++) A.LittleBlocks[Counter][cols(DIR)] = ... // store entries in A (doubles)
else
{ // setup information about size and position of the // blocks in A; then start the inner loop with the // next blocklevel of A: for (cols(DIR)=0; cols(DIR)<masklength; cols(DIR)++)
T. Barsch et al.
402
loop4rows(A.GreatBlocks [Counter], rows, cols, DIK, SpaceDim) ;
// Now the same for the other two parts of I_m: for (/, rows(DIK)
kin l_m^O */)
// ...
for (/, rows(DIK)
kin l_m^K */)
// ...
DIR-- ;
/ / e n d of r e c u r s i o n
In summary, the application of multiscale transformations, when applicable, is a fast process. Its effectivity depends to a large extent on the fact that the underlying structures are defined on a uniform grid. However, there are situations when a nonuniform grid may be preferred. For instance, wavelet-based adaptive schemes are designed to reduce the overall complexity by ignoring the basis functions that are not necessary for a good resolution [11]. This makes it necessary to work with the wavelet basis itself and to set up and apply the system of equations corresponding to the multiscale representation, which are in general not (9(#Ij) processes. Also, the optimal compression results for elliptic singular integral equations along the lines of [20, 21] are based on carefully weighted adaptive quadrature rules in terms of wavelets which cannot make use of the fast multiscale transformations. Moreover, when dealing with divergence-free wavelets, it could be more efficient to set up and apply the stiffness matrix with respect to the divergence-free wavelets. In this case, one starts by multiresolution spaces of vector fields with components in HI(Ft). The corresponding complement spaces are decomposed into the divergence-free part and a stable complement. That means that the single scale basis does not consist of divergencefree functions. Solving the system according to a compressed multiscale divergence-free basis might overcome this difficulty. 3.4
Multilevel preconditioners
The relevance of multiscale transformations in numerical analysis is that together with a diagonal matrix, they lead to asymptotically optimal condition numbers and thus to a fast iterative solution of linear systems (3.2.1)
independent of the refinement level j. To recall the basic results, let H 8 = HS(~), s 6 lR, be a scale of Sobolev spaces on the domain underlying the problem, where here H s for s < 0 is
Object Oriented Software Tools for Multiscale Methods
403
to be understood as (H-~) ~ and H ~ - L 2. The operator in question, A 9H ~+r --, H ~, for which our software is developed is typically boundedly invertible, IIAvllHs ... IlvlIH.+~, V E H s. (3.4.1) This, together with the norm equivalences
Z
Z
kEJj
j=0
L2
~ Ilvll +.,
+
e
(3.4.2)
which are assumed to hold for some ~,3' > 0 and which are valid for many known biorthogonal wavelet systems [12, 13], proves the following [15, 20, 24]. T h e o r e m 1. Let the operator A satisfy (3.4.1) with some r E lR and let (3.4.2) hold for s + r E (-~/, 7). If Ds denotes the diagonal matrix with entries
2 sm
(f(m,k),(m',k'),
m,m'
-
-
jo-1,...,j
_
1, k E Jm,
kI
E Jm,,
then one has c o n d ( D _ r / 2 A ~ D _ r / 2 ) - O(1),
j --4 c~,
(3.4.3)
where condB "- IIBll lIB-111 and II" II is the spectral norm. This shows the crucial role of multiscale transformations for preconditioning: on the one hand, one wants to store A~ because of its sparsity while on the other hand A~ together with a diagonal matrix gives rise to an optimal preconditioned linear system. Note that Theorem 1 is also valid for nonsymmetric operators A. If A has positive order and is symmetric it should be mentioned that closely related alternative preconditioners often referred to as BPX-preconditioners or multilevel Schwarz schemes are available [4, 15, 28, 29] that do not need to use the wavelet bases explicitly. In this case, the operator
j-1 Cjtt - A ; 1 ~ (u, ~jo,k)L 2 qPjo,k + ~ 2-2mr ~ (it, qPm,k)L2 ~m,k kEIj o m=jo kEIm
(3.4.4)
also gives rise to optimal condition numbers, i.e., cond(Cj1/2 A~jCr (9(1), j -4 c~. However, the application of this preconditioner requires some restriction and prolongation operators because information concerning all levels is to be summed up. In numerical experiments with different implementations for the restriction and prolongation operators, we achieved
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T. Barsch et al.
the best results when using block-banded matrices which were set up at the beginning and applied as matrix-vector multiplications in every step. The use of the class blband makes the preconditioner very fast since expensive index calculations can be avoided completely during the algorithm. Moreover, it also needs little memory (less than 100 KByte in two and three dimensions) due to its specific structure. The structure of the matrices is almost the same as above. The only difference is that, e.g. the restriction matrices look like the upper half of the matrix in (3.3.5). Therefore, the first loop over the generators and wavelets is obsolete. The rest is essentially the same as above. w
Outlook
We view the software described in this paper as a first step of the implementation of multiscale methods for "realistic" problems (for example those mentioned in [34]). Here we outline some of the future projects which are already in preparation. The Stokes problem can be seen as the linear core of the incompressible (stationary) Navier-Stokes equations which is a system of nonlinear partial differential equations. In [18] adapted trial functions and asymptotically optimal preconditioners for the time dependent Stokes problem have been constructed. Treating the nonlinearity by linearization then gives a solver in terms of a series of time dependent Stokes problems which should be reasonable at least for moderate Reynolds numbers. Another way to solve these equations is to consider a sequence of Oseen problems [31], which are nonsymmetric problems. Since multilevel preconditioners as described in Subsection 3.4 are also available for a class of nonsymmetric problems, developing multilevel schemes for these kind of problems, as a preliminary step towards a Navier-Stokes solver. On the other hand, this approach requires solving a convection-dominant problem, which will be considered in further research. The full Navier-Stokes equations including the nonstationary terms, requires then a solver for time dependent problems. Another important class of problems arising in fluid dynamics are hyperbolic conservation laws. There is already ongoing work concerning the implementation of multiscale methods using the above described software tools for the numerical solution of these systems [23]. One of the perhaps most promising features of multiscale methods is the availability of error estimators and adaptive schemes that already fulfill the saturation property [11]. Since adaptivity is crucial for the efficiency of a solver, the development of corresponding software is an important project. The implementation of these methods requires a very careful look at the data structure because adaptivity treats only the significant coefficients of
T. Barsch et al.
406
Table 1. Needed memory, number of iterations, residual and CPU time in seconds for the 2D mixed formulation. j
Unkn.
2 3 4' 5 6 7
A B 1 0 21 6 2 133 9 . . . . 13 645 2821 15 59 249 11781 28 48133 55 1624
Uz. pcg 3 4 13 156 21'331 25 454 24 496 20 487
Residual 3.16354e-15 1.93954e-06 2.81319e-06 5.84368e-06 1.25834e-05 3.07779e-05
Setup 0.138 0.158 0.198 0.307 0.644 1.776
Its. 0.005 0.185 1.927 ~ 15.330 73.566 288.111
Bj "- ( /~ Cj,k(X) div ~Pj,m(X)dx) keIjM'mEIjx fj
:=
( / f (X) CPj,k(X) dX) keljX
and the trial spaces Mj "- span({r 9k E IM}) and Xj "- span({~j,k 9 k E I x }) are built in such a way that the LBB condition is fulfilled. As a generator for the velocity trial space X j we have used a tensor product cardinal B-spline of order three. The trial space Mj for approximating the pressure pj is spanned by the biorthogonal functions ~52,4 from [9]. The construction of the corresponding trial spaces can be found in [18]. The results are obtained on a PC, INTEL PENTIUM 90 with 16 MByte memory using LINUX and the gcc compiler 2. In Table 1 we display the level j and the corresponding number of unknowns. The columns A and B contain the size of the needed memory in KByte for the corresponding stiffness matrices A, B. Uz. shows the number of iterations for the Uzawa algorithm, the iteration method which is used to solve the saddle point problem (5.1.1), and p c g means the total number of pcg steps for the particular level with B PX preconditioner (3.4.4). The last three columns give the size of the residual in the ~2 norm, the time in seconds needed for the setup, and the time for the whole iteration, respectively. The computation of the solution requires some additional effort, namely, the initial setup, the evaluation of the values of the solution, and the output into a file. For this example, these are displayed in Table 2. The initial setup includes the determination of the range of indices and the evaluation of the inner products. Furthermore, the setup of the restriction and prolongation matrices and the solution of the prob2Copyright by GNU software foundation.
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Table 2. Additional time requirements for the 2D example. Initial setup and coarse level solution Evaluation of the values of the soiution Output into a file
2.272 sec. 21.430 sec. 1.271 sec.
lem on the coarsest level by a direct method are also included, which are needed for the application of the BPX preconditioner (3.4.4). The evaluation of the solution values requires the evaluation of refinable functions on a grid of gridsize 2 -t, where l is given by the user. In this example we have chosen 1 = 6. After this evaluation, which can be done by using the routine i n t e g r a l s described in Subsection 3.2, the linear combination with the computed coefficients has to be determined. The displayed time in Table 2 corresponds to level j = 7 and includes the values for velocity and pressure. We displayed the resulting velocity field (here for the divergence-free case) in Figure 1. It shows the particular good resolution of the secondary vortices in the edges of the cavity.
0.0
0.2
0.4
o.6
0.8
1.0
0.00
0.~
0.10
0.I$
0.000
0.006
0.010
0.015
Figure 1. Vortices of the solution of the driven cavity Stokes problem in two spatial dimensions (zoom-in).
5.2
D i v e r g e n c e - f r e e f o r m u l a t i o n of t h e Stokes p r o b l e m in 3D
The three-dimensional test example concerns the divergence-free formulation which leads to a positive definite system
A j v j = f j,
(5.2.1)
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T. B a r s c h et al.
where Aj
"-
(/agradCt,vk(x)gradCt,Vm(X)dx)t=j ~..... j - l , k , m E I t
l'-jo,...,j-l,kEIt
and c vl,k denote the divergence-free wavelets. Note that this is already the multiscale representation of the equation. Computations were performed on a SILICONGRAPHICS R4400 workstation with 64 MByte memory under the SILICONGRAPHICS compiler (according to IRIx-version V.2, VI.0.1). As a trial function we took a tensor product cardinal B-spline of order three as starting point for the construction of divergence-free trial spaces as introduced in [27, 33]. The results are displayed in Tables 3 and 4.
Table 3. Needed memory, number of iterations, residual and CPU time in seconds for the 3D divergence-free formulation. Level 2 3 4 5 6
Unkn. 16 432 5488 54000 476656
A 7 424 601 955 1663
pcg 2 34 91 196 389
Residual 2.72656e-14 2.37998e-06 7.11948e-06 2.19192e-05 6.38941e-05
Setup 0.026 3.518 25.706 131.022 605.513
Iteration 0.003 0.505 16.154 365.267 7009.759
Table 4. Additional time requirements for the 3D example. Initial setup and coarse level solution Evaluation of the values of the solution Output into a file
75.545 sec. 124.792 sec. 1.407 sec.
Finally, the streamlines of the resulting velocity fields are displayed in Figure 2. A c k n o w l e d g m e n t s . The authors are grateful to Frank Knoben for many valuable hints concerning the implementation. The second author is supported by the Deutsche Forschungsgemeinschaft.
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Figure 2. Streamlines of the velocity field in the 3D case. References
[1]
Andersson, L., N. Hall, B. Jawerth, and G. Peters, Wavelets on closed subsets of the real line, in Recent Advances in Wavelet Analysis, L. L. Schumaker and G. Webb (eds.), Academic Press, Boston, 1994, pp. 1-61.
[2]
Barsch, T. and K. Urban, MLLib - software tools for multilevel methods, in preparation.
[3] Braess, D. and W. Dahmen, Wavelet methods for mixed formulations of second order elliptic problems, private communication. [4] Bramble, J. H., J. E. Pasciak, and J. Xu, Parallel multilevel preconditioners, Math. Comp. 55 (1990), 1-22. [5] Brezzi, F. and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, Berlin, 1991. [6] B/icker, M. and M. Sauren, Private communication. [7] Canuto, C., M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988. [8] Cohen, A., W. Dahmen, and R. A. DeVore, Multiscale decompositions on bounded domains, IGPM-Report Nr. 113, RWTH Aachen, 1995. [9] Cohen, A., I. Daubechies, and J. Feauveau, Bi-orthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560.
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[10] Cohen, A., I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal. 1 (1) (1993), 54-81.
[11]
Dahlke, S., W. Dahmen, R. Hochmuth, and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, IGPMReport Nr. 124, RWTH Aachen, 1996.
[12] Dahmen, W., Some remarks on multiscale transformations, stability and biorthogonality, in Wavelets, Images and Surface Fitting, P. J. Laurent, A. Le M~haut~, and L. L. Schumaker (eds.), A K Peters, Wellesley, 1994, pp. 157-188. [13] Dahmen, W., Stability of multiscale transformations, J. Fourier Anal. Appl. 2 (1996), 341-361. [14] Dahmen, W., Multiscale analysis, approximation, and interpolation spaces, in Approximation Theory VIII, C. K. Chui and L. L. Schumaker (eds.), World Scientific Publishing Co., 1995, pp. 47-88. [15] Dahmen, W. and A. Kunoth. Multilevel preconditioning, Math. 63 (1992), 315-344.
Numer.
[16] Dahmen, W., A. Kunoth, and R. Schneider, Operator equations, multiscale concepts and complexity, WIAS-Report 206, 1995, in: Lectures in Appl. Math., J. Renegar, M. Shub and S. Smale (eds.), AMS, Providence, RI. [17] Dahmen, W., A. Kunoth, and K. Urban, Biorthogonal spline-wavelets on the interval- stability and moment conditions, IGPM-Report Nr. 129, RWTH Aachen, 1996. [18] Dahmen, W., A. Kunoth, and K. Urban, A Wavelet-Galerkin method for the Stokes-equations, Computing 56 (1996), 259-302. [19] Dahmen, W. and C. A. Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal. 30 (1993), 507-537.
[20]
Dahmen, W., S. PrSssdorf, and R. Schneider, Multiscale methods for pseudodifferential equations, in Recent Advances in Wavelet Analysis, L. L. Schumaker and G. Webb (eds.), Academic Press, Boston, 1994, pp. 191-235.
[21]
Dahmen, W. and R. Schneider, Multiscale methods for boundary integral equations I: Biorthogonal wavelets on 2D manifolds in ~3, in preparation.
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[22] Girault, V. and P.-A. Raviart, Finite Element Methods for NavierStokes-Equations, Springer-Verlag, Berlin, 2nd edition, 1986. [23] Gottschlich-Miiller, B., A multiresolution scheme based on biorthogonal wavelets for scalar conservation laws in one space dimension, in preparation. [24] Jaffard, S., Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal. 29 (1992), 965-986. [25] Kunoth, A., Computing refinable integrals - Documentation of the program- Version 1.1, ISC-95-02-Math, Texas A&M-University, 1995, preprint. [26] Kunoth, A., Multilevel preconditioning- Appending boundary conditions by Lagrange multipliers, Adv. Comput. Math. 4 (1995), 145-170. [27] Lemari~-Rieusset, P. G., Analyses multi-r~solutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs divergence nulle, Rev. Mat. Iberoamericana 8 (1992), 221-236. [28] Oswald, P., On discrete norm estimates related to multilevel preconditioners in the finite element method, in Constructive Theory of Functions, Proc. Int. Cone Varna 1991, K. G. Ivanov, P. Petrushev, and B. Sendov (eds.), Bulg. Acad. Sci., Sofia, 1992, pp. 203-214. [29] Oswald, P., Multilevel Finite Element Approximation, Teubner Skr. Numer., Teubner-Verlag, Stuttgart, 1994. [30] Robichaud, M. P., P. A. Tanguy, and M. Fortin. An iterative implementation of the Uzawa algorithm for 3-D fluid flow problems, Internat. J. Numer. Methods Fluids 10 (1990), 429-442. [31] Turek, S., Ein robustes und effizientes Mehrgitterverfahren zur L6sung der instation/iren, inkompressiblen 2-D Navier-Stokes-Gleichungen mit diskret divergenzfreien finiten Elementen, dissertation, Universit~t Heidelberg, 1991. [32] Urban, K., Multiskalenverfahren fiir das Stokes-Problem und angepat3te Wavelet-Basen, Verlag der Augustinus Buchhandlung, Aachen 1995. [33] Urban, K., On divergence-free wavelets, (1995), 51-82.
Adv. Comput. Math. 4
[34] v. Watzdorf, R., K. Urban, W. Dahmen, and W. Marquardt, A Wavelet-Galerkin method applied to separation processes, in Scientific Computing in Chemical Engineering, S. Keil, W. Mackens, H. Vof~, and J. Werther (eds.), Springer-Verlag, Berlin, 1996, pp. 246-252.
Titus Barsch Institut fiir Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55 52056 Aachen Germany barsch~igpm.rwth-aachen.de Angela Kunoth Institut fiir Geometrie und Praktische Mathematik RWTH Aachen 52056 Aachen Germany kunoth~igpm.rwth-aachen.de Karsten Urban Institut fiir Geometrie und Praktische Mathematik RWTH Aachen Templergraben 55 52056 Aachen Germany urban ~igpm. rwt h-aachen, de
Scaling Function and Wavelet Preconditioners for S e c o n d O r d e r Elliptic P r o b l e m s
Jeonghwan Ko, Andrew J. Kurdila, and Peter Oswald
Abstract. In this paper we present a theoretical framework and numerical comparisons for multilevel solution procedures associated with both scaling functions and wavelets of second order elliptic boundary value problems for a simple class of bounded domains. In particular, we consider a multiwavelet formulation using AFIF elements. The advantage is in the simplicity of the boundary modification, and relatively small masks representing the differential operators, in contrast to other wavelet-based methods. A brief comparison to conventional finite element methodologies is included.
w
Introduction
It is by now well appreciated that wavelets and multiresolution analysis are useful as an analysis and discretization tool (see, e.g., [31, 15, 5, 17]). Particular attention has been paid to the incorporation of wavelet and multiresolution analysis in the study of integral, differential, and pseudodifferential operators. For example, Beylkin, Coifman, and Rokhlin in [3, 2] have derived estimates for the compression of pseudo-differential operators. Jaffard (see [24] and [25]) has obtained a number of interesting results applicable to elliptic boundary value problems including asymptotically optimal complexity results, local regularity, and refinement methodologies that retain optimal approximation order. Dahmen, PrSt3dorf, and Schneider [12, 13, 14] have investigated the various approximation theoretic issues in the representation of pseudo-differential operators in terms of wavelets. More recently, Dahlke, and DeVore [8] have utilized wavelet techniques to M u l t i s c a l e W a v e l e t M e t h o d s for P D E s W o l f g a n g D a h m e n , A n d r e w J. K u r d i l a , and P e t e r O s w a l d ( e d s . ) , pp. 4 1 3 - 4 3 8 . C o p y r i g h t (~)1997 by A c a d e m i c P r e s s , I n c . All r i g h t s of r e p r o d u c t i o n in a n y f o r m reserved. ISBN 0-12-200675-5
413
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derive the regularity of solutions to boundary value problems in the scale of Besov spaces as a prelude to the investigation of adaptive algorithms. One early result, derived, for example, by Jaffard in [24], that attracted the interest of many researchers active in computational mechanics and scientific computing, was the potential for asymptotically optimal complexity in wavelet G alerkin methods. Essentially, this result asserts that if an elliptic boundary value problem is cast in terms of a wavelet basis, there is a diagonal preconditioner that uniformly bounds the condition number of the matrix representation of the equations that must be solved. Some of the early papers in the use of wavelets for approximation of partial differential equations provided empirical verification of this fact for elliptic problems subject to periodic boundary conditions. Tables that illustrate that the condition number of the preconditioned matrix associated with elliptic equations does not increase with dimensionality can be found in [24] and [3]. In a philosophically similar result, Rieder et al. [34, 35, 36] show that multigrid methods can be derived using wavelets in which the contraction rate is independent of the number of levels of discretization. Meanwhile, a number of researchers have concentrated on practical implementations of wavelet formulations in a growing number of applications. An important question, the discussion of which is neglected in many publications, is how one goes about actually computing with some of the "esoteric" wavelet bases that have been derived by analysts. Along these lines, the work of Latto, Resnikoff, and Tenenbaum [28], and the later, more comprehensive paper by Dahmen and Micchelli [11] give a good account of the calculations required to implement wavelets in approximation of differential equations. Similar results may also be found in [1] and [3]. Alternatively, fictitious domain techniques have been employed by Glowinski et al. in [19, 20] to enable a unified treatment of different boundary conditions. In the present paper we focus on a class of domains which is still very close to the rectangular situation, and discuss multilevel preconditioners based on the use of scaling functions (C-algorithms) and wavelet functions (C-algorithms). In Section 2, we briefly state some general assumptions under which these multilevel algorithms lead to uniformly bounded condition number estimates, and, thus, to asymptotically optimal algorithms for second order elliptic problems. This material is essentially known. However, due to the assumption on the domains, no boundary modification is necessary. This also simplifies the implementation. In Section 3, the particular example of a multiresolution analysis using so-called AFIF elements is examined. These multiwavelets have been introduced by Hardin et al. [23, 18, 16], and are somehow intermediate to classical finite elements and wavelet families such as Daubechies' compactly supported orthogonal wavelets. The application to the discretization of boundary value problems has previously been discussed in [27, 26]. Numer-
Scaling Function and Wavelet Preconditioners
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ical comparisons, including AFIF elements, Daubechies scaling functions, and low order finite elements, are reported on in Section 4 for typical H 1problems in one and two dimensions. They confirm the theory outlined in Section 2 and show the asymptotical optimality of the preconditioners. In these experiments, the finite element solvers perform slightly better while the AFIF element preconditioner is more robust if low order terms are present. w
Multilevel framework
We start by introducing a multiresolution analysis on all of ]Rd. Since we will use second order elliptic boundary value problems as our prototype for comparing multilevel preconditioners, we make the following assumptions: (A1) The scaling functions Ct E L2(lRd), l - 1 , . . . , L, have compact support and belong to H t (IRd) for some t > 1. (A2) The set of integer shifts of the scaling functions has the property of local linear independence. That is, if A0 denotes the set of all index pairs (1, a), 1 = 1 , . . . , L, a E Z d, such that supp Ct (. _ a) N (0, 1) d # 0, then we can conclude that
[t
~ cl,~ (/,a)eho
r
(" -- a) t]L~((0,z)a)) -- 0 ===~ c t , ~ - - 0
V(l,a) eA0.
(A3) The space V0 = spanL2 {r (. _ c~)} contains linear polynomials. (A4) The scaling functions are refinable, i.e., L /~=1 aEZ d
for some finite sequences (a~ t'). The scaling functions corresponding to Daubechies' family of orthonormal, compactly supported wavelets, Coifman's compactly supported wavelets, affine fractal wavelets, and standard finite elements on uniform partitions satisfy these hypotheses. It is well known that multiresolution analyses defined via sufficiently smooth scaling functions (resp. wavelets) provide explicit characterizations for a number of function spaces including the Sobolev and Besov scales on
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IRd, see [31, 17]. For example, as was stated in [33, Sections 2 and 5.1], the system of functions -
- .).
.
e z
,j=O,
1,...,l=l,...,L,
generated from a set of scaling functions {r satisfying properties (A1) through (A4) yields a frame in HS(IRd), 0 < s < min(t, 2), after normalization. Criteria for the construction of wavelet and other multilevel Riesz bases in Besov-Sobolev spaces are discussed in [10]. Definitions for Riesz bases and frames, and their connection to multiresolution analyses and wavelets can be found in, e.g., [15, 6, 5, 33]. Extensions of such results to domains fl C IRd, d > 1, usually require a sort of boundary modification in the system {r t for each j >_ 0, those of the r with support intersecting with a certain neighborhood of the boundary 0fl have to be replaced by their boundary-adapted counterparts r while functions are simply dropped from the system if supp r Cl fl = 0. See [7, 33] for examples in the case d > 1 where local support of the functions and local polynomial reproduction in the resulting subspaces are taken as the important features to be preserved. In other papers (such as [24, 25]), the boundary adaption is incorporated via a Schmidt orthogonalization process and may lead to functions with global support. In this note, we considerably simplify the task by concentrating on the following class of domains. Let Q = (0,1) d denote the open d-dimensional unit cube, and introduce the notation Qj,~ = 2 - J ( Q - a) for the dyadic cubes of level j >_0 (c~ e zd). We say that a bounded open domain fl C ]R d is aligned with the cube structure if for some j0 _> 0 there is an index set Y_~C Z d such that fl = ~.J Q~o,~. aEIo In addition, we exclude the possibility of slits and cuts (what is actually needed is the extension property for HS(fl) (s > 0) to hold). The last assumption can be removed if the following construction is slightly changed, compare the discussion in [32] for the case of finite element multilevel schemes. After scaling, we may assume that j0 = 0, which we will do throughout the paper. For domains that are aligned with the cube structure, a boundary modification is not necessary, simply set (I)j,a -- {r
" supp r
N fl # q}},
Then, Cj,~ is an algebraic basis in Vj,a - (span{r which we call nodal basis in Vj,~.
,
j k 0,
j k O.
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1. Suppose that {r satisfies (A1)-(A4), and that f~ is aligned with the cube structure (jo = 0). Then {Vj,~ } is a multiresolution analysis with the following properties:
Theorem
(i) For any gj - E(l,~) cl,.r
c Vj,n,
ilgi[12ix(a) ~" .., Z
(l,~)
2 -jd [cl,,~I2 9
(2.1)
(ii) For 0 < s < min(t, 2), we have CX)
ilfll2H.(~) v
i?f
~.
oo
~
g~eyj,~: =~j=o gJ ~=0
22J~l]gj[125=(~)
(2.2)
for all f 6 H s (~). In other words, the system (X)
~, - (.J %,,
(2.3)
j=O
is a frame in Hs(Q) after normalization, i.e.,
Iifll~.. II ,d
~
(f, Cj,~)Hs(~) I 2
II _t'j[
j=o (l,~)
V f C HS(~).
(2.4)
L2(12)
In the above statements, • stands for a two-sided estimate, with positive constants that may only depend on ~, s, d, and on {r The summations ~(l,a) are with respect to index pairs representing the basis functions in ~j,a .
We sketch the proof of Theorem 1 for the sake of completeness only. By assumptions (A1)-(A2), and using the usual dilation arguments, we observe there is a function r with support that for each cube Qj,z c supp r in Qj,z, such that ][r l <-- c2Jd/2 (2.5) and
Q
r 3,~,f~(x)r l'
(x) dx - ~t,~'~,~'
(2.6)
for all a' E Z d, l' - 1 , . . . , L (here, and in the following, C denotes a generic positive constant). With these biorthogonal functions at hand, various quasi-interpolants can be introduced. For the domains under consideration, for any j _> 0 and any (/, a) corresponding to a nontrivial basis function
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r E Oj,n, we can fix the above cube Qj,~ to be inside ~ (here, the specific structure of the domain is essentially used). Define
(2.7) (l,o~)
~,~
Obviously, by (2.6) we see that Qj is a projection from L2(Ft) onto Vj,n, and the L2-stability of the basis (I,j,n expressed by Theorem 1 (i) follows from the local support assumption (A1) and (2,5) in a standard way. To prove assertion (ii) of Theorem 1, it is enough to verify Jackson-Bernstein inequalities for the sequence {Vj,n } as stated below, compare the approach of Dahmen [10], especially, Theorem 5.1.1 (i). Since (A3) impies at most reproduction of linear polynomials, the standard second-order L2-modulus of continuity w2(5, f)L~. - - sup IIA~filL~(a) o
-
h)
o
if I x - h,x + h] C otherwise.
We can prove a Jackson-type inequality
Ill -
QjfliL~(a)
<
Cw2( 2-j,
f)L2,
f e L2(fl),
(2.8)
by first establishing it for smooth functions f E H2(gt) in the form
IIf
-
Q.ifllL~.(a) < C2-2J IlfllH~(a),
(2.9)
and then using real interpolation (to this end, the L2-boundedness of Qj,n is needed which follows directly from the formula (2.7)). To show (2.9), the definition (2.7) of Qj,a, assumption (A3), and a Bramble-Hilbert argument have to be explored (here, the exclusion of slits is important for the proof, otherwise the union of the spaces Vj,a might not even be dense in HS(~)). The Bernstein inequality
W2(5, g~)L~ < C(min(1,2JS))~llg~llL~(a) , gj
e V~,a ,
(2.10)
holds for all 0 < s < min(t, 2), with a constant C depending on s but not on gj and j >_ 0. The simple proof uses the definition of the modulus of continuity via differences and (A1). The case ~ _> 2-J is trivial since w2(5, f)L2 _< 411fllL2(n) by the triangle inequality. For 5 < 2-J, the local support property of the scaling functions leads to L2
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Scaling Function and Wavelet Preconditioners
Now using the continuity of the embedding H t ( ~ ) C B~,~(Ft), where [If]lm~,~(n) = I]fl]L2(n) + sup 5-Sw2(5, f)L2 , 5>0
and a dilation argument, we get
< c(2'5) 2 for all index pairs (/, a) of interest. Substitution together yields with (2.1) the desired (2.10). For a more detailed exposition of these arguments, see [32]. Finally, the equivalence of (2.2) and (2.4) follows from (2.1) and general properties of frames (see [15, Section 3.2] or [33, Section 2]). Norm equivalences as stated in Theorem 1 play a crucial role in deriving multilevel preconditioning methods for operator equations in HS(~) (see [10, Section 5], [33, Section 2], and in a finite element context, [32, Section 4]). For the connection to multigrid theory, see [4, 22]. We quote some consequences of Theorem 1 for a generic HI-elliptic boundary value problem, i.e., we put s - 1 in the following. Let a(., .) be a symmetric, continuous, Hi-elliptic bilinear form, and b(.) a continuous linear functional on H 1(Ft). Denote by u the unique solution of the variational problem Y v e g 1( ~ ) ,
a(u, v) = b(v)
(2.11)
and by u g E Vj, a the solution of its projection to Vg, n" (2.12)
V v j E Vg, a .
a(u g, Vg) -- b(vg)
Introduce the following operator equation in Vj, a: P j u g = ~g ,
(2.13)
where
j=o (t,a)
d},a
This equation is called additive Schwarz formulation for (2.12) associated with the subspace splitting J
- Z Z vJ,o,o j=o (l,~)
into one-dimensional subspaces spanned by the individual functions r J,c~ [a from the finite section J
(~g,n - [,.J (I)j,n j=O
(2.14)
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J. Ko et al.
of the infinite system (2.3). The scaling factors in the definition of P j and bj are chosen such that d~,~ • 2 (2-d)j • 22JlIr
) • a(r
r
9
(2.15)
It is straightforward to check that (2.13) is equivalent to (2.12) and that 7~j is symmetric positive definite with respect to a(., .) since
J Z F_, j=o (t,~)
a(uj r '
[f~)a(vj r d~,~
'
,
and ~g,a is a generating system for Vj,~. The important consequence of Theorem 1 (ii) is: T h e o r e m 2. Let the conditions of Theorem 1 and (2.15) be satisfied. The spectral condition number of ~)g is uniformly bounded for J >_ O. The bound depends on the constants in the two-sided estimate (2.4) (or, equivalently, on the constants in (2.1) and (2.2)), and in (2.15). The proof can be found in [32, Section 4.1]. What makes (2.13) interesting for solving (2.12) numerically is that in addition to the J-independent behavior of the condition numbers, the action of P j can be implemented as a sparse operation (see also [33, Section 2]). From the above definition of P j, it becomes clear that the matrix representation of (2.13) with respect to the nodal basis r of Vj,~ takes the form
CjAj - Cjbj
,
(2.16)
where l' 9 A j is the stiffness matrix with the entries a(r162 (the dimension n j of this matrix coincides with the number of nontrivial
basis functions r
on level J),
9 bj is a vector with entries b(r 9 C j is a symmetric multilevel preconditioner which possesses a recur-
sive structure Co-Do,
Cj - I j C j _ I I T + Dj , j - 1 , . . . , J ,
(2.17)
where the matrices Dj are diagonal nj • nj matrices containing the scaling factors (d),~) l -1 on the diagonal, and the matrices Ij of dimension nj x nj-1 contain the elements of the refinement equations (A4) and represent the exchange between neighboring levels. I T is the transpose of Ij. Readers familiar with standard multigrid implementations [22] will recognize the analogy to a V-cycle with one smoothing step.
Scaling Function and Wavelet Preconditioners
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This allows us to solve (2.12) via the equivalent formulation (2.13) by a preconditioned conjugate gradient (pcg) method. According to Theorem 2, the number of pcg iterations to reach a fixed error reduction is bounded by a constant independently of J. The complexity of one iteration step can be estimated by O(nj) arithmetical operations, at least for typical HI-elliptic problems such as second order problems with Neumann or Robin boundary conditions. The corresponding algorithm is called a C-algorithm (associated with a scaling function preconditioner), in contrast to C-algorithms (associated with wavelet preconditioners) which will be briefly discussed next. What we call wavelets associated with the sequence {Vj,~ } are locally supported basis functions Cj,i, i = 1 , . . . , mj - nj - n j - 1 , for the L2orthogonal complement spaces
Wj,~ = Vj,~ OL2 Vj-I,~ ,
j _> 1,
(2.18)
leading to a L2-stable Riesz basis in Wj,a. Note that in other papers, such Cj,i are called semiorthogonal wavelets or prewavelets. Again, for domains aligned with the cube structure (and j0 - 0), the construction of such functions can be reduced to the case of 1Rd and a boundary modification which is essentially one-dimensional. In some cases, like for the AFIF elements discussed in Section 3, this boundary modification is trivial. What we gain in comparison with the frame concept, i.e., the use of the scaling function system ~ , is the Riesz basis property of the system
in Hs(fl) (resp. of the finite sections ~g,o in Vj, o), for a larger interval of Sobolev exponents, including s = 0, s = 4-1/2, and s = - 1 . The following result is a direct consequence of Theorem 1 and duality arguments; it is a particular case of [10, Theorem 5.1.1 (ii)]. T h e o r e m 3. Let the assumptions of Theorem 1 be satisfied. Assume that the wavelet bases ~j,~ of the spaces Wj,~ defined in (2.18) are uniformly L2-stable, i.e., mj
II~ ~r i--1
mj
• ~ 2-~dc~
(2.19)
i--1
for all coetticient choices and j _> 0. Then ~ a is, after normalization, a RJesz basis in H~(fl) [or ali - m i n ( t , 2) < s < min(t, 2). Again, the associated Schwarz formulation (with respect to sections of ~ a ) of symmetric HS-elliptic variational problems will lead to uniformly well-conditioned linear systems, and to preconditioners which possess a recursive structure similar to (2.17):
Co - Do,
Cj = I j V j _ l I T + b D j I T , j - 1,..., j .
(2.20)
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422
The new matrices /*j are of size nj • mj and contain the wavelet mask coefficients given by the representations
Cj,i
At =
t
,_.,
(t,,~)
as entries. The diagonal matrices /gj of dimension mj contain scaling factors (dj,i) -1 which approximately equal the energy of the corresponding wavelet functions (replace the scaling functions by Cj,~ in the assumption (2.15)). At the first glance, applying a C-algorithm seems to be more costly, on the other hand, one expects a better robustness, especially if lower order terms (like in a Helmholtz problem) are involved. {}3
AFIF elements
In this section we briefly describe a particular multiresolution analysis using so-called AFIF elements. This analysis is intermediate to classical finite element constructions and the Daubechies wavelets and is amenable to the multilevel theory presented in Section 2. The underlying scaling functions (resp. multiwavelets) have been introduced and investigated in [23, 18, 16] and are constructed using fractal interpolation functions. As for finite element examples, they 9 have local support, 9 allow for reproduction of low-order algebraic polynomials, 9 are interpolatory at nodes. Just as importantly, the AFIF scaling functions r tional finite element nodal basis functions in that
depart from conven-
9 they form an L2 orthogonal basis for Vj,~ if fl is aligned with the cube structure, 9 they have highly unusual local smoothness characteristics (in fact, they are "fractal"). Recall some definitions from the previously mentioned papers (see also [29]). An iterated function system is a complete metric space (X, d) and a collection of n strict contractions
wi : X ~ X ,
i=l,...,n.
Associated with these mappings, we define the set-valued map
W(A)-
(.j k:l,...,n
wi(A)
Scaling Function and Wavelet Preconditioners
423
acting on the complete metric space (H(X), dH), where H ( X ) is the collection of all compact subsets of X, and dH is the Hausdorff distance between sets g.
dH(A,B) = max l~sup inf d(x,y), sup xAjinfd(x'Y)~" xEA YEB
yEB
The following theorem (see [29]) establishes an important property of W: T h e o r e m 4. Let {X, wk : k = 1... n} be a (strictly contractive) iterated function system. Then the set-valued map W is a contraction on (H(X), dH) with contraction rate < 1. There is a unique fixed point of the equation G = W (G) given by
VA E H ( X ) .
G = lim Wi(A) i--~ oo
There is a lot of research on how this fixed point equation can be used to visualize strange attractors that arise in chaotic dynamical systems. What is of interest for us is that there exists a standard machinery for utilizing this theorem to generate attractors that are actually the graphs of functions on X. Consider the case X[0,1 ] ~ [0, 1] X ]1:~,
and define the mappings
i=1,2, by x
x
X
0 X
-1
1
-~
Clearly, each mapping is a strict contraction on X[o,1] so that by Theorem 4 there is a unique attractor G 1 E H(X[o,1]). It is easy to see that G 1 is actually the graph of a continuous function r : [0, 1] --+ ]R which can be extended by zero to a continuous function on lR with support in [0, 1]. Similarly, the choice
w2( )
[i
lO -g
0]{x} {0}
-g
Y
0
-Y6
424
J. Ko et al.
leads to another function r r (x) -
on [0, 1] which can be extended by setting r 0
if x e (1,2] if x r [0, 2]
to a continuous function on the axis, with support on [0, 2]. A detailed investigation of the above choice of scaling functions Ct, 1 = 1, 2 (and of a whole family of similar AFIF functions) can be found in [18, 16, 30]. We summarize the important properties in the following theorem. Theorem 5. The AFIF scaling functions r r properties (A1)-(A4) of Section 2 (d = 1, L = 2). for all t < 3/2. Moreover, the functions {r } in k), j >_ O. With the usual tensor product generalize to d > 1.
aS defined above satisfy In particular, Ct E Ht(lR) form an orthogonal basis construction, the results
The refinement equations for (A4) which are important for the evaluation of the values of functions from Vj at intermediate points and for the implementation of the multilevel algorithms (see the definition of the matrices Ij entering (2.17)) can be found in [16, Section III]. Figure 1 depicts the classical linear, Lagrangian finite element (a), the finite element created from AFIF scaling functions (b), and the classical quadratic Lagrangian finite element (c). (More precisely, for (b) the graphs of the nontrivial AFIF basis functions r r r on [0, 1] are shown, analogously for (a) and (c).) The linear finite element and AFIF element have similar approximation properties, both contain piecewise linear (but not quadratic) functions within their span. On the other hand, the AFIF element and quadratic finite element have similar cardinality. That is, both have three basis functions that intersect a single element in one dimension. This fact should be kept in mind when we compare the performance of algorithms based on the different choices in terms of numbers of iterations for a fixed error reduction or in terms of condition numbers of the multilevel preconditioned systems (see the next section). Though the AFIF functions do not have explicit expressions and are only implicitly defined via recursions involving the refinement equation, the accurate calculation of terms that typically arise, such as v'j,~'j,~, dx
or
Vr
vet',a, dx
[10 0]
can be performed efficiently (see [23] or, in general, [11]). For example, the elemental mass and stiffness matrices for the above AFIF case are given by [Me]=
0 0
25 0
Scaling Function and Wavelet Preconditioners
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in 1D
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J. Ko et al.
426
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Scaling Function and Wavelet Preconditioners
and
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respectively (see [27], the entries correspond to ~t - e - [0, 1], and to the three functions r r and r shown in Figure 1 (b)). According to Section 2, Theorem 5 implies the optimal condition number estimates for the C-algorithm (or scaling function preconditioner) corresponding to the AFIF discretization of second order elliptic boundary value problems on domains Ft which are aligned with the cube structure. It is easy to see that, due to the support and symmetry properties of the AFIF scaling functions, the restricted functions r la j > jo preserve the L2-orthogonality property for such domains in any dimension d k 1. Even though we did not explicitly deal with the case of essential boundary conditions in Section 2, zero Dirichlet boundary conditions can be incorporated in a similar way (by keeping only those r ~ with support in gt in the basis). This has been utilized in the study of multigrid methods in [27]. We finish this section with a short description of multiwavelets {r r associated with the AFIF multiresolution analysis. As derived in [16], the AFIF wavelet functions are constructed to be interpolants at the quarter integer points and orthonormal to each other and the AFIF scaling functions. The function space W0 generated by the two wavelet functions is the orthogonal complement of V0. Several choices are discussed in [16]. The one which we present here as most convenient for our purposes consists of a symmetric r - r and an antisymmetric r - r Both functions are supported on [0, 2], and are depicted in Figures 2 and 3, respectively. This particular choice of wavelet functions makes the analysis on a finite domain much easier. The expressions for the wavelet mask coefficients which enter the matrices/~j from (2.20) can be found in [16, section IV] for the 1D case (one needs to be careful with the normalization factors introduced in [16]). Wavelet systems restricted to the interval [0, 1] have been described in [16, Theorem 4.4]. For our case, the choice of a basis for the complement spaces Wj,[0,1] as defined by (2.18) consists of all nontrivial restrictions of the symmetric ~,~1[0,~] and only those antisymmetric Cj~,~ with support completely contained in [0, 1]. It should be emphasized that the resulting restricted function spaces Vj,[0,1] and Wj,[0,1] are still orthogonal complements. This fact generalizes immediately to d > I and rectangular domains via tensor product arguments. As in [15, Section 10.1], the multivariate wavelet spaces are defined as spans of tensor products of univariate scaling functions from Vj-1 and univariate wavelets from Wj (except for pure
J. Ko et al.
428
C-products). For a 2D-rectangle, this means to take the span of all
Cj-I (Xl)~j(X2), Cj(Xl)r (X2), ~j(Xl)~)j(X2), Cj--1, Cj denote the generic basis functions from the above defined
where Vj-1 and Wj on the respective 1D-intervals. This construction automatically ensures L2 orthogonality of the multivariate wavelet spaces for different j for rectangular domains. Domains f~ with re-entrant corners such as L-shaped domains need more care (compared with the above construction for the rectangle it is enough to modify a few functions in the vincinity of the re-entrant corner to preserve full orthogonality as required in Theorem 3; we leave this as an exercise to the reader). In any case, elementary rules guarantee the applicability of Theorem 3 for the AFIF elements, and justify the optimality of the corresponding C-algorithm as described at the end of Section 2. Since the underlying system r is an orthogonal basis in L2(f~), this C-algorithm should perform extremely well for problems with dominating L2-elliptic part. For the corresponding numerical experiments, see Section 4. w
Numerical examples
An important goal of this paper is to assess the numerical performance of the class of multilevel r and C-preconditioning methods introduced above. In particular, we include standard multilevel finite element solvers (linear and quadratic elements) into the comparison. A motivation is that until now relatively few empirical results on wavelet preconditioning methods are documented, and fewer still make a serious attempt to calibrate performance to standard finite element formulations. Careful studies of the numerical performance of these algorithms are critical to establish viable, worthwhile directions for future research. Our numerical studies are still preliminary, and concern 1D and 2D Neumann boundary value problems for the Poisson equation. Most studies of the numerical performance of wavelet and scaling function preconditioners for Galerkin formulations have utilized a model equation in one dimension [3, 1, 9, 19]. While studies of one-dimensional boundary value problems are seldom of interest in applications per se, there remain important conclusions that can be drawn from this class of problems. In particular, the numerical examples in one dimension set precedents in optimal complexity that are realized in some more general problems over classes of domains in higher dimensions. This is the reason we include them here. The two-dimensional tests concentrate on simple domains like the unit square and L-shaped domains. Finally, robustness with respect to singular perturbations caused by a zero-order Helmholtz term is investigated. Some related experiments comparing Cand C-algorithms for linear finite elements on square domains can be found
429
Scaling Function and Wavelet Preconditioners
in [21]. Our numerical experiments show that (i) the multilevel preconditioning techniques presented in this paper are amenable to both scaling function and wavelet constructions, (ii) all selections (AFIF scaling and wavelet functions, Daubechies scaling functions, linear FEM and quadratic FEM) achieve asymptotic optimal complexity without tailoring the underlying function systems to the domain, if the latter is well-aligned with the cube structure. Also, we see that for the standard elliptic problems, multilevel finite element preconditioning of BPX type yields, as a rule, a better performance than wavelet-based methods. This supports our opinion that the ongoing development of wavelet-like solution methods for PDEs should include a thorough testing and comparison with conventional methods for partial differential equations. 4.1
1D t e s t s
We consider the weak formulation in Hi(0, 1) of the two-point boundary value problem d2 u
dx 2 ~-u
- 2 x 3 + 3x 2 + 1 2 x - 6,
x e (0, 1),
u'(0)- u'(1)-0, which has the exact solution u(x) = 3x 2 - 2x a. For the discretizations and solvers, we employ (i) linear finite elements, (ii) quadratic finite elements, (iii) Daubechies scaling functions, and (iv) AFIF scaling functions as defined in Section 3. We exclusively use the C-algorithm (i.e., the pcg method with the multilevel preconditioner based on the scaling functions) discussed in Section 2. As scaling factors (see (2.15)) we choose
which corresponds to multilevel diagonal (or Jacobi) scaling, and seems to be the most reliable choice on the average. An analogous choice is made for the scaling factors dj,i in the C-algorithms below. The results of the numerical study are summarized in Table 1 and depicted graphically in Figure 4. A first interesting conclusion in considering the results of Table 1 is that the multilevel r for the AFIF and the Daubechies scaling functions yield nearly identical results. Both methods require just under 30 iterations to reduce the residual to a value of 10 -6 of its starting value. This fact is quite counter-intuitive in light of the fact that many authors have noted the extremely poor conditioning of the Gramian matrix associated with truncating the Daubechies wavelets to the interval. Because the performance of these two formulations are so
430
J. Ko et al.
10 3
o 10 2
10~ i : : .........................
10 0
0
100
200
300
l o..e $ -multilevel,DaubechiesFEM B--e $ -multilevel, AFIF FEM A..A $ -multilevel, Linear FEM v - - v $ -multilevel, Quadratic FEM
: i
' i
i i
i i
i i
i i
400
500 Dim.
600
700
800
900
1000
Figure 4. Number of Iterations (Error Reduction by 10-6), 1D, log- Plot.
Table 1. Number of Iterations to Decrease Residual by 10-6, 1D. [!
I/ Daubechies l[
AFIF
!1 .....L'inear .
[i J [i 3 4 5 6 7 8 9 10
b
!] Quadratic ]1
n j l i t e r I[ ....n j... l i t e r [[ n j ] iter [I n j ..... i 12 12 17 14 9 5 17 20 20 33 19 17 8 33' 36 21 65 22 33 11 65 68 24 129 23 65 12 129 132 24 257 25 129 12 257 ' 260 26 "513 25 257 12 513' ....516 29 10'25 24 513 13 1025 1028 29 2049 26 1025 13 2049 . . . . . . . . . .
iter li 12 15 16 16 17 17 18 17
similar, in the remaining numerical examples, we will only consider scaling functions and wavelets associated with the AFIF basis. In comparison, the classical linear finite element basis and classical quadratic elements require 13 and 17 iterations, respectively, to achieve the same decrease in the residual. Finally, as noted earlier, these numerical experiments should be evaluated keeping in mind the cardinality of the masks representing the elemental mass and stiffness operators. In Section 3 we showed that the AFIF
Scaling Function and Wavelet Preconditioners
431
10 3
~ 0
10 2
'5
10~ 0
5000
10000
Dim.
15000
Figure 5. Number of Iterations (Error Reduction by 10-6), 2D, Square Domain, log-Plot. elemental mass and stiffness matrices have 3 x 3 -- 9 entries. Of course, it is well-known that for the classical linear and quadratic finite elements we have 2 x 2 = 4 and 3 x 3 = 9 entries, respectively. In contrast, it is shown in [26] that in the order 3 Daubechies case these elemental matrices have 5 x 5 -- 25 entries. In higher space dimensions, the differences are even more significant. 4.2
2D
tests
In this section, we report on similar performance results for the model problem -Au+u-f
Ou On
= 0
in
~,
on
0~.
Table 2 summarizes the number of iterations required to reduce the initial residual by a factor of 10 -6 using the C-algorithm in the case ~t - [0, 1]2. For this particular problem, the number of iterations approaches a value of approximately 10, 20, and 30, for the linear, quadratic, and AFIF elements, respectively. The performance of the r in comparison to the Jacobi preconditioner (i.e., when diagonal scaling of the discretization matrix A j is used as preconditioner) is depicted in Figure 5. Similar results are summarized in Table 3 and Figure 6 for an L-shaped domain. Again, all calculations are carried out using the C-multilevel algorithm, and
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Table 2. Number of Iterations to Decrease Residual by 10-6, 2D, Square Domain. AFIF nj
3 4 5 6 7
289 1089 4225 16641 66049
Jacobi,
10 3
--9
m--m
Linear iter 81 9 289 11 1089 12 4225 13 16641 12
iter 28 29 30 31 31
AFIF
nj
Quadratic nj iter 289 19 1089 20 4225 20 16641 20 66049 21
FEM
Jacobi, Linear
FEM
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0
2000
4000
6000 Dim.
8000
10000
12000
Figure 6. Number of Iterations (Error Reduction by 10-6), 2D, L-Shaped Domain, log-Plot.
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Scaling Function and Wavelet Preconditioners
Table 3. Number of Iterations to Decrease Residual by 10-6, 2D, L-Shaped Domain. AFIF
7
J
nj
3 4 5 6
225 833 3201 12545 49665
iter 37 40 42 43 43
Linear nj iter 65 i3 225 14 833 i5 3201 15 12545 15
Quadratic nj iter 225 25 833 28 3201 28 12545 29 49665 29
each case asymptotically achieves the expected level-independent iteration count. For the L-shaped domain, however, the number of iterations to reduce the original residual by 10 -6 is about 50% greater than in the corresponding simulations for the square domain. This decrease in efficiency is expected, and is attributed to the re-entrant corner. 4.3
R o b u s t n e s s of r
and C-algorithms
In the following set of numerical experiments, we vary the scalar magnitude of the source term. First, we consider a one-dimensional problem. d2 u dx 2 ~- qu - f ,
x e [0,1],
u'(0)- u'(1)=0. We vary the positive constant q between 102 and 101~ and have tabulated the condition numbers of the preconditioned matrix representations C j A j (resp. C j A j ) with J - 10 corresponding to linear finite element nodal basis functions, to AFIF scaling functions (resp. to AFIF wavelet functions). That is, in the first two cases, we employ again the C-algorithm while in the last case the r discussed at the end of Section 2 is used. The results of the numerical comparisons are summarized in Table 4. Clearly, the multilevel preconditioner based on AFIF scaling function basis yields condition numbers that are about 10 and nearly constant. The condition numbers of the wavelet preconditioner applied to the AFIF case actually decrease with the strength of the source term, from a maximum near 10 to a minimum of 1 which is expected by the theory. The BPX preconditioning of the classical finite element method exhibits an increase of the condition numbers from 5 to about 30 over the same range of q (note, however, that an inexpensive modification of the BPX preconditioner would resolve
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434
this defect, see [32, Section 4.2.3]). As for any C-algorithm, the condition numbers will grow as ,~ J in the limiting case q --+ co. The better behavior for the AFIF element (compared to the BPX preconditioner for the linear elements) is due to the L2 orthogonality of the AFIF scaling functions within each level. In Table 5, we summarize completely analogous results
T a b l e 4. Condition Numbers, 1D, Varying Source Strength.
]1 q I[ L i n e a r F E M ,10 2 4.7 6.7 10a 104 9.23 10 5 11.5 , ,,10 6 14.0 10 7 18.3 ...10 . 8 29.4 10 9 31.9 1010 32.2
CAFIF 12. 11.4 10.2 11.0 11.4 9.6 10.6 10.9 11.0
C A F I F [I 10.6 10.2 9.20 7.45 4.88 2.24 1.20 1.023 1.011
T a b l e 5. Condition Numbers, 2D, Varying Source Strength.
I! q
10 2 10 3 104 10 5 10 ~ 10 7 l0 s '"109 10 lo
Linear FEM
r AFIF
7.15 11.3 15.7 36.6 56.5 60'4 60.8 60.8 60.8
14. 11.1 9.57 8.40 7.13 7.0 7.0 7.0 7.0
r AFIF I] 2.06 1.81 1.87 1.55 1.1i 1.00 1.00 1.00 1.00
...............
for the Neumann problem in two dimensions
-Au + qu- f
in f~ - [0, 1]2 '
Ou 0--n=0
on 0 f t .
Here, the final discretization level is J - 6. As shown in Table 5, the Cmultilevel preconditioner of the AFIF scaling function basis yields condition
Scaling Function and Wavelet Preconditioners
435
numbers that vary from 15 to 7, while for the C-multilevel preconditioning of the AFIF wavelet basis the condition numbers are again reduced if q grows. In contrast, the condition numbers for the BPX preconditioning of the linear finite element discretization increase from 7 to 60 as the strength of the source term is increased. References
[1] Alpert, B. K., Wavelets and other bases for fast numerical linear algebra. In: [5], pp. 181-216. [2] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Namer. Anal. 29 (1992), 1716-1740. [3] Beylkin, G., R. Coifman, and V. Rokhlin, Fast wavelet transform and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141-183. [4] Bramble, J. H., Multigrid Methods, Pitman Res. Notes Math. Ser., vol. 294, Longman Sci. & Tech., Harlow, 1993. [5] Chui, C. K. (ed.), Wavelets: A 7htorial in Theory and Applications, Academic Press, Boston, 1992. [6] Chui, C. K., An Introduction to Wavelets, Academic Press, Boston, 1992. [7] Cohen, A., W. Dahmen, and R. A. DeVore, Multiscale decompositions on bounded domains, IGPM-Report Nr. 113, RWTH Aachen, May 1995. [8] Dahlke, S. and R. A. DeVore, Besov regularity for elliptic boundary value problems, IGPM-Report Nr. 116, RWTH Aachen, August 1995. [9] Dahlke, S. and A. Kunoth, Biorthogonal wavelets and multigrid methods, Report, IGPM, RWTH Aachen, 1993. [10] Dahmen, W., Multiscale analysis, approximation, and interpolation spaces, in Approximation Theory VIII, vol. 2, C. K. Chui, L. L. Schumaker (eds.), World Scientific, Singapore, 1995, pp. 47-88. [11] Dahmen, W. and C. Micchelli, Using the refinement equation for evaluating integrals of wavelets. SIAM J. Namer. Anal. 30 (1993), 507-537.
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[12] Dahmen, W., S. PrSfldorf, and R. Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994), 583-620. [13] Dahmen, W., S. PrSfldorf, and R. Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993), 259-335. [14] Dahmen, W., S. PrSfldorf, and R. Schneider, Multiscale methods for pseudodifferential equations, in Recent Advances in Wavelet Analysis, L. L. Schumaker and G. Webb (eds.), Academic Press, New York, 1994, pp. 191-235. [15] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992. [16] Donovan, G., J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions. SIAM J. Math. Anal. (1996), to appear.
[17] Frazier, M. and B. Jawerth, Wavelet transforms and atomic decompositions, Dep. Math., Univ. South Carol., 1993, preprint.
[lS] Geronimo, J. S., D. P. Hardin, and P. R. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), 373-401.
[19] Glowinski, R., W. M. Lawton, M. Ravachol, and E. Tenenbaum,
Wavelet solution of linear and nonlinear elliptic, parabolic, and hyperbolic equations, Technical Report AD890527, Aware Inc., Cambridge, MA, 1989.
[20] Glowinski, R., T. W. Pan, R. O. Wells, and X. Zhou, Wavelet and finite element solutions for the Neumann problem using fictitious domains, J. Comput. Phys. (1996), to appear. [21] Griebel, M. and P. Oswald, Tensor-product-type subspace splittings and multilevel iterative methods for anisotropic problems, Adv. Cornput. Math. 4 (1995), 171-206. [22] Hackbusch, W., Iterative Solution of Large Sparse Systems of Equations, Springer, New York, 1994. [23] Hardin, D. P., B. Kessler, and P. R. Massopust, Multiresolution analyses based on fractal functions, J. Approx. Theory 71 (1992), 104120.
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[24] Jaffard, S., Wavelet methods for fast resolution of elliptic equations, SIAM J. Numer. Anal. 29 (1992), 965-986. [25] Jaffard, S. and P. Laurencot, Orthogonal wavelets, analysis of operators, and applications to numerical analysis, in [5], 543-601. [26] Ko, J., A. J. Kurdila, and M. S. Pilant, A class of finite element methods based on orthonormal, compactly supported wavelets, Comput. Mech. 16 (1995), 235-244. [27] Kurdila, A. J., Sun, T., P. Grama, and J. Ko, Affine fractal interpolation functions and wavelet based finite elements, Comput. Mech. 17 (1995), 169-185. [28] Latto, A., H. L. Resnikoff, and E. Tenenbaum, The evaluation of connection coefficients of compactly supported wavelets. Technical Report, Aware Inc., Cambridge, MA, 1991. [29] Massopust, P. R., Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, San Diego, 1994. [30] Massopust, P. R., Fractal functions and applications, in Special Issue of Chaos, Solitons, and Fractals, to appear. [31] Meyer, Y., Wavelets and Operators, Cambridge Stud. Adv. Math., vol. 37, Cambridge Univ. Press, Cambridge, 1992. [32] Oswald, P., Multilevel Finite Element Approximation. Theory & Applications, Teubner Skr. Numer., Teubner, Stuttgart, 1994. [33] Oswald, P., Multilevel solvers for elliptic problems on domains, this volume. [34] Rieder, A., Multi-level methods based on wavelet decompositions, East-West J. Numer. Math. 2 (1994), 313-330. [35] Rieder, A., R. O. Wells, and X. Zhou, A wavelet approach to robust multilevel solvers for anisotropic elliptic problems, Appl. Comput. Harmon. Anal. 1 (1994), 355-367. [36] Rieder, A. and X. Zhou, On the robustness of the damped V-cycle of the wavelet frequency decomposition multigrid method, Computing 53 (1994), 155-171.
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Jeonghwan Ko
Department of Aerospace Engineering Texas A&M University College Station, TX 77843 [email protected] Andrew J. Kurdila
Department of Aerospace Engineering Department of Mathematics Texas A&M University College Station, TX 77843 [email protected] Peter Oswald
Institute for Algorithms and Scientific Computing GMD - German National Research Center for Information Technology D-53754 Sankt Augustin, Germany [email protected]
Vo Multiscale Interaction and A p p l i c a t i o n s to T u r b u l e n c e
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L o c a l M o d e l s a n d L a r g e Scale S t a t i s t i c s of t h e Kuramoto-Sivashinsky
Equation
Juan Elezgaray, Gal Berkooz, Harry Dankowicz, Philip Holmes, and Mark Myers
A b s t r a c t . We investigate the ability of local models of the one space dimensional Kuramoto-Sivashinsky (KS) equation, obtained either from 'local' wavelet projections or by projection on a small set of Fourier modes supported on a short subinterval, to reproduce coherent events typical of solutions of the same equation on a much longer interval. We also show that an effective equation preserving the statistics of the large scales of the KS equation can be obtained from a coarse-graining procedure based on the wavelet decomposition of the KS equation.
w
Introduction
This paper overviews previous work of the authors [6, 14, 32, 11, 13] on the study of local models and large scale properties of the KuramotoSivashinsky equation [24]. However, our initial motivation comes from a more difficult problem, namely, describing local dynamics of the coherent structures in the wall region of boundary layer turbulence [34, 25, 3, 7, 18]. In this introduction, we briefly describe the ideas and techniques used in [3] to address this problem, as well as some difficulties raised by this approach. In the remaining sections of the present paper, we will discuss how a wavelet based approach of the problem can be used to overcome analogous difficulties in the case of the Kuramoto-Sivashinsky equation. The wall region of a turbulent boundary layer [31, 17] is a typical example of Multiscale Wolfgang Copyright
Wavelet
Dahmen,
Methods
for PDEs
Andrew
J. K u r d i l a ,
(~)1997 b y A c a d e m i c
All rights of reproduction ISBN 0-12-200675-5
441 and Peter
Press, Inc.
in a n y f o r m r e s e r v e d .
Oswald
(eds.), pp. 441-471.
442
J. Elezgaray et al.
an open flow in which large coherent structures [34] contain a major fraction of the energy. It has been demonstrated that axisymmetric turbulent jet mixing layers [15] also belong to this class. In the wall region of a turbulent boundary layer, large eddies are experimentally observed to be associated with violent bursting events in which they suddenly break up. These events also correspond to peaks in the turbulent energy production, and are thought to be the basic mechanism for the transfer of energy between the inner and outer regions of the layer. The study of an open flow faces several problems. First, it seems hopeless to account for the detailed (small scale) spatially chaotic aspects of fully developed turbulence using a low dimensional model. Moreover, it is natural to seek a decomposition of the flow which takes into account the existence of coherent structures. In other words, one would like to look for a decomposition of the flow which is, in some sense, well adapted or optimal. In [3], N. Aubry et al. used an unbiased technique for identifying such structures, originally proposed by Lumley [27] in the context of turbulence, although well known in probability theory as the Karhunen- Lo~ve decomposition [27, 5, 8]. The procedure is discussed at length in [18]. Essentially, the method extracts, for a given flow, the velocity distribution ~l,i(x), i = 1, 2, 3 which is the best correlated with the given velocity field in a statistical sense. More precisely, we look for the solution of the maximization problem:
Kuramoto-Sivashinsky Equation
443
and we may sort the eigenvalues in decreasing order: A1 _ A2 >__... _> An >_ )~+1 >_ .... Using the fact that Ri,j(x,x I) is a symmetric tensor, one can show that the eigenvectors ~n,i's are orthogonal, so the proper orthogonal decomposition of the velocity field ui(x, t) reads:
u i ( x , t ) - E an(t)~n,i(x),
i - 1,2,3.
(1.4)
n>l
Furthermore, it can easily be seen [18, 27, 4, 8] that the components an(t) are uncorrelated to the second order:
< an (t)am (t) > = ~n,m)~n.
(1.5)
One of the major difficulties raised by the above procedure is due to the fact that, for domains which are translationally invariant (homogeneous) in some directions (e.g., Ri,j(Xl,X2,X3,X'l,X'2,x'3) - Ri,j(Xl x~, x2, x~, x3, x~) if invariant along the Xl direction), the proper orthogonal decomposition is identical to the Fourier decomposition in these specific directions. In the boundary turbulence problem, for instance, the spanwise (xl) and streamwise (x3) directions are homogeneous [3, 8]. Accordingly, the proper eigenfunctions take the form: e 2 ~ ( k ~ + k ~ ) ~ ~i,kl,k3 (~) (x2) The ~(n) ~i,kl,k3 (x2) functions are often rather well localized for flows dominated by coherent structures. On the contrary, the harmonic part of the decomposition is not localized at all, and we are led to a paradox: we want to construct a low dimensional system (ODE) that will reproduce the behavior of localized coherent structures. Among all possible decompositions of the velocity field, we choose the one which is optimally adapted (in the L 2 average sense) to the given velocity field. However, in the presence of translation symmetry, this results in nonlocal modes, and consequently nonlocal models. A possible solution to this is simply to choose an appropriate length scale containing 'enough' structures to afford a reasonable interaction, and to impose periodic boundary conditions in the homogeneous direction(s). Then we obtain the dynamical model by Galerkin projection onto the most energetic modes coming from the proper orthogonal decomposition. This is the strategy adopted in the paper by N. Aubry et al. (we refer the reader to [3, 7, 18] for further details concerning in particular the modeling of the energy transfer to small scales as well as the interaction with the mean flow). Basically, what is postulated when using such an approach is the fact that interactions are primary local and that the influence of missing neighboring structures at the boundary of the modelled subdomain can be supplied by the coupling with the 'opposite end' afforded
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by periodization. This approach is in fact widely used in computational fluid dynamics in order to reduce problem size (see [19] for a study of the periodization procedure in the context of channel flows). Due to their good spatial and scale/wavenumber localization properties, it seems natural in this context to use wavelet bases [29] to replace Fourier bases in homogeneous directions. In order to investigate this conjecture, we chose the one space dimension Kuramoto-Sivashinsky (KS) equation as a suitable model problem. With periodic boundary conditions on sufficiently long intervals (length L), this PDE exhibits rich spatio-temporal behavior [24, 9], in which zeros of the scalar variable u(x, t) are created and destroyed in irregular yet structured events (see Figure la) closely reminiscent of the burst/sweep cycles of the turbulent boundary layer. The rest of the paper is organized as follows: in Section 2, we discuss the fact that, while wavelet bases are not optimal in the L 2 sense, representations of the KS solutions in terms of the finitely many scale generations of the Perrier-Basdevant [33] periodic wavelets do almost as well as the optimal Fourier mode representations of comparable dimension in capturing the average energy (L 2 norm)[6]. Thus, it seems reasonable to exploit the spatial localization property of wavelet projections in an attempt to extract local models which capture the relevant dynamics in a limited part of a large spatial domain. Such local models can be extracted by Galerkin projection onto subspaces spanned by "close" wavelets, but, while short term transients display power spectra comparable to that of the full system [14], typical solutions eventually decay to fixed points. Exploratory analyses indicate that the drastic loss of symmetry (typically, only a 292 symmetry survives from the initial rotational 0(2) invariance in phase space) is responsible for the absence of long term recurrent dynamics in such local models. In Section 3, we show [32] that provided that periodic boundary conditions are used on the short subdomain, the loss of symmetry induced by wavelet projections is sufficiently mild that key global features of the dynamics are preserved. Section 4 explores the ability of local models of the KS equation, obtained with a small set of Fourier modes supported on a short subinterval, to reproduce coherent events typical of solutions of the same equation on a much longer interval [11]. Finally, Section 5 explains how wavelet decompositions afford a natural connection between large scale properties of the KS equation in a large domain (extended system limit), and its behavior in a small domain [13]. 1
1We assume the reader is familiar with orthogonal wavelet bases [33, 29]. In all the numerical simulations below (except in the simulations reported in Table 1), we used periodic fifth order spline wavelets, as described in [33].
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100
8O t 60
40
20
0
50
100
150
200
250
300
350
400
100 10-1 10-2 (b)
I
10 -1
k
10~
F i g u r e 1. (a) A numerical simulation of Equation (2.1) with periodic boundary conditions on the interval of length L - 400. (b) The spatial power spectral density of the solution obtained by time averaging. (Note" logarithmic scales on both axes).
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w Near optimality~ of wavelets and symmetry breaking local models The Kuramoto-Sivashinsky equation was originally proposed as a model for instabilities on flame fronts and 'phase turbulence' in chemical reactions. In this paper, we will use the formulation"
Ut + L-2uzz + L-4uxzxz + 2L-luu~ - 0, 0 _< x _ 1,
(2.1)
with periodic boundary conditions. The Fourier transform of the linear part of (2.1) with respect to x, --
(k,t)
(2.2)
shows that the uzz term in (2.1) is destabilizing and acts as an energy source, while u ~ is a stabilizing or energy sink term whose effect increases with (spatial) wavenumber. More precisely, (2.2) shows the existence of a range of linearly unstable wavenumbers ]27rk[ < L, with a peak at 27rlk[ = L/~f2, corresponding to the wavenumber of the most rapidly growing linear mode. The form (2.1) has the advantage over the form considered in, say [2] or [23] that it preserves the spatial average
ft(t) =
/o 1u(x, t)dx,
(2.3)
which is therefore customarily set to zero. When the KS equation is integrated on a sufficiently long domain a characteristic pattern emerges, which we illustrate in Figure la for the specific case L = 400. Here the value of u(x, t) is indicated by a gray scale (black = minima, white = maxima) over the (x, t) plane. A typical spatial wavenumber emerges, as indicated also by the clear peak in the power spectrum of Figure lb, which occurs near the maximum linear growth rate 27rlk[ - L/v~. This average wavenumber is maintained largely by interactions among a range of active wavenumbers up to and above the linear stability cut off 27r]kl - L, causing the continual appearance and disappearance of peaks and troughs, or equivalently zeroes, in u(x, t). These coalescence and creation events, and the dominant wavenumber 27r]k[ - L/x/~, are the coherent structures of our model problem. A physical explanation for the appearance of the chaotic state depicted in Figure la is as follows: the linear stability analysis above shows that the system tends to create 'cells' of a preferred wavelength. However, the creation of these cells takes place randomly in space, so that the cells are either compressed or stretched by neighboring cells [9]. This results in the creation or annihilation events clearly visible in the space-time representations of the solutions of the KS equation.
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Comparing Fourier and wavelet decompositions in the L 2 a v e r a g e sense
As stated in the introduction, one clear advantage of wavelet bases is their spatial localization. Unfortunately, when the equation is translationally invariant, wavelet basis are not optimal in the L 2 average sense. Let us be more precise. If
u(x, t)
-
E
qEZ
uq(t)eq(X), eq(x)
-
e 2iTrq=
(2.1.1)
denotes the Fourier decomposition of any solution of the KS equation, and
u(x, t) -
E
aj,k(t)~j,k(X),
(2.1.2)
j>_0,ke[0,2J-1]
a wavelet decomposition, then it can be proved [8, 4] that, for any n >_ 1, n
n
(l~(~)(t)i 2) >_ ~ ,
i=1
(2.1.3)
i=1
where we have sorted the wavelength q indices (resp. the (j, k) wavelet indices) in decreasing average energy order:
(]ttq(i) (t)! 2) _> (lUq(i+l)(t)!
(2.1.4)
2)
and
(a2(i),k(i) (t)) >-- (a2(i+l),k(i+l) (t)).
(2.1.5)
Now, as shown in [6], it is possible to obtain an estimate of the amount of energy lost (in the average) in going from the Fourier {uq(t)} decomposition to the wavelet {aj,k(t)} decomposition. Namely, if for a given e, we need N(e) Fourier modes in order to satisfy
N(~)
~(TUq(~)(t)i 2) - ~ (luq(~)(t)i 2) <_ c, i--1
(2.1.6)
i=1
then, for some constant C, depending only on the equation parameters (and not on e), we have ~(~) i=1
(laj(,),k(,) (t)l =) - ~ (laj(~),k(,)(t)l =)
~_ Ce,
(2.1.7)
i=1
for some _N(e) > N(c) slightly bigger than N(e). We refer the reader to [6] for a detailed discussion of this estimate, and here merely give some numerical results which confirm the above estimate. Let us first note that, due
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448 Table
Number of modes 64 (j = 6) 96(j = 6, 5) 127(0 <_ j <_ 6) 255(0 <_ j _< 7)
Spline wavelets (m -- 6) 70.84% 79.1% 84.1% 99.9%
1.
Spline wavelets (m = 8) 71.5% 79.43% 84.9% 99.9%
Fourier 72.2% 83.3% 89.7% 99.9%
2 to translational invariance, < aj(i),k(i ) (t) > is independent of the position parameter k. From a numerical simulation of the KS equation (2.1) for a large value of the length parameter L(= 400), we numerically computed 2 Rj - 2J < aj(i),k(i ) (t) >, the average energy contained in the scale 2-J We used periodic spline wavelets of degree m, as described in [33]. We recall that the greater the value of m, the bigger the "numerical support" of the wavelets, and the faster the decay of the Fourier spectrum at short wavelengths. Although spline wavelets [33, 29, 26] are not of compact support, they decay exponentially fast at infinity, and can be considered from a numerical point of view as having a compact support.
Using the ordering j(i) of the scales, we compare in Table I the average percentage of energy contained in the most energetic scales (j(1) = 6,j(2) - 5,etc . . . . ), with the average percentage obtained in a Fourier decomposition using the same number of modes (sorted according to equation (2.1.4)). The above figures show that for reasonable values of m (i.e., for sufficiently smooth splines), the wavelet projection captures almost the same amount of energy as the Fourier (optimal) decomposition (within 5
%). 2.2
S y m m e t r y b r e a k i n g in local m o d e l s
The results in the preceding section, prompt us to conclude that from an average energy point of view, wavelets are a reasonable candidate for a model decomposition of the Kuramoto-Sivashinsky equation. In order to get a model of the local dynamics, we use a Galerkin projection as in [3] to extract a subset of equations from the infinite set of equations
&j,k -- Z l (j, k, j', k')aj,,k, + ~ n(j, k, j', k', j", k")aj,,k, aj,, ,k" . j' ,k' j' ,k' ,j" ,k"
(2.2.1)
These are obtained in the usual way by replacing the wavelet decomposition (2.1.2) into the KS equation, and taking the scalar product with each
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wavelet mode ~j,k(x). The linear and nonlinear coefficients are:
l ( j , k , j ' , k ' ) = - L -2
/o 1~j,k(x)[~j,,k,(X)x x + L-2~j,,k,x~xxldx,
(2.2.2)
and
n ( j , k , j ' , k ' , j " , k " ) - - 2 L -1
/o 1~j,k(X)~j,,k,(X)~j,,,k,,~dx.
(2.2.3)
In order to obtain a reasonable truncation, several points need to be considered. We want to give sense to the evolution equation defined on a subset B (called a 'box' in the sequel) of indices of the form
((j,k),jmin <_j <_jmax,k = 0,... ,2 j - j m i n -- 1} which correspond to the wavelet modes of scales between 2 -jmin and 2 -jma~ the 'supports' of which are concentrated on an interval centered around k - 2 -jmi~-l, and of length 2 -jmi~ . We expect that the knowledge of the dynamics of the aj,k'S included in such a box will provide a reasonable model for the local dynamics in the physical domain x E [0, 2 -j~'n] associated with the box. In truncating the equations (2.2.1), we need to take into account unresolved modes corresponding to indices (j, k) not included in the box B. Without further justification (besides the fact that large scales do not contribute much to the global energy of a typical solution of the KS equation, see Figure lb), we will ignore the interaction with the variables aj,k for j < jmin, and refer the reader to Section 5 for a discussion of the interaction between small (j >_ jmin) and large (j < jmin) scales. In this section we will then assume that j m i n is small enough to neglect the aj,k'S for j < jmin. Taking into account unresolved small scales (i.e., j > jmax) is not difficult provided the energy containing scales (for L - 400, j - 5 and 6) are included in the range [jmin, jma~]. Under such conditions, we can apply an inertial manifold technique [20], replacing the effect of scales j > jma~ by an affective nonlinear damping. This is a nonlinear analogue of the Heisenberg dissipation model used in [3]. Finally, we have to model the interaction with wavelet modes jmin <_ j <_ jmax (of comparable scales to those in the box B), but with k > 2j-jm~n - 1, i.e., their centers are located outside [0, 2-Jm'~]. For this purpose, we conjecture [6, 14] the existence of a dynamica[ly relevant length scale Lc such that interactions between physical regions separated by more than Lc are dynamically insignificant. Assuming the existence of such an Lc, we take the length of the box LB to be greater than 2Lc, so that we can 'periodize' the small model using resolved 'distant' modes instead of unresolved ones. To summarize, our derivation of a model for the local dynamics involves three parameters:
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(jmin, jmaz) to define the box, and Lc in order to periodize the equations. Numerical simulations of such model systems, first reported in [14], show that their dynamical behaviors are largely independent of the choice of jmax. A typical choice for L = 400 is jmaz = 7. A typical value for jmin (and L = 400) is jmin = 3. This corresponds, roughly speaking, to a length of the box B such that an average of four maxima are found inside B. Decreasing the value of jmi~ below jmi~ = 3 overconstrains the size of the dynamical model. On the other hand, jmin > 3 restrains considerably the phase space of the model, and the dynamics so obtained differ radically from the dynamics in the simulation of the KS equation. The value of Lc plays also a crucial role. Figure 2 shows two examples of dynamical behavior obtained for jmi~ = 3, jmax = 8 (the box B containing 63 modes), and the values Lc = LB/4 and Lc = 3LB/8. Figure 2a shows the spatiotemporal evolution of the model with Lc = 3LB/8, which is in qualitative agreement with the nature of the dynamics in the full system. However, after a very long period (~ 1000 times the characteristic time scale of the most energetic scales), the system goes to a stationary steady state. The temporal evolution for Lc = LB/4 (Figure 2b) is drastically different: the system eventually settles down to a state where no interaction between the localized structures is observed. This state is reached after a short transient in which the behavior of the solution is similar to that of Figure 2a. In analyzing the difference between the Lc = LB/4 and Lc = 3LB/8 models, one realizes that the significant interactions added are mainly nonlinear couplings between physical locations at distance 3LB/8. A more quantitative comparison [14] between the energy spectra of the model solution and that of the full KS equation shows that, although the model is able to capture the value and position of the maximum of the spectrum, the agreement is rather poor, especially in the long wavelength (~ 27rLB1) range. It will be shown in Section 5 that the model equations obtained using the 'periodization' trick correspond, up to an adequate rescaling of the variables aj,k, to a wavelet projection of the KS equation for a value of the length parameter L' = L/2 jm~n. However, the procedure given in this section yields, in general, a system of equations which are only ~2 and not fully 0(2) equivariant, that is, they only preserve the reflection and not the translation-reflection symmetry of the KS equation with periodic boundary conditions. Earlier studies [2, 1] of the KS equation projected onto a Fourier basis, which automatically preserves 0(2) invariance, had indicated that the heteroclinic cycles and modulated traveling waves which occur for relatively short length parameters, and which occur robustly in 0(2) symmetric systems, play a crucial role in the dynamical behavior. These attracting sets are destroyed by 'large' 292 symmetric perturbations and replaced by sinks or stable periodic orbits. We therefore suspect that
t
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F i g u r e 2. Spatio-temporal evolution for a symmetry-breaking model with 1 1 and (a) L c = LB X -~, 3 (b) L c = LB x ~. ~,
x LB
=
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the trivial asymptotic behavior displayed by the models discussed in the present section is due to this symmetry breaking. Section 3 further explores this point.
w
Wavelet projections of the KS equation for short systems
This section summarizes the results contained in [32]. Our goal here is to study in a systematic way the effect of the symmetry breaking induced by a Galerkin projection onto wavelet subspaces of the form W0 @ W1 @ . . . @ W , - I , where we exclude the 'constant functions' subspace V0 = span{l}, due to the condition f~ u(x, t)dx - 0 imposed on the solutions of the KS equation (see equation (2.3)). We study values of the length parameter L close to L = 41r, where the linearized operator - u x x x ~ - ux~ has at most four positive eigenvalues [2, 1]. This parameter region has been much studied in the past, both numerically and analytically. Our purpose here is to investigate with methods similar to those in [2, 1] the bifurcation diagram of the 15-dimensional set of ODEs obtained by projecting KS onto Wo G W1 @ W2 @ W3. The first observation is that a projection of KS onto the first n scales (j = 0, 1 , . . . , n - 1) yields a set of 2" - 1 ODEs which are D2- invariant. We use the notation D2- for the dihedral group corresponding to discrete translations of the form f(x)~f(x-2-np),
p=0,1,...,2 n-1.
The D2- invariance holds because of the equality Vn = Wn-1 0 . . . @ Wo @ Vo, and the fact that Vn = span{~on(x - 2-np),p = 0, 1,... ,2 n - 1} for some function ~on(x). In this section, we will only use the fact that V2, V3 and V4 are each invariant under D4 (notice that V3 is additionally invariant under Ds and V4 under D16). Using the vector notation a = (ao,o, al,o, a1,1, a2,o, a2,1, a2,2, a2,3) for the wavelet decomposition of any function f = ~ aj,k ~j,k(x) E Wo 9 W1 G W2, the matrix representation of the 'shift' operator S : f(x) ---+ f ( x - 1/4) and the 'reflection about x = 1/2' operator I t : f(x) --+ f ( 1 / 2 - x) can be easily shown to be (remember that we always use the Perrier-Basdevant wavelets):
! S _~_
0
-1/v
1/v
-l/v
-1/2 -1/2 -1/2 -1/2 1 0 0 0j
(3.1)
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Kuramoto-Sivashinsky Equation and
/1 0 1 R=
1 0 0 0 0 1
\
0 0 0 1 1 0 0 0
1 0 0 0
(3.2)
One can easily extend this representation to high dimensional spaces of the form ~)~=o Wl using the fact that ~j,k(X - 2 -2) - ~ j , o ( x - 2-J(k + 2J-2)) - ~j,k+2J-2(x) and the evenness of the wavelet functions around their center 2-J (k + 0.5). Using vectorial notation, the Galerkin projection of the KS equation reads: = La + N(a, a). (3.3) Then, D4 equivariance implies that the following relationship holds: Sh
=
L S a + N ( S a , Sa)
Rh
=
L Ra + N ( Ra, Ra ) .
These two vectorial relations impose a drastic reduction of the number of (nonzero) nonequal linear and nonlinear coupling coefficients. As an example, the general form of a 2-scale D4 invariant wavelet projection is: 31
--
0 0
32
/11
112
al
/12
/11
32
+ d
a0(3a2 - 31) a0(a2 -- 3al)
(3.4)
(we used the notation aj,k = a2i+k-1). The potentially 9 linear and 27 nonlinear coefficients are reduced to two and one, respectively. Further simplification of the projected equations is possible using a change of basis, from {~o,o,..., ~3,7} to {Xo, Xl,...}, where the X21(x) ~ v~cos(27r(/+1)x) and X21+~(x) ~ v~sin(27r(/+ 1)x) are linear combinations of the ~r j < 3. For instance, Xo = ~o,o, Xl = 1/v~($1,o - ~1,1), etc . . . . (see the Appendix of [32]). The Xl functions are by construction very close to the trigonometric functions (the L 2 norm of their difference is O(10-3)), and one expects that rewriting equation (3.3) in these new coordinates will simplify considerably their form. This is indeed the case, as exemplified by the form taken by the 2-scale wavelet projection (3.4), which is transformated to
(bo) [oo 0 bl
b2
-
0 0
C~11 0
0 0/22
bl b2
+ 2d
bob2 -2bobl
(3.5)
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The fact that appropriate combinations of functions within each scale yield almost pure Fourier modes derives from the rapid (algebraic) decay of the mother wavelet ~ in Fourier space [33, 26]. Until now, we have used the invariance properties of both the KS equation and the wavelet functions in order to simplify the projected equations (3.3). To further reduce the complexity of this set of 15 ODEs, one has to perform a reduction to a center-unstable manifold [22] (this is possible for values of L ~ 4r). We direct the reader to Section 5 of [32], where a thorough discussion of such a procedure is given. The output of such a computation is a D4-invariant four-dimensional set of ODEs, which appears to be a very mild perturbation of the O(2)-equivariant four-dimensional system studied by Armbruster et al. [2, 1]. This result demonstrates that the wavelet projection and subsequent reduction to a center-unstable manifold is almost as successful as the projection onto Fourier modes in capturing the major features of dynamic and bifurcation behavior of 'short' KS systems (small differences appear for specific bifurcations, see Section 6 of [32]). However, wavelets do not preserve the crucial symmetries exactly, and require a significant additional computational effort. One is prompted to ask: Why wavelets? The main motivation for the study of [32] was to understand whether appropriately 'symmetrized' wavelet projections restore the correct asymptotic behavior of naive projections (such as those in Section 2) where only a 292 symmetry is present. The results suggest that this is indeed the case. Equipped with our present knowledge, we will go back in the next two sections to our original motivation: the extraction of local models for turbulent processes on large spatial domains. However, in view of the small differences between Fourier and appropriately periodized and symmetrized wavelet projections, one has the choice between windowed Fourier transforms or correctly periodized wavelet projections. Section 4 studies the former; the later will be considered in Section 5.
w
O ( 2 ) - s y m m e t r i c local models for the KS equation
In this section, we consider projections of the KS equation onto small sets of Fourier modes supported on a short subinterval (or subdomain) S. According to the previous section, these can be viewed as perturbations of the wavelet projections considered in Section 2. We are interested not only in the ability of short subsystems to reproduce the statistics of long ones, but also in their ability to correctly capture instantaneous dynamical events involving coherent structures and, more generally, to track the solution of the full system in phase space. The material outlined here appears in greater detail in [11]. We will denote by ul (x, t) a solution of the KS equation for L = 400, obtained using a pseudo-spectral code containing 1024 elements (512 com-
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455
Equation
plex Fourier modes). The solution Ul(X, t) is regarded as a standard to which solutions of local models may be compared. We consider two types of models, in one of which the length L s of the subdomain S is fixed while in the other it varies in time. For fixed length subdomains, we integrate the truncated set of Fourier mode equations: ak -- \ L s ,1
1-
-~s
ak -- 2--~S E
J
jajak_j,
(4.1)
and take as initial condition the Fourier coefficients of ul (x, t) computed on the subdomain S = [ 2 - Ls/2,2~ + L s / 2 ] 1 ~~+Ls/2
ak(t = O) -- -~S
~-Ls/2
ul (x, t - O)e -i2~lx/Ls dx.
(4.2)
In deriving these equations, we implicitly enforce periodicity on the short domain S. This choice preserves the translation invariance of the full system, but at the expense of poorly approximating the correct values of Ul (x, t) at the ends of the subdomain. The inverse Fourier transform (based on the subdomain S) of the set {ak(t)} will be denoted u 3 ( x , t ) . By construction, u3(x, t) is a Ls-periodic function which at t = 0 corresponds to the projection onto the space of L s periodic functions of the restriction of ul (x, t) to S. In order to obtain the equations for the varying length model, let us denote the time varying boundaries of the subdomain S by l(t) < r(t), so that L s ( t ) = r(t) - l ( t ) . Rescaling the KS equation on [/(t), r(t)] via y = (x - l ( t ) ) / L s , we obtain: Ls 1 1 1 1 u t - y -L~s Uy - ~ s U + -~s u u y + -L-~su u y y y + 2 -L-ssu u y - o, 0 _< y _< 1, (4.3) in which L s ( t ) and l(t) may be either supplied as forcing functions or to which an additional equation modeling the evolution of L s may be added. The projected ODEs are as in (4.1), but the right hand sides contain the additional terms: Ls -k -~s
i Trk a k + E jr
i [27rika~] " j _J k aj ] + -~s
(4.4)
Intuitively, allowing L s to vary mimics the influence of neighboring domains, something which is completely neglected in the fixed length model. It should also be noted that specific choices of L s exert a crucial influence on the nature of long time solutions available to the fixed length short
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subsystems, so that allowing Ls to vary should permit a broader range of behaviors from which to build the space-time events seen in the full system. In the case where Ls(t) is taken from the full simulation, we simply choose r(t) and l(t) as being the position of two zeroes of the solution ul(x,t). This definition will hold until one (or both) zero(es) vanishes. Providing an autonomous model for the evolution of Ls(t) of the form Ls
-
l [ f ( a l , a 2 , . . . ;) T
-
Ls]
(4.5)
without reference to the full solution ul (x, t) is a more difficult task. We do not have any argument to show that such an equation even exists, and we have to proceed in an intuitive way. In equation (4.5), T is a time scale on which Ls tends to relax towards an 'equilibrium' value, f = 0. We want the function f to decrease as the amplitudes la~(t)l increase and vice-versa. This has the effect of stabilizing 'overactive' systems because, as can be seen from the bifurcation studies of [2, 1], viewed as an external parameter, increases in Ls lead (in general) to successive destabilization of linear modes. A specific choice for length variation is (we take l(t) = 0)-
II0 u ( ,t)l12
Ls - 1T A]i0-~u3~:tii[2 - Ls
),
(4.6)
where T and A are parameters to be chosen appropriately, and I1" ]12 denotes the L 2 norm based on the subdomain S. If the ratio IlO~u3112/]lOx,u3112 is interpreted as the inverse of the mode number of the most active mode, equation (4.1) yields the desired behavior for Ls: an increase in activity among higher modes results in a decrease in the above quantity, which leads to a decrease in Ls, and vice versa. In order to compare the solutions of the local models with the full simulation, one can either focus on the ability of solutions of the short system to follow the 'full' solution Ul (x, t), or examine the long term statistics of both solutions. We will focus here on short time tracking estimates, and refer the reader to [11] for a more detailed comparison of long time statistics (we found that local models which exhibit good short time tracking capabilities are generally also reliable with respect to long time statistics). As an estimate of the instantaneous difference between the solutions ul (x, t) and u3(x, t) (the solution of the local model) we took the L 2 norm of the difference computed in a central window of length d <_ L s as
-
t)
-
tll2dx
(4.7)
(u2(x, t) is, for any t, the projection onto the space of Ls-periodic functions of the ul (x, t) function). In all the computations reported here, we
Kuramoto-Sivashinsky Equation
457
included ten Fourier modes in the truncated system; computations with twelve modes gave indistinguishable results. In Figure 3 we show surface plot comparisons of the full Ul (x, t) and local u3(x, t) solutions on (short) time intervals of length 20 for several fixed choices of Ls. The midpoint 2 = 235 of the subinterval is chosen so that for t ~ 15, a well defined coalescence event takes place in the subdomain. It is immediately apparent from Figure 3 that, at least for some choices of Ls, the qualitative and even quantitative features of the solution are captured well by the local model (see Ls = 14.06 and Ls = 30.47). Other choices, such as Ls = 20.31, yield poorer results, and there is no simple trend toward improvement as L s increases. This is clear in both panels of Figure 4 where the instantaneous error e(t) and the average e0 - ~-1 .f~ e(t)dt error are shown for various values of L s. These results seem fairly typical and independent of the choice of 2. We conclude that there is some evidence of a 'resonance' effect: subdomains of lengths 16, 20-22, 24-27, 29-31, and 32 gave superior results. Broadly similar conclusions are reached when one includes the forcing terms Ls(t) and l(t) from the full simulation (see Figure 5). When equation (4.5) for the autonomous variation of Ls(t) is used, only a restricted choice of the parameter values T and A yields a behavior typical of the original KS equation. Again, the local model exhibits similar dynamics to the full system (this is further confirmed by an excellent agreement of the long time statistics, see Figure 7d of [11]). Of course, the specific model (4.6) is somewhat ad hoc. The fact that realistic results are found suggests that the local dynamics are not much influenced by the boundary variations, except for the continuous triggering of new events provided by the feedback term. These observations suggest that the coherent structures in the KS equation draw their energy primarily from local interactions and that long distance or 'external' inputs to an isolated short system are not necessary for the spontaneous production of realistic instantaneous events, although they may be important in keeping the system "stirred up." These conclusions are further supported by simulations on coupled systems of the type (4.1) reported on in [10]. The coupling is derived by considering two adjacent boxes and projecting each box onto Fourier modes. It is found that even very weak coupling can result in the appropriate long-term statistics, while the uncoupled systems settle onto traveling waves.
J. Elezgaray et al.
458
(b) u]
u~
1
1
0
0
-1
-1
20
20
(c) u::
u~
1 0
1 0
-1
-1
2O
20 L5
0
L5
0
F i g u r e 3. Comparison of full ul(x,t) and local u3(x,t) solutions for various s u b d o m a i n lengths: (a) u~(x,t) on Ls = 30.47; (b) u3(x,t) on Ls = 14.06; (c) u~(x,t) on Ls = 20.31; (d) u3(x,t) on Ls = 30.47.
459
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F i g u r e 4. Differences in windowed L 2 norms between the full and local solutions: (a) e(t) as functions of time for various L s and d = 12 (Ls = 14.06 dashed, L s = 20.31 dotted, and L s = 30.47 solid); (b) ee as a function of L s for tO- 20. Squares indicate data from 2 - 235 runs, crosses from ~ - 285. w
S t a t i s t i c s of the large scales of the KS equation
In this section we will focus on the statistical properties of the large scales of the KS equation, as well as the statistics of their couplings with the small scales. As can be seen in Figure lb, large scales do not contain much energy (on the average), so that they do not play an important role from the point of view of the proper orthogonal decomposition (the L: average sense). However, since the pioneering work of Yakhot [36], much attention has been devoted to the study of effective equations for the large scales of the KS equation, particularly in view of connections with the KPZ equation for interface growth [21, 35, 28]. More generally, considerable effort has been made to derive effective models for deterministic dynamical systems with many degrees of freedom, of which the KS equation is a particularly 'simple' example. We believe that the methods to obtain coarse grained model equations presented in this section and in more detail in [13] could replace other existing methods (such as the renormalization group method (RNG)), which require fitting parameters or rely on strong assumptions about the behavior of small scales. In [36], Yakhot used the RNG approach to show that the KS equation can be described at long wavelengths by the stochastic Burgers equation: Otu - v L - 2 u x x -F L - i u u ~
- ~(x, t),
v > 0,
(5.1)
where ~(x, t) is a stochastic forcing. However, the introduction of unknown parameters in the description of the small scales makes the effective corn-
J. Elezgaray et al.
460
U2
U3
1 0 -1 2O
1 0 -1 20 0
0
o~,
0
o~,
Figure 5. Local model with varying length: (a) The periodized window of the full solution u2(x, t); (b) Solution of a local model supplied with Ls(t) and l(t). putation of u unclear. A completely different approach is that of Zaleski [38], later pursued by Hayot, Jayaprakash, and coworkers [16] in one and two dimensions. These authors use a coarse-graining procedure to integrate out short wavelength degrees of freedom k > A, for a suitable cutoff A. If u(k, t) denotes the Fourier transform of u(x, t), then equation (2.1) may be rewritten as: k2
~(k, t) - - ~ 4 . ~ ~ ( k , where
t) + g(k, t) + / ( k , t),
(5.2)
g(k, t) - -2i 27rk
Z u(q,t)u(k-q,t) (5.3) Iql
-
1 47r2 47r2k2u(k,t) (u 4 L2 L2 ) L2 27rk -2i L • ~ ~(q, t)u(k - q, t)
(5.4)
Iql>_A or Ik-ql>_h
involving the coupling between at least one small scale mode and any other mode, is uncorrelated with the large scale modes for time differences longer than the (shortest) characteristic time scale of the large scales Tt: T -1
:f(k, t)u(-k, t - r)dt ~ 0 as T
-~ ~ ,
7- > rl, Ikl < i .
(5.5)
Kuramoto-Sivashinsky Equation
461
In other words, the assumption that only the linear term is renormalized yields a closed expression for u in terms of three point correlation functions mixing short and long wavelength modes. It then remains to check that the above definition of u is independent of the parameters A, k and the time delay T, and that the 'stochastic forcing' f(k, t) has the expected correlation properties. Although this method works well, it is not constructive, in the sense that one effectively needs to integrate the full KS equation in order to compute these three point correlations, and eventually u. In order to show how wavelet projections can yield an effective equation for the large scales, we use an approach reminiscent of that in the previous section, although here the strategy is rather to start from the 'full' equations, without any approximation, and successively drop coupling terms until an already known equation is reached. Hereafter, we will use alternatively the notations u(x, t) = ~ aj,k(t)ff~j,k(X ) and u(x, t) = ~ a s ~ s ( x ) for the wavelet decompositions (a = (j,k)), and split the wavelet coefficients into small ({a > }) and large ({a < }) scales, corresponding respectively to the values of j > jo and j <_ j0 respectively for some cutoff scale j0. We first study the small scales equations. Let us write equation (2.1) in terms of the as variables (see equation (2.2.1)):
i~s = lsza~ + nsz.yaza.y,
(5.6)
where summation over repeated indices is implied. The evolution equation for the small scales is therefore:
a">s - [Is~ > > + 51s~ > > (t)]a~ + ns~ >>~ > - >~ a.y > +
(t)
(5.7)
where ~ls>~ (t) - 'tsZT'>><>-<~7is a slowly varying perturbation of the linear term, and f> (t) - n s~7 ><< a~a < + />< s~ a~ is a slowly varying additive forcing term. Let us first consider the small scales equation (5.7) without the forcing terms 51s>~ and f>s, i.e., in isolation from the large scales. The key observation is that the equations governing the statistical properties of the a > variables satisfying these unforced equations (5.7) are identical to those of the KS equation with parameter f , - L2 -(j~ up to rescaling and small corrections. Namely, if we define the "boxes" Bk, k = 0 , . . . 2j~ - 1, specified by the sets of indices (j, k') with j _> j0 + 1 and Ik2-(Jo+l) - k'2-J I <_ 2 -(j~ we can decouple the a > equations into 2j~ independent sets of equations, each one including only variables a > belonging to the same Bk box, in a manner that preserves the statistics but not necessarily the dynamics. To prove this claim, we rely primarily on the fact that the evolution equations of the moments < as(t)a~(t + T) > (< 9 > is again a time average) can be periodized, and on the scaling relations between the
J. Elezgaray et al.
462
coupling coefficients la~ and na~7 for equation (2.1) with parameter L and L = L2-(J~ which are:
Iaz(L) n~z~(L)
-la~z:(L)(l+O(exp-2~"'"/s)) -
2-(J~
(L) (1 + O(exp -2j'~''/S)),
where a = (j, k), c~1 = (jl, kl), jl = j - (jo + 1), kl = 2 -(j~
(5.8) etc., and
jmin - min(j, j',j"). See [13] for details. Assembling these observations, approxione can easily see that the rescaled variables aal = 2(J~ mately satisfy the moment equations arising from equation (2.1) for length parameter L. Consequently, one gets the relation claimed above, between the equation governing the statistical properties of solutions of (2.1) with parameter L and those of (5.7) with no forcing. Let us now investigate how the approximations performed on the unforced small scale equations (5.7) survive, when one includes the slow forcing terms coming from the interaction with the large scales. Our intuition is that these approximations can hold provided the amplitude of the forcing is small and its evolution is well separated in time from the small scales (a similar idea constitutes the main ansatz of the work by C. C. Chow et al. [9] ). If jM corresponds to the scale of the most energetic Fourier mode k = L/27rv~, i.e., 2 -(jM+I) < 27rv~/L < 2 -jM, we expect that the above scaling relations between statistical quantities obtained for two different values of L will hold even in the presence of the slow forcing terms 8/>> and f > , provided that jo < jM. In order to illustrate this point, we present in Figure 6 the second moments and the probability distribution functions of the wavelet coefficients aj,k for several scales j and for the two parameter values L = 50 and L = 400 (that is, jM ~ 5 and jo = 2), with the appropriate rescaling. The most energetic scale j = 6 as well as the second moments of these distributions are in a rather good agreement. This means that (i) the scaling relations obtained for the unforced equations do imply approximate scaling relations for the statistical properties of the solutions for length values L and L, and (ii) the forcing is weak enough to preserve the relations obtained for the unforced equations. We note however that the same comparison with j0 -- 3 would yield very poor agreement. The reason is that the asymptotic dynamics for the KS equation with L = 25 is a fixed point (see [24]): this system is simply too "short" to exhibit sufficiently rich dynamics in isolation (but see below). We conclude that the choice of the cutoff j0 is actually dictated by at least two conditions: (i) the order of magnitude of the forcing terms 8>> and ]>, and (ii) the (asymptotic) dynamics of the KS equation for the parameter value L. Let us now focus on the large scale equations of motion: 9<
<<
+
<<
(t)la
+
<
(5.9)
Kuramoto-Sivashinsky Equation
'''
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-4
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'
'
I
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-1 10
- ....
I ....
I ' ' " ' l ' " ' '
,~,~
_
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. . . .
2
4
, 1,,,
~ ,
J 6
8
10 -4
[ I , , I ....
-0.1
-0.05
..-1 10-1
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I '
10-3
lu
I ....
10
-0. i
0 aj,k
0,I
I ,,, 0.05
aj'k I' '''
I,
.........
-0.2
1
l'''
,I
a~1^-3 ~
10 -4
I ....
I .... 0
-0.I
~ .........
0
0. I
0.2
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F i g u r e 6. a) T i m e average of a s2 as a f u n c t i o n of j: ( A ) , L = 400, (o) L = 50, divided by 8 - 2 j ~ b), c), a n d d) p r e s e n t respectively t h e p d f ' s P(aj,k=o) of t h e wavelet coefficients aj,k=o for scales j -- 4, 5 a n d 6 c o m p u t e d from t h e KS e q u a t i o n w i t h L - 400 (solid line), a n d j = 1, 2, a n d 3 for L - 50 ( d a s h e d line), rescaled by a factor 2x/~.
J. Elezgaray et al.
464
where 61~<~(t) - -'~<<> a> + -'~< > < -'*>'Y and f< (t) - ~<> > a.>r + /<>-> '~ > af~ a~ ~ " Following our initial picture of the small scales as a set of 2 j ~ independent boxes slowly driven by the large scale variables, we wish to check its validity with regard to the statistics of the forcing terms 61,<~ and f<. With this in mind, we performed three kinds of simulations. Simulation 1 is just the integration of the full KS equation for L = 400. Simulation 2 is integration in time of 2r176 8) independent KS equations with L = 50, each forced by the set of large scale (j < jo) components a < computed from simulation 1. Namely, for k = 0 , . . . , 2 / ~ -- 1, we integrate the equations
a> --
E
la>~(t)a~+ 2-(i~
+ +
(o>><
E
_
'~
E
">>>'~(t)a~a~>
,yl~<>(S))a~ (5.10)
a 7< 9
(Here we momentarily drop the implicit summation convention, and make use of the approximate scaling relation (5.8)). The comparison between simulations 1 and 2 will substantiate an important point of this section, namely, that the statistics of couplings between small and large scales in equation (2.1) can be computed from the interaction with a collection of independent low-dimensional systems. Simulation 3 no longer takes the large scale evolution from Simulation 1, but instead uses
l.<;(L)a.< + Z +
E
+E
'"a~ (L)a~ a<
, ' ~a ~.y ( L ) + ~, o,~.y~ ( L ) ) a ~ a.y>
+
E
, o,~~.y
a~ a~
la~ (L)a~
a.y> (5.11)
as governing equations for the large scales. The ensemble formed by equations (5.10) for k = 0 , 1 , . . . , 2j~ - 1 and (5.11) constitutes a completely autonomous model for the statistics of the KS equation (neither fitting parameters other than the cutoff scale j0 nor external forcing terms are needed). We considered the statistics of the two ( a ~ ( t ) a ~ ( t - T)) and three point correlations ( a ~ ( t ) a ~ ( t ) a ~ ( t - T)). These enter, for instance, in the energy budget equation (T = 0)
da <2 dt
la~ a~< a~< + ~<<
Kuramoto-Sivashinsky Equation i
!
!
!
(
i
|
!
!
~
t
I
,
i
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!
!
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10
-1
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-2
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10
-3
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i
465
i
!
J
-4. 10 -5
t
i
t
,
L,*
[
i
t
0 T>>j=0
|
l
J
!
|
!
!
4 10 -5
-1.
10 -4
0
IO-
T>>j=I
Figure 7. a) Probability distribution function of the transfer T> > term for j - 0, corresponding to the integration of the full KS equation (solid line), simulation 2 (long-dashed line) and simulation 3 (short-dashed line), for j0 = 2. b) Same as in a) for the j - 1 transfer T> >. 9< < < << --tazaaa~+T~ +Ta< >
+Ta
>>
,
(5.12)
and have been studied in the RNG approach to the Navier-Stokes equations [39, 12]. The time average of the transfer T >> is shown [39, 12] to correspond to a negative linear correction to the l~a < a 2s > term. As can be seen in Figure 7, the average < T >> > for a = (0, 0) is negative, although very frequently the "instantaneous" transfer is actually positive. Simulations 1 and 2 seem to agree fairly well for any j _< j0. The same remark is also valid for the other values of a, as well as for the "cross" transfer term T <>. The good agreement between the two statistics stresses the fact that all the information needed to compute the interaction between small and large scales (up to a good approximation) is actually encoded in the dynamics of the individual Bk boxes. However, the comparison of the undelayed three-point correlation functions obtained in Simulation 3 with those of simulations 1 and 2 shows clearly that, as one could expect, the agreement deteriorates as j approaches j0. The pdf's of the transfer term are close for j = 0, deviate for strong (but rare) values of the transfer for j = 1, and differ by a factor -~ 4 for j = 2 (not shown in Figure 7). Such disagreement should be expected. In fact, the periodization approximation involved in our model neglects nonlinear couplings of the type n<~> , where the modes ~Z and ~ belong to different boxes, and a corresponds to some large scale mode. Simulation 2 shows that this approximation is correct as soon as the small scales are forced with the right statistics. On the other hand, Simulation 3 shows that these missing interactions can significantly change the statistics of the large scales close to the cutoff scale j0.
J. Elezgaray et al.
466
The conclusion of these numerical simulations is clear: forcing a set of 2j~ independent subsystems with the large scales computed from the full simulation of equation (2.1) adequately reproduces energy transfer between large and small scales, provided the phase space of each of the uncoupled boxes Bk is large enough. However, when the large scales are generated autonomously by a "closed" large scale model coupled to the independent subsystems, the agreement is satisfactory only for scales well separated from the cutoff scale j0. w
Conclusion
In this paper, we have reviewed several recent approaches to the problem of modeling the local dynamics of extended turbulent systems in which most of the energy is contained in coherent structures. We first showed that, in an L 2 average sense, wavelets can do almost as well as Fourier modes in systems with translation invariance, in which the latter are optimal. However, a naive extraction of model equations from the wavelet projections of the original equation can be unsuccessful if the symmetries of the problem are not preserved (at least approximately). The study of wavelet projections of the KS equation on short intervals revealed that, from the point of view of capturing the correct dynamics, wavelets are comparable to Fourier modes, although the resulting equations are more massively coupled than the corresponding Fourier projections, which in addition preserve exactly the 0(2) symmetry of the KS equation. Moreover, in the analysis of the resulting models in phase space we are driven to consider combinations of wavelets which represent 'global' functions almost indistinguishable from Fourier modes. This led us to consider local models obtained by projecting onto a small set of Fourier modes supported on a short subinterval. Somewhat surprisingly, short systems having periodic boundary conditions on both fixed and varying lengths subdomains can reproduce the characteristic local events of much longer systems. Furthermore, we find strong evidence for a resonant effect, in which particular length ranges give superior tracking and long term asymptotic behavior. Finally, we considered a coarse-graining procedure to obtain an effective equation for the large scales of the KS equation. The method is very natural when written in wavelet bases, and no fitting parameters are needed in order to model the behavior of the small scales. It should be evident from all the above results that much remains to be done in order to understand why such local models behave so well. A first hint could be given by the short term tracking estimates suggested by A. Mielke [30], which shows that for parabolic equations such as KS, finite disturbances only make their presence known within a nonlinear cone of influence in the (x - t) plane (see [11]). Thus, we expect the effect of
Kuramoto-Sivashinsky Equation
467
neighboring 'boxes' to be important for local models only after a finite time has elapsed. One can also give a probabilistic estimate on short term tracking in phase space [11], and estimate the average (over initial conditions) divergence between the solution of the full equation and our local models. However, despite such open questions, and from a physical point of view, our initial conjecture that short systems possessing the appropriate dynamical 'components' can reproduce the coherent structure interactions of extended systems, appears to be validated.
Acknowledgments This work was partially supported by DoE Grant DE-FG02-95ER25238 (PH) and NATO 92-0184. HD also thanks the Swedish Fullbright Commission and the Department of Mechanics at the Royal Institute of Technology in Stockholm, Sweden, for their support.
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[22] Kelley, A., The stable, center stable, center, center unstable and unstable manifolds, J. Diff. Eqns. 3 (1967), 546-570. [23] Kekrevidis, I. G., B. Nicolaenko, and J. C. Scovel, Back in the saddle again: a computer assisted study of the Kuramoto- Sivashinsky equation, SIAM J. Appl. Math. 50 (1979), 760-790. [24] Kuramoto, Y., Diffusion-induced chaos in reaction systems, Suppl. Prog. Theor. Phys. 64 (1978), 346; Sivashinsky, G. I., Nonlinear analysis of hydrodynamic instability in laminar flames, Part I: Derivation of the basic equations, Acta Astronautica 4 (1977), 1176-1206; Hyman, J. M. , B. Nicolaenko, and S. Zaleski, Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces, Physica D 23 (1986), 265-292. [25] Leibovich, S., Structural genesis in wall bounded turbulent flows, in Studies in Turbulence, T. Gatski et al. (eds.), Springer, New York, 1992, p. 387. [26] Lemari4, P. G., Une nouvelle base d'ondelettes de L2(R'~), J. Math. Put. Appl. 67 (1988), 227-236.
[27]
Lumley, J. L., Stochastic Tools in Turbulence, Academic Press, New York, 1971.
[2s]
L'vov, V. S. and I. Procaccia, Comparison of the scale invariant solutions of the Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations in d dimensions, Phys. Rev. Lett. 69 (1992), 3543; V.S. L'vov and V.V. Lebedev, Interaction locality and scaling solution in d + l KPZ and KS models, Europhys. Lett. 22 (1993), 419; V. L'vov, V. Lebedev, M. Paton and I. Procaccia, Proof of scale invariant solutions of the Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations in 1+ 1 dimensions: analytical and numerical results, Nonlinearity 6 (1993), 25-47.
[29] Y. Meyer, Ondelettes et Operateurs, Hermann, Paris, 1990. [30] Mielke, A., personal communication, Montr4al, September 1993.
[31]
Moin, P., Probing Turbulence via large eddy simulations, AIAA 22nd Aerospace Sciences Meeting.
[32]
Myers, M., P. Holmes, J. Elezgaray, and G. Berkooz, Wavelet projections of the Kuramoto-Sivashinsky equation. I. Heteroclinic cycles and modulated traveling waves for short systems, Physica D 86 (1995), 396-427.
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[33] Perrier, V. and C. Basdevant, Periodical wavelet analysis, a tool for inhomogeneous field investigation- Theory and algorithms, La Rech. Aeros. 3 (1989), 53-67. [34] Robinson, S. K., Coherent motions in the turbulent boundary layer, Ann. Rev. Fluid Mech. 23 (1991), 601.
[35]
Sneppen, K., J. Krug, M. H. Jensen, C. Jayaprakash, and T. Bohr, Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation, Phys. Rev. A 46 (1992), 7351.
[36]
Yakhot, V., Large-scale properties of unstable systems governed by the Kuramoto- Sivashinsky equation, Phys. Rev. A 24 (1981), 642. See also H. Fhjisaka and T. Yamada, Theoretical study of a chemical turbulence, Prog. Theor. Phys. 57, (1977), 734 for related work.
[37]
Yakhot, V. and S. A. Orszag, Renormalization-group analysis of turbulence, Phys. Rev. Lett. 57 (1986), 1722; Renormalization-group analysis of turbulence, I.: theory, J. Sci. Comput. 1 (1986), 3-51.
[3s]
Zaleski, S., A stochastic model for the large scale dynamics of some fluctuating interfaces, Physica D 34 (1989), 427-438.
[39]
Zhou, Y. and G. Vahala, Reformulation of recursive-renormalizationgroup-based subgrid modeling of turbulence, Phys. Rev. E 47 (1993), 2503.
Juan Elezgaray CRPP-CNRS, Av. Schwietzer 33600 Pessac, France [email protected] Gal Berkooz BEAM Technologies, Inc. 110 N. Cayuga St. Ithaca, NY 15850 gal~cam. cornell.edu
Kuramoto-Sivashinsky Equation Harry Dankowicz Department of Mechanics Royal Institute of Technology S-100 44 Stockholm Sweden [email protected]
Philip Holmes PACM, Fine Hall Princeton University Princeton, NJ 08544-1000 [email protected]
Mark Myers
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T h e o r e t i c a l D i m e n s i o n a n d t h e C o m p l e x i t y of S i m u l a t e d Turbulence
Mladen Victor Wickerhauser, Marie Farge, and Eric Goirand
Abstract. A global quantity called "theoretical dimension" is defined which is roughly proportional to the number of coherent structures that expert observers count in simulated two-dimensional turbulent viscous flows. This paper reviews some previously published computations of this quantity for a few academic examples and for a small number of flows computed from random initial vorticity fields.
w
Introduction
Evolution equations describing complicated phenomena like turbulence and nonlinear wave propagation sometimes produce coherent features such as shock fronts and traveling vortices. These coherencies permit an approximate description of the evolving state by relatively few parameters, regardless of how many free parameters were initially used in the numerical resolution of the equation. The goal of this paper is to discuss automatic methods for extracting such low-rank approximations to complicated phenomena, and to present results of one such method applied to two simple examples: Burgers equation with dissipation, as previously computed in one spatial dimension [8], and the incompressible Navier-Stokes equation, previously analyzed in two spatial dimensions [7]. New data is contained in Figures 4 and 5, and Tables 2 and 3. The rank reduction method will be a kind of lossy compression; the solution at any instant in time will be written as a superposition of orthogonal phase atoms,, defined below, and then only those component atoms whose amplitudes exceed some threshold will be retained. Coherence will M u l t i s c a l e W a v e l e t M e t h o d s for P D E s W o l f g a n g D a h m e n , A n d r e w J. K u r d i l a , and P e t e r O s w a l d ( e d s . ) , pp. 4 7 3 - 4 9 2 . C o p y r i g h t (~)1997 by A c a d e m i c P r e s s , Inc. All r i g h t s of r e p r o d u c t i o n in a n y f o r m r e s e r v e d . ISBN 0-12-200675-5
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be detected by counting the number of retained components; when this count is low, the instantaneous state will be considered coherent. To count the relative importance of the retained components in such phase atom expansions, their amplitudes will be weighted using the entropy functional defined as follows. For any nonzero sequence a = {a(n) : n = O, 1, 2 , . . . } with I[all2 - ~ n la(n)l 2 < c~, put
H(a) de.._f Z 'a(n)'2 .
ilal12
log
('a(n)'2) ilali2
.
(1.1)
As usual, 0 log 0 is evaluated by continuous extension as 0. This is called the "entropy functional" because it is the entropy of the discrete probability distribution given by p(n) = la(n)12/llall2. In [19, p.278], and many other places it is shown that if M > 0 is the count of nonzero elements a(n), then 0 <_ H(a) <_ log M, and the maximum value is achieved when all nonzero a(n) have the same magnitude. Thus H(a) measures the flatness of the component amplitudes; it will be low when the amplitudes are not flat, i.e., when they are concentrated into fewer than M large components. Now define the theoretical dimension TD(a) of the sequence a by
TD(a)
def
exp H(a).
(1.2)
This quantity is used to boost intuition about the sequence a, since it can be said to measure the number of significant amplitudes rather than their logarithm, which measures the number of bits required to encode them. Both H(a) and TD(a) are computable for both finite and infinite sequences, so long as the sequences have slightly faster decrease as n -~ c~ than required for square-summability. In the experiments below, all sequences a are finitely supported with at most some large number M of nonzero coefficients; in that case, coherence will mean simply that TD(a) << M. Phase atoms are smooth functions which are well localized in both position and momentum in the sense of quantum mechanics. Namely, a phase atom r = r must have the following properties: 9 Finite energy"
ii~3112def /I~)(X)I2 dx < c~. Without loss, it can be assumed that I i r
(1.3)
1.
9 Smoothness and decay: both r and r are smooth, where r is the Fourier integral transform of r
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9 Position and momentum: xo def - f xlr
) [2 dx < co,
~o def - f ~l~b(~)[2 d~ < cc ,
(1.4)
these are respectively called the position and momentum of r 9 Localization in position and momentum:
AX de._f ( f ( X
d )1/2
--
A~ def (f(~
~0)2I~(~) 12d~) 1/2
(1.5)
these are respectively called the position uncertainty and momentum uncertainty of r 9 Concentration: r must be approximately as well localized in position and momentum as the Heisenberg uncertainty principle allows, that is, Ax A~ ~ 1. (1.6) The theoretical dimension of a function f is the minimum value of TD(a) achievable for any sequence a for which -
n
(1.7)
and {Ca} is a collection of phase atoms. Call that quantity T D ( f ) . It is obviously difficult to compute, since there is no simple parameterization of phase atoms over which to optimize, so it must be estimated using some particular, easily computed subset of the phase atom decompositions. The matching pursuit algorithm [12] is one effective way to search over a large library of phase atom decompositions, the Gabor bases. Adapted wave/orm analysis [10, 18] is a fast approximation of matching pursuit which uses wavelet packets [4] rather than modulated Gaussians as phase atoms. Both are examples of meta-algorithms [16] which fit a decomposition with good analytical properties to a function. The best basis expansion [5] of a function is a further simplification and speed-up of adapted waveform analysis; it is the phase atom decomposition used here to obtain an approximation for T D ( f ) . To estimate the evolving complexity of a numerical simulation using the notion of theoretical dimension, let f = f(x, to) be the solution at a fixed time to. Then T D ( f ) is estimated using a reasonable library of phase atoms and the result is plotted as a function of to. Progress through states
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of instantaneous coherence will be seen as local minima, and incoherence will be seen as local increases of T D. The remainder of this paper is divided into three parts. In the first, the techniques used to compute solutions to two evolution equations are described, as well as the algorithm for approximating T D ( f ) for each instant in time using wavelet packet best-basis expansions. In the second, numerical results from two simulations are presented. In the third, there is a brief discussion of the interpretation of the results and comments on how the technique might be improved.
w 2.1
Techniques
Wavelet packet best basis e x p a n s i o n s
Wavelet packets are generalizations of the compactly supported wavelets introduced by Daubechies, Mallat, and Meyer [6, 11, 15]. They constitute an over abundant set of basis functions with remarkable orthogonality properties, namely, that very many subsets form orthonormal bases. The one-dimensional functions were first described in [4]. Each basis element r is characterized by three attributes" scale s, wavenumber k, and position p, so they may be labeled Cskp. By the Heisenberg uncertainty principle, it is not possible to localize a function to arbitrary precision in both p and k. In other words, Ap. Ak >_ 1 in normalized units, where Ap is the uncertainty in position and Ak is the uncertainty in momentum. In the wavelet packet construction, Ap ~ 2" and Ak ~ 2 -8 in the same normalization, so that the product of the uncertainties is roughly as small as possible. Such functions, which cannot be significantly better localized in phase space, are evidently phase atoms. Fourier analysis with such waveforms or atoms consists of calculating the wavelet packet transform Wskp(f) -- (r f). Certain subsets of the indices (s, k, p) give orthonormal bases B, and for these subsets there is an inversion formula:
f-
E
(s,k,p)eB
(r
flr
(2.1.1)
Wavelet packets are rarely constructed explicitly. More often, one simply applies the fast discrete algorithm described in [4] to the sampled values of f, and thereby produces the coefficients W,kp(f). The underlying functions r can, however, be developed as follows. Introduce two (short) finite sequences {ha} and {g,~}, called conjugate quadrature filters, which satisfy
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the relations: 1
Eh2n-Eh2,~+l= n
v/~,
gn-(-1)'~-lhl_n
VnEZ;
(2.1.2)
n
5:
E
hnhn+2m =
n
gngn+2m =
{,,0, ifotherwise; m- o,
(2.1.3)
n
E hngn+2m - O,
(2.1.4)
Vm E Z.
n
Next, define a family of functions recursively for integers k _> 0 by:
W2k(X) W2k+l (x)
= --
V~ E n hnWk(2X -- n), Vr2 ~ n gnWk(2x -- n).
(2.1.5)
Note that W0 satisfies a fixed-point equation. Conditions (2.1.2) through (2.1.4) ensure that a unique solution to this fixed-point problem exists, and that {Wk :k E Z} forms an orthonormal basis for L2(]R). The quadrature filter pair h, g can be chosen (see [6]) so that the solution has any prescribed degree of smoothness. Equations (2.1.2) through (2.1.5) all have periodic analogs as well, which can be used in the case of periodic boundary conditions. The experiments in this article used a periodic algorithm with the so-called "C 6" coefficients, given as hn and gn in Table 1. Table 1. "Coiflet 6" coefficients for orthogonal wavelet packets.
nl
< -2
-2 -1 0 1 2 3 >3
Low-pass filter coefficient hn 0 3.85807777478867490 x 10 -2 -1.26969125396205200 x 10 -2 -7.71615554957734980 x 10 -2 6.07491641385684120 x 10 -1 7.45687558934434280 x 10 -1 2.26584265197068560 x 10 -1 0 ..........
High-pass filter coefficient gn
0 -2.26584265197068560 7.45687558934434280 -6.07491641385684120 -7.71615554957734980 1.26969125396205200 -3.85807777478867490 0
x x x x x x
10 1 10 -1 10 1 10 -2 10 -2 10 -2
One-dimensional wavelet packets are defined from these Wk by the formula: ~)skp(X)
--
2-s/2Wk(2-Sx -- p).
As described in [4], taking those functions {r " (s, k,p) E Z} for which " (s,k,p) e Z} form a disjoint the half-open dyadic intervals { [ k , ~ ) cover of the unit interval gives an orthonormal basis subset Z.
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The library of basis functions in two dimensions consists of all possible tensor products of the r functions with both factors sharing the same scale s. The definitions and formulas for this two-dimensional case may be found in [20]. Certain basis subsets can be described by disjoint tilings of the unit square, as follows. Let I be a half-open dyadic square [ ~ , 2,k|+1 )X [2, ,k-Y- kv2` +1 ) and put r174 (x, y) -- 2 - ' s W k . ( 2 - S x - - p x ) W k , ( 2 - S y - - p y ) . Then every basis in the library, for functions on the 2 s x 2 s grid, corresponds to a set of the form: {r
" I E Z, px E Z,py E Z, 0 <_ Px < 2 s - s , 0 _ py < 2s-s},
where Z is a disjoint cover of the unit square by such dyadic squares I, for 0 <_ s _ S and 0 <_ kx,ky < 28-1 . A graph basis is a collection of wavelet packets corresponding to some disjoint cover Z with squares no smaller than a fixed minimum. Computation of inner products with all such functions is performed recursively, with recursion depth controlled by the minimum square size. The best basis for a function f, chosen from among graph bases, is the one minimizing the entropy functional H(a) of the expansion coefficients of f. The implementation of graph basis expansions and the best basis search algorithm is described in detail in [19, Sections 7.2 and 8.2]. The entire procedure has complexity O ( N log N) where N is the rank of the problem, and N - 22s for the original grid-point formulation. The function f can be approximated by if, a superposition of just the largest components of its best-basis expansion. Call the best basis Z,. The projection onto the top few coefficients is defined as follows:
Here I E I , , al = (f, l/)l) and e is some predetermined threshold. The summation over all integer translates (Px,Py) is suppressed for compactness. The approximate value to be used for the theoretical dimension T D ( f ) will be TD(a) = expH(a). It makes little difference whether the full sequence for f or the truncated sequence for ff is used; truncation is mainly useful when analyzing infinite sequences.
2.2
Burgers' evolution equation with viscosity on the circle
Burgers equation is the first part of the following initial value problem:
OF (x t) - -10---F2(x,t) + v A F ( x , t ) Ot ' 20x F(x, O) - Fo(x) for all x, ~, F(x + l,t) - F ( x , t ) for allt>_O. f
(2.2.1)
(2.2.2)
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-0.5
I
l
I
I
I
I
I0
20
30
40
50
60
F i g u r e 1. Burgers' evolution from sin(2rx) at t -- 0 to t -- 1 in increments of 0.05. The constant u is the viscosity of the fluid and the function F0 = Fo(x) is the initial state at time t = 0. Consider one classical example: Fo(x) = sin(27rx). The graph in Figure 1 shows the evolution from this initial function at times 0, 0.08, 0.16, 0.32, 0.5, 0.75, and 1.00. The two arcs of the sine, one positive the other negative, are propagating in opposite directions to produce a steep slope at x = 1/2. The dissipation term A F produces the vanishing effect: the total energy in the solution tends to 0 as time increases. Without dissipation the slope at x = 1/2 would become infinite and a discontinuity would appear; the viscosity term controls how close the solution gets to singularity before dissipating. The apparition of a near-discontinuity means that the amplitudes of small-scale components in the solution are increasing, since they contribute the large derivatives. This phenomenon is better seen in Figure 2, which depicts the amplitudes of wavelet components of the signal arranged by scale. The evolution was computed with a Godunov scheme applied to the 1-periodic signal, using a space-step of 1/64 and a time step of 1/100. In [8], it was shown that the energy is decreasing in the biggest-scale wavelet components of the evolving function, while the energy in the smallest-scale ones is increasing. It was observed that one of the big-scale amplitudes already begins to decrease at time zero. The maxima of the smaller-scale amplitudes are reached later and later with decreasing scale. This last aspect can be better seen on Figure 2 which shows the absolute value of
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Abs[wavelet coef]
initial condition:sin(2*Pi*x)
60
50
40
30
20
10
0 .
.
.
.
.
.
2'0 .
.
.
.
40
,
.
.
.
.
.
a
60
,
,
,
o
|
................
80
i
I00 time/0.01
Figure 2. Amplitudes of the wavelet coefficients of Burgers' evolution from the initial function sin(21rx), shown in gray scale. the wavelet coefficients in gray scale: white is zero, black is the maximum. The graduations between 0 and 100 represent time; the others show the index of the wavelet coefficients. The first wavelet coefficient is the mean of the signal (actually 0), the second is the biggest-scale difference coefficient, the third and fourth are next largest difference coefficients, and so on. The last 32 are the smallest-scale difference coefficients, since there are a total of 64 samples of the signal. The analysis was done with "Coiflet 30" wavelets [6] because they have a large number of vanishing moments and are nearly symmetric. Ultimately, through dissipation, the function and thus all its wavelet coefficients decrease to 0. 2.3
Two-dimensional incompressible Navier-Stokes simulations on the torus
The classical simulation of two-dimensional decaying turbulent flows uses the incompressible Navier-Stokes equation with small viscosity. The Kraichnan-Batchelor theory in this situation [1, 9] postulates homogeneous mixing within the flow and supposes that the whole vorticity field is involved in the "cascade process" that transports enstrophy from large eddies to small ones, while energy is transferred from small to large scales. In contrast to this explanation, we believe that two-dimensional turbulent flows are generically inhomogeneous, and propose to model them
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as a superposition of coherent rotational vortices embedded in a random quasi-irrotational flow. We have observed, in numerical simulations of twodimensional Navier-Stokes equations with random initial conditions, that isolated vortices result from the condensation of enstrophy into localized, well-separated structures. These structures are stable as long as they do not interact with one another, but during close encounters they experience strong deformations, which then excite some internal degrees of freedom. This gives rise to a local cascade or transfer of enstrophy toward small scales and to its concomitant dissipation. Consequently, only a limited active portion of the vorticity field, correlated to the coherent vortices, is responsible for the turbulent cascade. The remainder, or background portion of the field, is passively advected and plays a negligible dynamical role. The atomic view may be compared with the vortex methods of Winckelmans and Leonard [22], Marchioro and Pulvirenti [14], and Saffman [17]. It generalizes the simplest model used to approximate two-dimensional flows, that of superposed point vortices, by considering the flow to be a superposition of atoms that are chosen from among a library of smooth localized functions such as wavelet packets [4] or localized cosine functions [3, 13]. The additional parameters available to these atoms enable us to take into account the internal degrees of freedom of each vortex, which can be considered as a molecule. The goal will be to compute the number of "significant" atoms in a turbulent flow, i.e., those components whose amplitudes exceed a preset threshold. Those that correspond to the same locale can be interpreted as the principal components of a coherent structure. Their number evolves in time, with a generally decreasing trend due to the decay in enstrophy caused by dissipation, and gives a quantitative estimate of the number of coherent structures and the complexity of the turbulent flow. The analysis begins with a direct numerical solution of the NavierStokes equation describing the dynamics of a two-dimensional incompressible viscous flow. In the periodic plane S = (0, 27r) • (0, 27r) C ]a 2 and in the absence of external forcing, these take the following form: 0u cg----t-+ (u. V)u + V P -
uV2u = 0
in S • ]R+,
V.u=0
inS•
u(x, O) = Uo(X)
+,
in S.
Here u is the velocity field, P is the pressure field, and v is the kinematic viscosity. Periodic boundary conditions are also imposed. The equations are rewritten in terms of vorticity w and stream ]unction r defined by u
-
(ux u2
-
_
OXi
-
'
Oxl
Ox2
(2.3.1)
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The Navier-Stokes equations then become + J(r
Ot
- vV2w - O, -
V2r
(x, t) ~ S x IR+,
(x, t) ~ S x la +, xES.
~(x, 0) - ~0(x),
Again, periodic boundary conditions are imposed. The Jacobian operator in terms of these new variables is: J(r w)-
0r Ow
0r 0_~w
'
(~Xl 0X2
0X2 0Xl"
(2.3.2)
The functions w and r can be expanded in their Fourier series over the periodic domain S" w(x, t) - E &(k, t)e ik'x,
&(k, t) = ~1 fx es w(x, t)e -ik'x dx,
k
r
t) - E r
t)e ik'x,
r
t) = ~1 fx es r
t)e -ik'x dx.
k
A turbulent vorticity field such as the one depicted in Figure 3 develops from a random initial vorticity field w0(x) which is integrated for many time steps in the presence of time-periodic external forcing (at very low wavenumbers), until the vorticity field reaches a statistically steady state. Forcing is subsequently turned off and the same integration is continued in the decaying regime. A pseudo-spectral Galerkin method was used to integrate the NavierStokes equations; at each time step, all differentiation was performed in ~b, r coordinates and all multiplication in w, r coordinates. Both w and r are represented as finite Fourier series, or superpositions of the Fourier modes at wavenumbers 0 <_ ]k I < kr, where kr is the cutoff wavenumber which gives some fixed resolution. The time integration was done using an Adams-Bashforth scheme. The periodic plane S was sampled on 1282 grid points in the simulation. This is not terribly fine, so the commonly used mechanism of modeling subgrid dissipation was employed to increase the apparent resolution. The subgrid scale model was a hyperdissipation operator, (-V2) 4, which replaced the Laplacian operator in the Navier-Stokes equations. This caused the vorticity field produced by the direct numerical simulation to decay more rapidly in regions of high local variation than it would in a simulation using Laplacian dissipation. The disproportionally fast decay produced a flow that acts "as if" energy losses from subgrid scales were included in each time step through aliasing.
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F i g u r e 3. Vorticity field at an instant of time, scaled to fill an 8-bit dynamic range.
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484 Approximate |
/
3.5
2.5
theoretical |
dimension
as i
a
function
of i
time
1.5
0.5
-
0
0
i
I
I
I
0.2
0.4
0.6
0.8
1
Figure 4. Theoretical dimension of a solution of Burgers equation from an initial state sin(2~x) at t = 0 to t = 1. The program ran for 6000 time steps At -- 10 -4 in units of T 1, which corresponds to about 30 initial eddy turnover times, starting from the statistically steady state. The vorticity fields analyzed here are time slices spaced 20 time steps apart. These may be considered to be typical snapshots of a fully developed turbulent flow whose enstrophy is slowly decaying.
w 3.1
Results
One-dimensional results: Burgers equation
The periodic solution to the 1-periodic Burgers equation from an initial state Fo(x) = sin(21rx) was computed using a viscosity of u -- 0.01/lr, from t - 0 to t - 1. The results are plotted at time intervals of 0.05 in Figure 1, and the amplitudes of the associated wavelet coefficients are depicted as gray levels in Figure 2. The number of dark streaks in the latter figure give a crude estimate for the theoretical dimension of the solution. The theoretical dimension was approximated somewhat better with the best basis expansion using Coiflet 6 wavelet packets as phase atoms. This was done at every time step, with At = 0.01. The result is plotted in Figure 4. A longer-term plot of theoretical dimension is seen in Figure 5. It is readily noticed that the estimated theoretical dimension jumps from about 2 to about 3.5 at first, as the shock begins to form at spatial position 32, then decreases back to 2 as dissipation smooths out the function
Theoretical Dimension of Turbulence Long-time i
evolution
of a p p r o x i m a t e , .......................
485 theoretical
!
dimension !
3.5
J
2.5
1.5
0
......
0
i
I
I
I
1
2
3
4
5
Figure 5. Theoretical dimension of a solution of Burgers equation from an initial state sin(21rx) at t = 0 to t = 5. and removes the large derivatives. Extra phase atoms of small position variance and large frequency variance seem to be needed to represent large derivatives near the shock, whereas the phase atoms which represent the two smooth lobes of the initial sine curve remain as a kind of background, persisting even to long times when the solution has dissipated nearly to 0. The peculiar feature at time t - 0.08 may be the result of a sudden change of the basis in which the atomic decomposition is performed. The best orthogonal basis changes at that instant from a 21-subband decomposition to a 10-subband decomposition. Even crude approximations of theoretical dimension provide some clue to the number of degrees of freedom required to approximate a solution to a complicated evolution equation. In the one-dimensional Burgers equation with dissipation, even the wavelet decomposition provides a reasonable estimate of the number of phase atoms in the minimal decomposition. As the shock begins to form, extra wavelet components appear at the small scales indexed by ordinate values near 50 in Figure 2. These decay as the energy dissipates and the sharp slope near abscissa 32 in Figure 1 becomes smoother.
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350 300 250 200 150
0
| ........
0
I
i
i
i
I
0.I
0.2
0.3 time t
0.4
0.5
Figure 6. Evolution of estimated theoretical dimension for the original vorticity field.
3.2
Two-dimensional results: Navier-Stokes equation
Now consider a vorticity field, similar to that depicted in Figure 3, representing what we believe is a generic time slice of a homogenous, isotropic, fully developed two-dimensional turbulent flow. The experiment segmented it into high-enstrophy and low-enstrophy components in the wavelet packet best basis. All experiments began with an initial condition consisting of a fully developed two-dimensional turbulent flow sampled on 1282 grid points. This gave a "reference initial flow field" which was then evolved for an additional 6000 time steps between t = 0.0 and t = 0.6, using the Navier-Stokes model described in the previous section. In the chosen normalization, this interval is approximately 30 eddy turnover times, or the time it takes for an average vortex to make 30 rotations. The resulting evolution may be called the "reference flow field evolution". Vorticity fields were computed at 300 equally spaced future times and then their theoretical dimension was estimated in the wavelet packet best basis defined by the "Coiflet 6" filters listed in Table 1. In Figure 6 one can see the evolution of theoretical dimension for this reference evolution. There are fluctuations in the estimated theoretical dimension which cause the graph to depart from its course of smooth decay. These are caused both by sudden changes in the basis choice and by the lack of shiftinvariance of the orthogonal wavelet packet expansion. The theoretical dimension estimate starts at approximately 400 for a
Theoretical Dimension of Turbulence
487
i i
Figure 7. Evolution of estimated theoretical dimension for the vorticity field reconstructed from: (left) the top 50%, or 8192; (middle) the top 5%, or 819; (right) the top 0.5%, or 82 of the original's wavelet packet components. Table 2. Subjective count of vortices (VCt) compared with theoretical dimension (TDt) at time t for 2-D decaying evolutions from increasingly simplified initial turbulent vorticity fields. Times t - 0.020 and t - 0.598 were chosen slightly inside the simulation interval [0.0 0.6] to avoid artifacts. Components 16384 8192 819 82
Fraction
VC.020
TD.o2o
100% 5o% 5% 0.5%
26 26 21 20
381 381 340 135
VC.598 TD.598 15 15 17 19
151 143 127 95
field with 26 distinguishable vortices and decays to 150 at the 301st time slice when there are 15. The count of vortices is necessarily subjective, and no attempt was made to include the contribution of vortex filaments which also evolve and decay in the simulation. The evolution of theoretical dimension measures the quality of an approximate evolution from a projected initial state [7]. Figure 7 shows the evolution of estimated theoretical dimension from initial states approximated by 50%, 5%, and 0.5% of the original components. These represent 8192, 819, and 82 degrees of freedom, respectively. Table 2 shows how initial and final estimates of theoretical dimension compare with the subjective count of significant vortices in the original and the three approximations. By contrast, the portion of the initial vorticity field which was discarded by the projection onto strong wavelet packet components contains a very large number of local vorticity extrema. The theoretical dimension estimate for those weak remainder fields are plotted in Figure 8, which shows the 50%, 95%, and 99.5% leftovers from Figure 7. Notice that the remainder theoretical dimensions are much larger than those for the approximations. Table 3 compares the estimate of those theoretical dimensions with a crude subjective count of significant vortices. Small fluctuations of the theoretical dimension estimate are due to the lack of shift-invariance of wavelet and wavelet packet decompositions. This
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!
,!
i
!
i 9
o.~
9
,.~
i
i ~
o,
Figure 8. Evolution of estimated theoretical dimension for the vorticity field reconstructed from: (left) the bottom 50%, or 8192; (middle) the bottom 95~163 or 15564; (right) the bottom 99.5%, or 16302 of the original's wavelet packet components. Table 3. Subjective count of vortices (VC) compared with theoretical dimension (TD) for 2-D decaying evolutions from the increasingly energetic remainders of simplified initial turbulent vorticity fields. Components 16384 8192 15564 16302
Fraction
100% 50% 95% 99.5%
VC.o2o TD.o2o 26 381 600 1160 350 1583 200 927
VC.598 TD.598 15 151 100 534 150 915 477 60
problem can be alleviated by computing TD(f) as the minimum of the information costs of the best basis wavelet packet expansions of f ( x - xo), where xo ranges over several small spatial shifts. That algorithm seems to have O(N 2) complexity, though there is a well-known shift-invariant wavelet expansion [2, 19] with complexity O(N log N) which may be used when wavelet phase atoms suffice.
w
Discussion
This paper describes very crude approximations to the theoretical dimension of a complicated evolution, yet even these provide some clue to the complexity of the flow. Still, many improvements of the computation are possible. The most basic improvement would be to compute theoretical dimension with a larger library of phase atoms. For example, Gabor functions could be used as in the matching pursuit algorithm. However, this would raise the complexity of estimating TD on an N-point grid to O(N2). A faster improvement would be to use best basis with multiple wavelet packet libraries, possibly combined with adapted local cosine libraries or other modern basis sets. This would result in an algorithm of complexity O(N[log N]2), with a constant that grew with the number of distinct
Theoretical Dimension of Turbulence
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libraries searched. Moving still closer to matching pursuit, the requirement of using a best orthogonal basis could be relaxed and a best atomic decomposition could be sought instead, using the adapted waveform metaalgorithm. Since wavelet and more generally wavelet packet algorithms are not shift-invariant, their estimates of T D will always contain small fluctuations depending upon details of grid spacing and the motions of coherent parts of the analyzed function. There are several ways to avoid this, all of which increase the complexity of the computation by finding the minimal estimate of TD over a family of shifts. When wavelet phase atoms are used, the added complexity is minimal. Phase atom decompositions provide a tool for locating and measuring coherent parts of a flow. A coherent structure is said to be present at a point when a small number of large phase atoms are supported near that point; the number of these atoms gives an estimate for the number of degrees of freedom in the coherent structure. This definition, together with techniques from wavelet packet analysis, provides an algorithm to extract portions of flows that human observers see as "coherent." Theoretical dimension is useful in deciding both how many degrees of freedom are actually present in the coherent part of a function, and to determine the minimal rank of a projection onto a good approximate solution. Furthermore, the theoretical dimension of components discarded by such a projection is an indicator of the quality of the approximation. When the theoretical dimension of the discarded components is too low, it means that the discarded portion contains some coherent part. Since the computation of theoretical dimension is relatively cheap, it may be done alongside simulations and computed evolutions simply as a guide to some global properties of complicated phenomena. A c k n o w l e d g m e n t s . The work of the second and third authors was supported in part by the NATO program Collaborative Research, contract CRG-930456. The first author was supported in part by AFOSR contract F49620-92-J-0106, NSF grant DMS-9302828, and a private grant from the Southwestern Bell Telephone Company. The Burgers equation simulation was prepared with Mathematica on a NeXT computer at Washington University by Fr@d@ric Heurtaux and Fabrice Planchon, who were supported in part by the Internship Office of the I~cole Polytechnique. The NavierStokes equation simulations were performed on the Cray 2 at the Centre de Calcul Vectoriel pour la Recherche, using the incompressible Navier-Stokes code of Claude Basdevant. Theoretical dimension and best-basis wavelet packet expansions in one and two dimensions were performed using two versions of the Adapted Wavelet Analysis Library [18, 21].
490
M. Wickerhauser et al. References
[1] Batchelor, G. K., Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. Fluids 12 (Suppl. II) (1969), 233-239.
[2] Beylkin, G., On the representation of operators in bases of compactly supported wavelets, SIAM J. Namer. Anal. 29 (1992), 1716-1740.
[3] Coifman R. R. and Y. Meyer. Remarques sur l'analyse de Fourier fenStre, C. R. Acad. Sci. Paris Sdr. I Math. 312 (1991), 259-261.
[4] Coifman, R. R., Y. Meyer, S. R. Quake and M. V. Wickerhauser,
Signal processing and compression with wavelet packets, in Progress in Wavelet Analysis and Applications, Toulouse, 1992, Y. Meyer, S. Roques (eds.), Editions Fronti~res, Paris, 1993, pp. 77-93.
[5] Coifman, R. R. and M. V. Wickerhauser, Entropy based algorithms for best basis selection, IEEE Trans. Inform. Theory 32 (1992), 712-718.
[6] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 61 (1988), 909-996. [7] Farge, M., E. Goirand, Y. Meyer, F. Pascal, and M. V. Wickerhauser, Improved predictability of two-dimensional turbulent flows using wavelet packet compression, Fluid Dynam. Res. 10 (1992), 22925O. [8] Heurtaux, F., F. Planchon, and M. V. Wickerhauser. Scale decomposition in Burgers' equation, in Wavelets: Mathematics and Applications, J. J. Benedetto, M. Frazier (eds.), Stud. Adv. Math., CRC, Boca Raton, FL, 1992, pp. 505-523. [9] Kraichnan, R. H., Inertial ranges in two-dimensional turbulence, Phys. Fluids 10 (1967), 1417-1423. [10] Majid, F., Applications des paquets d'ondelettes au d~bruitage du signal, Department of Mathematics, Yale University, July 1992; Rapport d'Option, Ecole Polytechnique, 1992, preprint. [11] Mallat, S. G., A theory for multiresolution signal decomposition: The wavelet decomposition, IEEE Trans. Pattern Anal. Much. Intelligence 11 (1989), 674-693. [12] Mallat, S. G. and Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE Trans. Signal Process. 41 (1993), 3397-3415.
Theoretical Dimension of Turbulence
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[13] Malvar, H., Lapped transforms for efficient transform/subband coding, IEEE Trans. Acoustics, Speech, Signal Process. 38 (1990), 969-978. [14] Marchioro, C. and M. Pulvirenti, Vortex Methods in 2D Fluid Dynamics, Lecture Notes in Phys. 203, Springer, Berlin, 1984. [15] Meyer, Y., Orthonormal wavelets, in Wavelets: Time-Frequency Methods and Phase Space, J.-M. Combes, A. Grossmann, and P. Tchamitchian (eds.), Springer, Berlin, 2nd edition, 1989, pp. 21-37.
[16] Meyer, Y., Wavelets: Algorithms and Applications, SIAM, Philadelphia, PA, 1993.
[17] Saffman, P., Vortex interactions and coherent structures in turbulence, in Transition and Turbulence, R. E. Meyer (ed.), Academic Press, New York, 1981, pp. 149-166,
[is] Wickerhauser, M. V., Adapted Waveform Analysis Library, v2.0, Fast Mathematical Algorithms and Hardware Corporation, Hamden, Connecticut, June 1992.
[19] Wickerhauser, M. V., Adapted Wavelet Analysis from Theory to Software, AK Peters, Ltd., Wellesley, MA, 1994.
[2o] Wickerhauser, M. V.,
Comparison of picture compression methods: Wavelet, wavelet packet, and local cosine transform coding, in Wavelets: Theory, Algorithms, and Applications, C. K. Chui, L. Montefusco, L. Puccio (eds.), Academic Press, San Diego, CA, 1994, pp. 585-621.
[21] Wickerhauser, M. V., AWA 3: Adapted Wavelet Analysis Library, version 3. Fast Mathematical Algorithms and Hardware Corporation, Hamden, Connecticut, June 1995. [22] Winckelmans G. S. and A. Leonard. Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows, J. Comput. Phys. 109 (1993), 247-273.
492 Mladen Victor Wickerhauser Department of Mathematics, Campus Box 1146 One Brookings Drive Washington University Saint Louis, Missouri 63130 U.S.A. [email protected]
Marie Farge LMD-CNRS Ecole Normal Superieure 24 Rue Lhomond F-75231 Paris France farge~lmd.ens.fr
Eric Goirand LMD-CNRS Ecole Normal Superieure 24 Rue Lhomond F-75231 Paris France goirand(~lmd.ens.fr
M. Wickerhauser et al.
VI. Wavelet Analysis of P a r t i a l Differential O p e r a t o r s
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Analysis of Second Order Elliptic O p e r a t o r s W i t h o u t B o u n d a r y Conditions and W i t h V M O or HSlderian Coefficients Jean-Marc Angeletti, Sylvain Mazet, and Philippe Tchamitchian
Abstract. We study second order elliptic operators in divergence form or in nondivergence form with coefficients satisfying mild regularity conditions. We prove optimal Sobolev estimates. We also study the convergence of the Galerkin scheme when applied to divergence operators in multiresolution analysis.
w Introduction We consider partial differential operators, defined on lRn, in divergence form: L--div AV , or in nondivergence form: M - -a~OaO~,
with the usual summation convention. In both cases we assume that A-
A(x) - ( a ~ ( x ) ) l < a , Z < n
is a matrix-valued function of x E lRn, satisfying the classical ellipticity estimates [8AI[~ - sup []A(x)][ < ~ , (H1) x
Multiscale Wavelet Methods for P D E s Wolfgang Dahmen, Andrew J. Kurdila, and Peter Oswald (eds.), pp. 495-539. Copyright 0 1997 by A c a d e m i c Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-200675-5
495
496
J. A ngeletti et al.
where IlA(x)ll is the matrix norm of A(x) acting on the Euclidean space
IR n, and
3 5>0
V x,~ e ira
A ( x ) ~ . ~ >_ 5 I~!2 .
(H2)
We also assume some weak regularity of the coefficients aa~, the weakest of which is being in VMO(IRn). What this precisely means will be explained further, but at the present time, we only vaguely describe it as a property of uniform continuity in the mean. These operators are naturally defined on appropriate Sobolev spaces" L is continuous from W I'p (= WI'p(IRn)) to W -I'p, and M is continuous from W 2,p to L p , 1 < p < c~. Our first task is to describe and to discuss the resolvents of L and M on these spaces. We will prove the following. Theorem A. Assume that the coefficients a ~ 1 0 , (L + r
are in V M O .
Then, if
is continuous from W -1,p to WI,P;
ii) there exists a ~p > 0, such that if r ___Cp, then (M +r from L p to W 2'p.
is continuous
We will also give explicit constructions of the resolvents in question, with the help of wavelet bases. As an application, we will recover interior estimates for the solutions of inhomogeneous problems, due to Chiarenza, Frasca and Longo in the case of M ([5], see also [3]). These results form the content of Section 3. In Section 4, we consider again the above-mentioned resolvents when the regularity hypothesis on the coefficients is strengthened. In such a case it is natural to expect the invertibility of L + ~ and M + ~ to hold on a wider range of functional spaces, and it is our purpose to confirm and to describe this phenomenon. For simplicity, we only quote here a less precise version of the result to be proved: Theorem B. Assume that the coefficients are in the HSlder space C 8~, with O < so < l. Then, if l < p < oo, i) for every ~ > 0 , (L + ()-1 is continuous from W -l+s,p to W l+s,p when ii) if ~ >_ r (defined in Theorem A), ( M + r to W 2+s,p when 0 <_ s < so.
is continuous from W 8,p
This theorem is analogous to previous results of M. Taylor [15] in the case of L, and Chiarenza, Frasca and Longo [5] in the case of M. Our study will be more complicated for L than for M. This is not surprising, since L is the most singular operator. It has, however, the advantage of being associated with a variational formulation. Given some
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Analysis of Elliptic Operators
wavelet spaces of approximation, we can approximate the resolvent of L by using the well-known Galerkin scheme. It has been necessary for us, in constructing ( L + { ) -1 , to investigate some of the properties of this scheme. This is the subject of Section 5, which is of independent interest. The main theme of this study is to know for which topologies the Galerkin scheme converges. Indeed, the Galerkin scheme is constructed in such a way that its W 1,2 convergence is given for free. Looking for the range of p and s such that its W s,p convergence holds, we will obtain the following. T h e o r e m C. Given a multiresolution analysis of regularity r >_ 3, the GMerkin scheme, applied to an equation of the form Lu + {u - f, where f E C~~ ~) and ~ > O, will converge in the W I'p topology for 1 < p < ce, if the coemcients are in V M O , and in the WI+s,P topology, 1 < p < cc , 0 <_ s < so, if the coetticients are in C s~ . All these results are obtained by using one single common key idea, coming from [18], which we now roughly describe. The first step to invert a partial differential operator is to construct a parametrix, which in general is some approximation of the inverse operator on some space of functions having only high frequencies. In the framework of a wavelet basis, denoted by (~h)her, this amounts to constructing a suitable approximation of uh - (L + ~)-1 ~h or
vh = (M + ~)-l~x,
being fixed, when the scale associated to ~h is small. But in such a case, h has a small support (at least heuristically, if not exactly). It is therefore natural to use the old idea of freezing the coefficients on the support of ~h, and to define Oh by Oh - (Lh + ~ ) - 1 ti/h - (Mh + {)-I @h, where L h ( - Mh) is the frozen operator. Our parametrix P will then be defined by the formula P ~h--Sh 9 Now, the point is that, thanks to the cancellations of ~h , 0h is, up to a renormalisation, a vaguelette. In other words, it shares most properties of the wavelet ~I,h, except that of being deduced by translation and
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dilation from a finite number of functions. This crucial fact (which will be properly stated when necessary) allows us to use the Calderbn-Zygmund machinery, and the so-called paraproducts. These are operators which approximate the pointwise multiplication by a function, provided it has some mild regularity, and which can be composed with differential operators. The weakest property of regularity of the multiplier which is convenient for such an approximation to hold is being in V MO. Hence, under the V M O hypothesis, our parametrix will eventually approximate (L + ~)-1 and (M + ~)-1, for suitable topologies. This will be the key step in all our constructions. The background which is necessary to understand the definition and the analysis of our parametrix is given in Section 2. Let us mention that this text is not self-contained: every nonclassical result to be used is explicitely stated, but is not always proved, especially when it belongs to the Calderbn-Zygmund machinery. Our purpose is to emphasize how this machinery can be used to obtain simple proofs of nontrivial results, not to describe it in detail. For such a development, the reader is referred to Meyer's book [13], or other texts [15,16,17].
w Preliminaries After having defined the notation, we will recall some definitions and results on vaguelettes, constant coefficient elliptic operators and V MO. We will also study the product between the pointwise multiplication operator by a V M O function and a singular integral operator, stating a theorem which will play a key role in the sequel. 2.1 Notation We consider a multiresolution analysis on L2(IRn) of regularity r0 _> 3, denoted by (Vj)jez, with the associated orthonormal wavelet basis ( ~ ) ~ e I . In this notation, I is the set of all e ) 2_ j where j E Z, k E Z n, and ~ E {0, 1}n, e r 0. The correspondence ~
(j,k,e)
is bijective, and we will sometimes use the self-explanatory notation j(A), k(A), ~()~). Moreover, we define Ij by /j={1EI;
j (~) = j } .
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499
For each A E I we have
)~(x) -- 2Jn/2~ ~ (2 jx -- k) ,
x E IRn ,
(2.1.1)
where the 2 n - 1 functions ~ satisfy the following properties: ~ is Cro-1, 0 a ~ is defined almost everywhere and bounded when [a[ - r0, and each 0 ~ , [a[ <_ r0, is rapidly decreasing at infinity. Moreover, we assume zero moments up to degree ro, i.e., for every polynomial P on ~ n of degree less than or equal to ro with respect to each variable,
P(x)~(x)
dx-O.
n
The set ( ~ ) ~ e I is an orthonormal basis of L2(lR ~) and an unconditional basis of Ws'P(IR n) when 1 < p < c~ and lsl < r0. We denote by
")
Yj
the orthogonal projection on Vj, and by .
5
9
the orthogonal projection on Vj• The adjoint operator 7r~ is thus the canonical extension operator from Vj to L2(]Rn), and analogously for z@*. We will often need to extend the subspaces Vj and the associated lrj, 7@, 7r~, 7rJ-* to the L p (IR n) setting or to other functional spaces, such as W s,p (IRn). We will keep the same notation, for simplicity; the topology which will be referred to will be clear from the context, or explicitly stated. Finally, the letter Q will always denote a cube in ]Rn, and, especially, Q~ will be the unique dyadic cube containing A and of side length 2-J. We will also use the notations f~, 0~, a~ and so on, which have only the ordinary meaning, and do not imply that any relation like (2.1.1) holds. We adopt the standard convention that the letter C is devoted to constants whose dependence on other parameters is not indicated and whose values are allowed to change. Of course, all relevant information on constants will be explicitly quoted. 2.2 V a g u e l e t t e s
Definition 1. A family of functions ( 0 ~ ) ~ i is called a family of vaguelettes when there exist real numbers C > 0, q > 0, r > 0, and an integer m with 0 <_ m < q, such that for every A E I, we have V x e lRn
[0h(x)] _< C 2in/2(1 + 2 J [ x - )t[) - n - q ,
(2.2.1)
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J. Angeletti et al.
Ox is of c/ass C [r], and it" a is a multi-index, 1~1 ~ [r], then v x ~ Ft =
IO"0x(~)! < C 2J(~r+iai)(1 + 201x- AI) -=-q 9
(2.2.2)
Moreover, if a is a multi-index of length [r], 0a0x is r - [r] H61derian, with
lO~0~(~ + h ) - o~ox(~)! _< c 2J(~+~)lhl~-[~](1 + 2Jlz- AI)-"-q, (2.2.3) for ali x 6 IRn, h 6 IRn, [h] <_ 2-J. Finally, for every polynomial P of degree less than or equa/to m, P ( x ) 0x(x) dx = O.
R
(2.2.4)
n
The best constant C in (2.2.1-3) is called the size constant, while q, r and rn are respectively called the envelope index, the regularity index and the cancellation index of the family. The above inequalities are ca/led vaguelettes estimates on (0x)xei. When m = 0, q > 0, and (2.2.2) is replaced by a HSlderian condition of degree a E (0, 1), the analogous definition one obtains is that of an (a, q) smooth molecule of Frazier, Jawerth, and Weiss [9]. If (8~)xei is a family of vaguelettes, we define an operator T on the space of finite linear combinations of wavelets by the formula V AEI
T~x=O~.
We will often encounter such an operator. Here is a summary of its main properties. The previously defined operator T extends to a CalderbnZygmund operator. It is continuous on W',V(IR ~) when 1 < p < cr and [s[ < inf (r0, r, m + 1, q). Moreover, its norm on each of these spaces satisfies
Theorem
1.
[ITl[w.,. < C(s, p, q, r, m) Co, where Co is the size constant of the family (Ox)xei. This theorem will be fundamental for us. Its proof can be found, e.g., in [13]. 2.3 T h e L e m a r i d a l g e b r a Definition 2.
We denote by s the space of all operators T such that the a family of vaguelettes.
family (T~,x),xeI form
Let us quote the following results on L:.
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Theorem 2. i) s is independent of the choice of the wavelet basis, provided its regularity r0 is nonzero; ii) s is an algebra, stable under taking the adjoint; iii) s is exactly the set of all Calderbn-Zygmund operators T such that T(1)=T*(1)=O; iv) s is not stable under taking the inverse. The first three assertions are due to Lemari6 [11]. The characterization given in iii) has to be related to the celebrated David and Journ6 theorem, see [7] and [13]. We remind the reader that any Calderbn-Zygmund operator automatically maps L ~ to B M O , so that T(1) and T*(1) make sense as functions in B M O . The last point is the bad news in this theorem. It means that there exist some operators T in s which are invertible in L2(lRn), but that their inverses are not in s In fact, they even can be noninvertible on LP(]R n) for p arbitrarily close to 2. This comes from the fact that the size constant C of the family (T~x)xei and the L 2 norm of T are not equivalent: the C may be arbitrarily large. This and related results (in particular, ratio I-~ positive results about stability under taking the inverse) are discussed in some detail in [17]. We mention it to the reader to appreciate the interest of the foregoing statements. The Lemari6 algebra admits another characterization, which shows how numerically interesting its elements can be. It is given in terms of matrices, and first requires some definitions. Definition 3.
If A, # E I, we define d(A, #) by
d ( A , p ) - - i n f {log 26 : 6 such that 6 Qx D Qt, )
if j(A) _ j(#), and by d(A,#) = d(#, A) if j(#) :> j(A). It is amusing to check that Lemma 1. d is a distance on I. Definition 4. We denote by it4 the set of all matrices M = (mx,t,)x,t, ei for which there exists C > 0 and e > 0 such that
k/ )k,# E l
]m~,t~ [ _~ 2~[j(~)-j(t~)[ 2-(n+~) d(X,~).
(2.3.1)
We can now state the Lemari6 characterization of s Theorem 3. A linear operator T belongs to s if and only if its matrix (< T ~ , ~ >)~,~el belongs to A4. The interest in matrices M E M comes from the fact that they can be well approximated by band-limited matrices. We say that M = (ff~)~,.) is
J. A ngeletti et al.
502
band-limited if there exists D > 0 such that ~x,z - 0 if d()~, #) > D. Let M E f14. Given any D > 0 we define MD, a truncated version of M, by
(MD)~,,
S mx,z 0
if d(,~, #)
_< D
otherwise.
Then, MD is band-limited by construction, and it can be shown that the estimates (2.3.1) imply
IIi-
MDI] <_ C D2 -~D.
Hence, M is, in the matrix norm, arbitrarily close to a band-limited matrix. Let us finally point out that, in view of numerical computations, the value of e in (2.3.1) is very important as the last estimate shows. If we go back to Definition 1, and denote by q, r, m, the envelope index, the regularity index, and the cancellation index of the family (T~.x).xeI, respectively, then we can take e = min(r0, q, r, m) (2.3.2) in (2.3.1). 2.4 V a g u e l e t t e s a n d elliptic o p e r a t o r s w i t h c o n s t a n t c o e m c i e n t s Let
P(D) = -a~cg~cO~ + ao be a partial differential operator with constant coefficients. Its symbol is, by definition, the function
a(w) = a ~ w ~ w ~ + ao , where w E lR~, so that we have
P ( D ) f ( x ) - (21r)- n Inn eiX'~a(w)fA(w) dw
V f e 8(lRn) .
We assume that the matrix (aaz)l
aj - 4j + ao 9
(2.4.1)
L e m m a 2. For each ~ E I, let O~ be defined by the equation
P(n)o~ = ~ . Then, the functions aj(~)O:~ form a family of vaguelettes, with envelope index q = ro - 1, regularity index r = ro + 1, and cancellation index m - ro - 2. Its size constant C satisfies the estimate C <_ C ( a a z ) ,
(2.4.2)
where C(aa~) depends on the constants in (H1) and (H2), on the chosen multiresolution ana/ysis, but not on ao. We will make frequent use of the following immediate corollary.
Analysis of Elliptic Operators
503
Corollary 1. If c~,/~ E { 1 , . . . , n } , the functions OaO~8~, A E I, form a family of vaguelettes, with envelope index q = ro - 1, regularity index r - ro - 1, and cancellation index m - ro - 2. Moreover, when j(A) - j, the function Oa OZ 8~ satisfies estimates (2.2.1-2.2.3) with C - C o
4j aj
m
,
for some constant Co independent of A, j, and ao. To prove Lemma 2, we consider G(x), the fundamental solution of the operator P(D). In dimension n > 3, it satisfies the classical estimates
]O~G(x)l
(2.4.3)
for some constant 7 > 0 and every multi-index a E INn. In dimensions 1 or 2, the usual modifications have to be done. We thus have
ajt?~ (x) - aj /R n G(x - y) ~ (y) dy.
(2.4.4)
We at first observe that we can assume k(A) = 0, since P(D) has constant coefficients. We then consider two cases : ao _> 4 j, the low frequency case, where aj is equivalent to ao, and ao _ 4J, the high frequency case, where aj is equivalent to 4j. In the first case, ao _ 4j, we simply write ~jl0~(~)l < cN
~ ao
!~ - yl ~ - ~
2in~2 (1 + 12Jyl) - N dy,
using the assumed properties of { ~ } and (2.4.3). This gives, after computation, a j l ~ ( X ) l ~_ CN 2 jn/2 (1 + [2Jxl) - g , where CN is independent of ao. We estimate the derivatives of a j ~ in the same way, by differentiating ~ in (2.4.4) up to the regularity order of the wavelets. In the second case, a0 _< 4j, we use the cancellation properties of the wavelets and integration by parts in (2.4.4). To this end we remark that each "mother wavelet" ~6 , ~ E {0, 1} ~ , ~ ~ 0, is the derivative of order ro + 1 of a nice, rapidly decreasing function a6 :
J. A ngeletti et al.
504
where lael = ro + 1. Using a partition of identity, we obtain
(2.4.5) leg ~
where supp ae,l C l + [-1, 1]n
Ilae,tlloo <_ CN (1 + Ill) - N for every N E IN.
(2.4.6) (2.4.7)
By scaling we obtain from (2.5.5) and (2.5.4)
ajOx (x) - aj 2 in~2
G ( x - y)[0a~(~)ae(~),~] ( 2 J y ) d y .
For x fixed, we apply direct estimates to the terms with I1- 2Jxl <_ 10 in the sum above, and we use integration by parts for the others. Hence, if [l- 2Jxl ) 10, we transform the latter into
crj2Jn/2fit" Oa~(~)G(x-
y) 2 -j(r~
ae(~),l(2Jy) d y ,
which by (2.4.3) and (2.4.6) is comparable to
2Jn/2 12ix -
fin+to-1 '
uniformly with respect to a0, A and 1. By summation, this gives the size estimate (2.2.1), with the envelope index r o - 1, as desired. The other estimates are obtained similarly. Lemma 2 has the following interpretation. Let A be the (unbounded) operator defined by A~;~ = 2 J ( ~ ) ~ . If s E IR, we define A8 by A'~;~ = 2 ' J ( ~ ) ~ . In particular, the operator ao + A2 is such that
(ao + A 2) ~x - (ao + 4 j(x)) ~x - a~(x)~x. Let T be the operator defined by
T ~ x = aj(x)Ox ,
(2.4.8)
Analysis of Elliptic Operators
505
and recall that the wavelet basis extends to a basis in Lp, 1 < p < oo, with
Ilfll~ ~ I1(~ I< f,~A > 12 I~x12)i/211~. By Lemma 2 and Theorem 1, T is continuous on L 2, on L p if 1 < p < oo, and other spaces. By Lemma 2, the operator P ( D ) -1 admits the factorization P ( D ) -1 - T(ao + A 2 ) - 1 . (2.4.9) The operator (a0 + A2) -1 is diagonal in the chosen wavelet basis and of order -2, associated in an obvious way to the symbol (a0 + 4J) -1 (viewed as a function from I to IR). Were the wavelets eigenvectors of P(D) -1, this symbol would have given the eigenvalue associated to ~ for each ,k. Of course, this is not the case, and P ( D ) -1 is diagonalized only through the Fourier transform, the eigenvalue associated to e ~ ' * being a(w) -1. We see in (2.4.9) that the price to pay in going from the Fourier to the wavelet transform is the operator T. Indeed, if we heuristically think of the spectrum of ~x as being contained in a corona of the type 1
C
2j _< Iwl
_ c
2j
(which is exact for the Meyer wavelet, see for instance [13]), then we understand the action of (ao + A2) -1 as the multiplication by some mean value of a(w) -1 on this corona. The action of the operator T as prescribed by (2.4.9) has the effect of the spreading in frequencies of ~x. However, the vaguelettes estimates on the family (aj O)~))~eI show that the operator T does not distort too much the spatial localization of the wavelets (estimate (2.2.1)) nor their spectral localization (estimates (2.2.2-3)). This is also the content of estimates (2.3.1), which, by Theorem 3, is fulfilled for the set of coefficients < aj(,)O,, ~ >, )~, # E I. The operator T has, moreover, an additional interesting property. L e m m a 3. T is invertible on L2(]Rn), and its inverse belongs to ~.. Proof." Define the functions T~ , ,k E I, by
P(D)*g~ - r~. It is easy to see that the family (a~lTX)Xel satisfies the vaguelettes estimates, with envelope index as large as we want, regularity index r0 - 2, and cancellation index r0 (or r0 + 2 if a0 - 0). Thus, defining the operator U in the Lemari~ algebra, by UltI/ik - - O - 2 1 T , k ,
J. A ngeletti et al.
506 we have the factorization formula
P(D)* = U(ao + A2).
(2.4.10)
Then, formulas (2.4.9-10) show that TU* = Id, which proves the lemma. m R e m a r k 1. All the preceding results extend straightforwardly to pseudodifferential operators with a positive symbol depending only on w. In particular, we can factorize ( - A ) s/2, if Isl < r0, as ( - A ) 8/2 = TsA s ,
(2.4.11)
and the operators T8 are in the Lemari~ algebra, as well as their inverses. Hence, we recover the well-known formulas
) 1/2 11(- 1 / /il2
~
4J l <
> I
,
(2.4.12)
where ~ stands for the equivalence between two norms. We analogously recover
ll(--A)s/2 fllp
(2.4.13)
"~ il As flip
when 1 < p < c~, since every operator in the Lemari~ algebra is continuous in all such L p spaces. 2.5 T h e s p a c e VMO Definition 5. (Sarason, [14]). The space VMO (vanishing mean oscillation) is the closure in B MO of the space of uniformly continuous and bounded functions. With this definition, VMO strictly contains the closure of C0(lRn). If a E B M O and r > 0, we set, following Sarason: M~(a)=
sup
Q:l(Q)<_e
(~Q~ /Q la(x) - mQ(a)[ 2 dx)
1/2
where l(Q) denotes the side length of the cube Q, and mQ(a) the mean value of a over Q. We also set
Mo(a) = lim M6(a). e-+O
In [14], Sarason has proved
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507
L e m m a 4. a E V M O if and only if a
6_.
B M O and Mo(a) - O.
Note that In Ix[ does not belong to VMO, nor does any nonconstant step function. On the other hand, I ln IxII1/2 or In [ln IxII belong to VMO, as well as any Riesz transform of any uniformly continuous and bounded function (see [14]). We will need the wavelet characterization of VMO, which is as follows. We define
(1
N~(a)-sup
~
O.l(Q)~_e
E
I < a , ~ > 12)1/2
x: QncQ
and N0(a) = lim Ne(a). ~--+0 L e m m a 5. a E V M O if and only if a E B M O and No(a) = O. The proof is left to the reader. The space V M O is in some sense the limit of the H61der spaces. We recall that a function a belongs to C a (IR~) (0 < a < 1) when it is bounded and satisfies
C > 0
V x, y E ]Rn
la(x) - a ( y ) l _< C I x - yi ~.
(2.5.1)
For such a function it is well known that
l< a, ~ , >1<_ C 2-jn/2-ja, whence N~(a) - O(~a), and of course No(a) = O. 2.6 V M O
and vaguelettes
We will often encounter a product of the type MaT, where Ma denotes the pointwise multiplication operator by a, which will be some bounded function, and T is an operator in L. IfO~=T~x, A E I , wewrite
ag~ = m x (a)9~ + (a -
m,x
(a))9~,
where mx (a) stands for the mean value of a over the cube Qx 9
m~(a) =
1
IQ I
/Q
In terms of operators, we have
M~T-P+R,
a(x) dx.
J. A ngeletti et al.
508 where P and R are defined by
P ~ x = mx(a)Ox ,
Rq~x = ( a - mx(a))Ox.
In view of estimate (2.2.1), P appears to be some rough version of Ma T, and R some rest. Of course, we have
IIPll <__Ilall~ IITII,
(2.6.1)
and at least in particular cases (for example if T - Id), this inequality may become an equality. We also have
Ilnll ~ 2 Ilall~ IITII,
(2.6.2)
but this time it is not a sharp inequality, as the following result shows. T h e o r e m 4' i) There exists C(T), a constant depending on T and on the wavelet basis, but not on the function a, such that
JlRJl ~ C(T)[[a][BMO.
(2.6.3)
ii) For every j _ 0, we have the more precise inequality
JlRTrJ-*Jl _< C(T)N2-j(a).
(2.6.4)
We recall that rJ-* is the canonical extension operator from V/• to L2(lRn). The constant C(T) in the inequalities above is not the L 2 norm of T, and is not equivalent to it, but is equivalent to the size constant of the family (Tq~x)Xel (see the comments after Theorem 2). Hence, the estimate (2.6.3) is not necessarily sharper than (2.6.2) regarding the numerical values. But it is sharp in that it describes how the operator R actually depends on a. It is easy to see that, when T = Id, [IRI[ is equivalent to [[aI[BMO , proving the sharpness of (2.6.3). Theorem 4 is close to the commutator estimate of Coifman, Rochberg and Weiss [6] which we can recover and extend with our method (see also [5] for a similar result). Indeed, using the orthogonality relations given by the multiresolution structure of the wavelet basis, we can prove a slightly strengthened version of this theorem, which immediately gives the L p estimates. T h e o r e m 4 ~. The operator R belongs to the algebra generated by the Calderbn-Zygmund operators. Consequently, it is continuous on LP(IR n) if 1 < p < c~. Moreover, estimates analogous to (2.6.3) and (2.6.4) hold:
J]RJJv <_ C(T,p) ]Ial]BMO ,
(2.6.5)
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Analysis of Elliptic Operators
[[nTr~*[Ip <_ C ( T , p ) N 2 - j ( a ) .
(2.6.6)
Notice that R does not belong to the Lemari~ algebra. In fact, there does not exist, at the present time, any description of the algebra generated by the Calderbn-Zygmund operators. We will not prove here Theorems 4 or 4', their proofs use tools and ideas which are beyond the scope of this chapter. Instead, we mention two corollaries. The first one is an extension of the Coifman-Rochberg-Weiss result. Corollary 2. If T is a Calder6n-Zygmund operator, the commutators [[... [IT, Ma]Ma] . . .]Ma] are all bounded operators on LP(~n), 1 <: p ,: c~, when a E B M O . The second corollary is immediate. Corollary 3. If a E V M O , then lim IiRTr]-* lip - 0
j-+c~
for every p E (1, oc).
w The VMO case In this part, we consider L = -div AV and M = -aa~OaOZ,
under hypotheses (H1) and (H2) with the additional assumption that V a,~ E {1,...,n}
aaz E V M O .
(H3)
We already noticed that L is continuous from W 1,p to W -1,p, and M from W 2'p to L p. We want to investigate the invertibility of L + ~ and M + ~, with the aim of proving Theorem A.
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3.1 T h e r e s o l v e n t o f M Let ff > 0 be fixed. For every A E I we define 0x by - m x ( a a ~ ) OaO~O,x + ~ Ox - ~x ,
(3.1.1)
where we recall that mx(aa~) is the mean value of aa~ on the cube Qx. We define the operator PC by P~ ~x - 0x,
(3.1.2)
and the operator S~ by the relation (M + ~)P~ = I d -
S~.
By Theorem 1, Lemma 2, and Corollary 1, P~ maps L p into W 2'p if 1 < p < oe. Hence, Sr is well defined on L p. T h e o r e m 5. For every p E (1, c~) we have
lim IISr
0.
This shows that P; is a parametrix, in some sense, of M + ~. Before proving this theorem, we state two corollaries. Corollary 4. /fp E (1, c~) and if ~ is large enough, then M + ~ is invertible from L p to W 2,p. Moreover, we have the formula (M + r
_ p~ ( I d - Sr - t ,
and the estimates [I(M + C:)-II[w~.,~L~ < C
II(M+C)-lllp <
C ~.
(3.1.3)
(3.1.4) (3.1.5)
This corollary is a consequence of Theorem 5, and of the estimates
IIP~IIw~.,~L~ ~ c,
(3.1.6)
C lIP;lip --- iTr, Iql
(3.1.7)
valid for every r > 0, which are themselves consequences of Theorem 1, Lemma 2, and Corollary 1. Obviously, (3.1.6) implies (3.1.4), and (3.1.7) implies (3.1.5) if ~ is large enough.
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511
Corollary 5. Let u E W 2'q and f E L q such that - a ~ O ~ O ~ u + a~O~u + aou = f ,
for some bounded coet~cients an, a E {0, 1 , . . . , n} and for some q E (1, cr If f E L p , q < p < oe, then u E W 2,v, and
Iiuilw=,, ~< c (IfUltq + Ilfllq + Ilflip),
(3.1.8)
where the constant C depends on p, q, and the coefficients, but not on f or U.
Proof: We begin by proving (3.1.8) for p for (3.1.4) and (3.1.5) to hold. We have u - (M + r
q. Let ( > 0 be large enough
( f _ a~O~u - aou + ~ u ) .
(3.1.9)
Since, by interpolation, we know that
II(M + we obtain
C ~)-1 ilwl,q_+Lq <~_ I~11/2'
C II~]lw,,~ < ,~,1/2 (lI/ll~ + Ilullw~,~ + r Ilullq) 9
Choosing i(I = 2C2, this gives the first estimate i]uliw1,q < C ([Ifilq -I-Iluilq).
(3.1.10)
Then, we apply (3.1.4) to (3.1.9)-(3.1.10), giving (3.1.6) in the case p Let {qk, 0 _ k _ 1 + 1} be the set defined by 1
1
=---q0 qk
k
n
ql+l=Oe
q.
n
if k < - - , q0 n if l < - - _ < l + l . q0
(3.1.11)
It suffices to prove that, provided f E L p, (3.1.8) is valid for p <_ qk+l as soon as it is valid for p <_ qk, k <_ 1. To this end, we write (3.1.7) with a real ( such that (3.1.2) holds when p <__qk+l. By hypothesis, u E W 2'qk which implies that u E W l'qk+l, by (3.1.9) and Sobolev embeddings. It is therefore obvious that (3.1.8) holds for p _< qk+l, which finishes the proof. I
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J. Angeletti et al.
R e m a r k 2. If u is compactly supported, we recover the interior estimates of Chiarenza, Frasca, and Longo:
Ilullw~,~ < c (llfllp + Ilullq). We now prove Theorem 5. By definition of Sr we have for every ~ E I the representation
s~
= [~
- m ~ ( ~ ) ] o~o~e~.
We split the set I into two parts: jo being an integer to be chosen, we consider first those )~ for which j(A) _< jo. By Corollary 1, the functions (O,~O~8~,)j(~,)<jo satisfy vaguelettes estimates, with size constant 4Jo C~o < c 4Jo + ~ 9
Using Theorem 1 and the boundeness of the matrix A, we obtain
4r IIS~r~llp <_ C(p)IIA]I~ Mo + ~ "
(3.1.12)
When j(A) > jo, we use Theorem 4'. Indeed, we know by Corollary 1 that the functions (0a0~8~)j(~)>jo satisfy vaguelettes estimates with a size constant uniformly bounded with respect to jo. Hence, inequality (2.6.6) gives us
IlSc=~*II ~ _< C(p) N2-~o(A),
where
N~-,o (A) - ~
(3.1.13)
N~-~o (~,).
Summing up (3.1.12) and (3.1.13) we obtain the estimate IOSr
_ C(p)
We now choose jo = jo(r
(
IIAlloo Mo + r + N2-~o(A)
)
.
such that 1 4 -j~ g2-Jo (A)_> ~ ,
which is possible provided (: is large enough, in such a way that lim jo (r = c~. By construction we have
liSr
< c(p) (1 + liAlioo) Y~-~o(,)(A),
which proves Theorem 5.
(3.1.14)
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Analysis of Elliptic Operators 3.2 T h e r e s o l v e n t of L
3.2.1 S o b o l e v e s t i m a t e s We begin by proving the invertibility results that we announced in Theorem A. They rely on a study of the resolvent of L very similar to the previous one. We again consider the functions 0h and the operator PC defined by (3.1.1) and (3.1.2), but we now view PC as a continuous operator from W -I'p to W I'p , 1 < p < co. This is possible thanks to Theorem 1, Lemma 2 and Corollary 1 once more. We then define the operator R; by the relation (L + ~)P~ = I d - R~ , where Ri is acting (and continuous) on W -I'p. The analog of Theorem 5 is the following. T h e o r e m 6. For every p E (1, co) we have lim [[Rt][w-l,p = 0 .
~--+~
(3.2.1)
Proof: We compute
For each a E { 1 , . . . , n}, let Ta be the operator defined by =
[a.z
-
(where we dropped the subscript r Then, (3.2.1) will be proved if we show that, for every integer j0 and for each a, ][Tal[Lp,W-I,p ~_ C(p)
4j ~
IIA[[~ 4Jo + [r + N2-~o (A)] ,
(3.2.2)
following the reasoning after (3.1.14). It is a consequence of Remark 1, in particular, of relation (2.4.13), that the operator Id + A is an isomorphism from L p to W -I'p , 1 < p < co. Thus, (3.2.2) is equivalent to 4j ~
[[T~(Id + A)[lp _< C(p) [[A[I~ 4Jo + [r + N2-J0 (A)] .
We have To,(Id + A) ~ , --[a~z - m x ( a ~ z ) ] (1 + 2j('x)) 0f~O~, .
(3.2.3)
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J. Angeletti et al.
Since, by Lemma 2, the families ((1+2J(~))0/30~)~i, fl E ( 1 , . . . , n ) , satisfy the same vaguelettes estimates as the families (OaOf3Ox)Xel, with the same behaviour of the size constants, we can proceed to prove (3.2.3) exactly as we did for (3.1.14). The details are left to the reader, m As in the case of M, Theorem 6 implies that for every p E (1, c~), the operator (L + () is invertible from W -I'p to W I'p if ( is large enough. We can even prove a better result which is a slight improvement and an extension to the V M O case of a similar result due to Taylor [15]. T h e o r e m 7. For every ( > 0 and every p E (1, oc), the operator L + ( is invertible from W -1,v to W I'p. Proof: The case p - 2 follows from Lax-Milgram lemma. For the other values of p, we argue first with a weak version of the result to be proved. L e m m a 6. For every ( > 0, (L + ~)-1 extends to a continuous operator on L p 1 < p < oc This lemma is well known. It can be proved, for example, by writing (L + ~ ) - 1 _
e.-tL e - t ( dr,
~0~176
and noticing that the semigroup e -tL is uniformly bounded on L 1 and L ~ , by Aronson estimates [1]. Now let ( > 0 and p E (1, c~) be fixed, and let us choose ~ so large as to ensure the invertibility of (L + ~) from W -1,p to W 1,p. By the resolvent identity, we have (L + ~ ) - 1 _ (L + ( ) - 1 _ (( _ () (L + ()-1 (L + ( ) - 1 = (L + ~)-1 _ (r _ ~)(L + ~)-2 + (( _ ~)2 (L + ~ ) - I ( L + ()-1 (L + ~ ) - 1 . By Lemma 6 and the definition of ~, each term on the right maps W -1,p in W I'p. m At this point, we have proven Theorem A. We however would like to explicitly construct (L + ()-1 for every ( > 0, in the same spirit as we constructed (M + ()-1 for large (. Here also we have a formula analog to (3.1.3), that comes from (3.2.1) and Theorem 6, and reads (L + ~ ) - 1 _ P ; ( I d -
R;) -1
(3.2.4)
if ( is large enough. But we cannot reach the small values of ( with this method. We are thus led to a modification of our argument, and this is the reason why we now need some results about the Galerkin operators associated to (L + ().
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Analysis of Elliptic Operators 3.2.2 Galerkin
operators
To begin with, we at first consider L + ~ as acting on W 1'2 9 If jo is an integer, we define (3.2.5) (L + ~)jo - 7rio ( i + ~) 7rio , the Galerkin operator associated to L + ff on Vjo. By the Lax-Milgram lemma, this operator is invertible on Vjo (with the L 2 topology), and we define Fjo,; = 7rio* [(L + ~)jo] -1 7t'jo. (3.2.6) This is the natural extension to all of L 2 of [(L + ~)jo] -1. L e m m a 7. r'jo,( is continuous from W -1'2 to W 1'2, and uniformly bounded with respect to jo, satisfying
Ilrjo,r
1
<__min(5, if)
(3.2.7)
(we recall that 5 has been defined in (H2)). Proof: The essence of Lemma 7 is in the uniformity with respect to j0. It is indeed obvious that Fjo,; maps W -1,2 into W 1,2, since the topologies induced on Vj by these spaces are equivalent to the canonical one. But this cheap argument does not give the uniform estimate (3.2.7). To prove that, we consider f E Vjo, and write the definition of F jo,~. This gives, for every g C Vjo,
f ~ AV(rjo,CY). vg + r fR~ (rio,Of) y - fro y y"
(a.2.8)
Taking g - Fjo,r f, we obtain
5 IlV(rjo,r
+ ff Ilrjo,r
~ Ilfllw-~,= Ilrjo,r
(3.2.9)
It is now easy to deduce (3.2.7) from (3.2.9). m Lemma 7 is classical and well generalisation to W I'p spaces holds heavy a notation, we extend (3.2.5) the L p topology, and keep the same
known. A nontrivial fact is that its true. In order to state it without too and (3.2.6) when Vjo is equipped with notation.
T h e o r e m 8. For every p E (1, c~) and every ~ > O, the operators Fjo,i , acting from W -1,p to W I'p, are uniformly bounded with respect to jo. This theorem will be proved in Section 4, independently of what follows now. We quote here a simple consequence of it.
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J. Angeletti et al.
L e m m a 8. For every p E (1, c~) and every ~ > O, there exists a constant C(p, ~) such that V jo E N
_< C(p, r
II(L + r162
(3.2.10)
Let us explain the meaning of this inequality. By construction we have 7rjo (L + r Fjo,r = 7rio.
(3.2.11)
Hence inequality (3.2.10) is about the operator 7r~0 (L + r Fjo,;. This operator is well known in numerical analysis, since it measures the residual associated to the Galerkin scheme. Indeed, if f E W -I'p (classically p - 2 is only considered, but we may here consider other values of p) and if U E W I'p are such that (L + ~) u = f, the Galerkin method in Vjo consists of approximating u by Ujo = Fjo,r f . The associated residual is then f - (L + ~) Ujo = 7rfof - 7rfo (L + ~) Ujo , because of (3.2.11). We thus see that inequality (3.2.10) means that this residue is controlled in the W -1,p norm. We will see later that it eventually tends to zero when jo tends to infinity, but not uniformly with respect to
f. 3.2.3 C o n s t r u c t i o n of the resolvent of L for e v e r y ~ > 0 We will use Fjo,r as a first approximation of (L + r on Vjo , following a general scheme described in [18]. The preceding discussion shows why it is natural, and why it is neither too bad nor very accurate, in operator norm. On the orthogonal complement Vjo3-, we will use P; lr~0, the restriction of our parametrix P~ defined in (3.1.1) and (3.1.2). By (3.2.1) and the proof of Theorem 6, we have ( L + ( ) pc rr30 .3_ -- 7r~ - R~Trjo 3_
(3.2.12)
and [IRCr~o [[w-~,, <_ C(p, r
(A).
(3.2.13)
Our candidate to approximate (L + ~)-1 is then the sum rjo,r + P;Tr~o. We thereby define the operator Ujo,; , continuous on W -I'p, by the relation (n + r162
+ P~lr~o) = I d -
Ujo,~.
(3.2.14)
By (3.2.10), (3.2.12), and (3.2.13) we have
IIUjo,r
< C(p,
Explicit calculations with L - - A show that this inequality cannot be improved. However, we have
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517
T h e o r e m 9. For every p E (1, oc) and ~ > 0 [IU]o,r
<_ C (p,r
(A).
This proves that ( I d - Ujo,r -1 is invertible when jo is sufficiently large, and that in such a case we have • ) ( I d - Ujo,; ) - 1 . (n + ~ ) - 1 _ (Fjo,; + R Crjo
(3.2.15)
In this formula, ( I d - Ujo,;) -1 is a continuous operator on W -l'p, and (Fjo,r + P(7@o) as well as (L + r are continuous from W -1,p to W I'p. The proof of Theorem 9 is not difficult. We compute Ujo,( , and find that Ujo,i = -Trio (L + r + nr , (3.2.16) using (3.2.11), (3.2.12), and (3.2.14). Since, by definition, Fjo,r have Ujo, 2
-
-
•jo (L + r
r •
-
• (L + ~)Fjo,; +
- 0 , we 2.
Theorem 9 now follows directlyly from estimates (3.2.10) and (3.2.13). w The HSlderian case
We now assume that the coefficients aa~ are more regular, and examine the improvements of the preceding results that can be deduced. We begin with the assumptions we want to make on the coefficients, which involve some functional spaces close to the usual HSlder spaces, and then we return to our partial differential operators. 4.1 T h e s p a c e s B M O s If 0 < s < 1, we denote by B M O ~ the Triebel-Lizorkin space/~;2(IRn), which, as we will see, may be defined in the following way. Definition 6.
A locally square-integrable function a belongs to B M O ~ if there exists a constant C such that, for every cube Q,
f[h
I
/Q [a(x + h) -a(x)[2 ihln+2 s dx dh < C21QI.
(4.1.1)
The best constant C above defines the norm in B M O s. To give an example, note that the function x ,~ ; Ix] 8 belongs to B M O s. As usual, B M O s is in fact a space of classes of modulo constant functions. We will not insist on this point anymore. In this section we will assume the coefficients aaz to be in one of these spaces, for this hypothesis gives the most precise statements. However, since these spaces are probably less familiar than HSlder spaces, we think a discussion of some of their properties deserves a few pages. To begin with, let us compare B M O 8 to the HSlder spaces.
J. A ngeletti et al.
518
Proposition 1. If 0 < s < s' < 1, we have i) C 8' C L~176B M O 8, ii) B M O s C C8 (the homogeneous H51der space). Proof: Only a proof of the second statement is needed. We start with the observation that, if Q1 and Q2 are two cubes such that
l(Q1) = l(Q:) = d(Q1, Q:), then ImQ1 (a) - mQ2 (a) [ <_/(Q2) s
(4.1.2)
holds for a e B M O s (recall that mQ, (a) is the mean value of a on Qi). To prove this, it suffices to write (4.1.1) for the smallest cube Q containing Q1 and Q2, and to restrict the integral to the points (x,x + h) such that x E Q1 and x + h E Q2. It gives
11/Q/Q [Qll tQ21
1
2 ]a(y) - a(x)l 2 dydx <_ C/(Q2) 2",
which implies (4.1.2). We then extend (4.1.2) to the case where Q1 and Q2 are adjacent: we cut each Q i into 2 n smaller cubes canonically, and put the two partitions into correspondence by translation. We write (4.1.2) for each couple of small cubes, and find the desired estimate by summation. Hence, (4.1.2) is also valid when Q1 c Q2 and l(Q2) <_ 2/(Q1). An iteration argument finally shows that (4.1.2) holds whenever Q1 c Q2. Now let x, y be two fixed points in IRa, and let Q', Q" be two cubes such that e Q', y e Q", l(Q') = l(Q") < I~ - yl. There exists a bigger cube Q satisfying
Q' c Q, Q" c Q, l(Q) < C I x - y[ for some absolute constant C. We thus have, by (4.1.2),
ImQ, (~) - mQ,, (~)1 _< Imq, (~) - mQ(~)! + lmQ(~) - mQ,, (~)1 _< C I~ - Yl ". If x and y are Lebesgue points of a, we can conclude that
I ~ ( z ) - ~(y)l < c Ix,- yl ~, which finishes the proof,
m
Let us now consider two characterizations of B M O s, including the wavelet one. In order to state them, we denote by (I) a C ~176 function supported in [-1, 1]n and equal to 1 on [_1, 89 If Q is a cube centered at XQ and of side length l(Q), then CQ denotes the function (I)Q(X) = (~(x-xq~(Q)).
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T h e o r e m 10. A function a belongs to B M O s if and only if one of the two following conditions is fulfilled: (i) for every cube Q the function (a - aQ)~Q belongs to ff-2,~, and
I1( -
Q) QII w,, -
c IQI 1/2,
(4.1.3)
where aQ is so chosen that
R
n
(a -- aQ)(~Q
-- 0 ;
(ii) for every cube Q,
E
l < a, ~I'x> 12 4j(x)s <_C21QI .
(4.1.4)
QxCQ
Moreover, the best constants C in (4.1.3) or (4.1.4) define equivalent norms on B M O ~. Proof: We begin by some preliminary observations which we will use frequently. Both (4.1.3) and (4.1.4) imply a E (~s. To prove this for (4.1.3), argue as in Proposition 1, and for (4.1.4), just use the wavelet characterization of CS, which writes l < a , ~ x > 1 < C 2 -j(x)(~+s).
(4.1.5)
It is known that if f is measurable,
Ill
IIw=,,
- a~
/.L ~
~
If(x + h) f(x)l 2 ihl~§ ~ dxdh.
(4.1.6)
We will use, because it simplifies the proof, an orthonormal basis of compactly supported wavelets of regularity at least equal to 1, denoted by (G~,):~eI. The associated scaling function is F. The reader need not be acquainted with the properties of F and the wavelets G x. It will suffice for the reader to keep in mind that supp F is compact, F is at least Lipschitz, F defines a partition of unity: V x E lRn
E
F(x-
k) - l,
kEZ n
the mean value of F is 1, and F is orthogonal to the wavelets at scales smaller than 1: V A E I, j(A) > 0, < F, Gx > = 0. (4.1.7)
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520
Finally, we note that condition (4.1.1) is equivalent to
flhl
_
/i~
~
la(x + h ) - ~(~)1 ~ I~Q(~)I ~ dxdh <_C IQI. h}n+2~
(4.1.8)
We now prove that belonging to BMO 8 is equivalent to (4.1.3). Let a E B MO s, Q be a cube, and assume that aQ = O. Then we have
/R /,~ la(x + h),~Q(X + h) -a(x)OQ(X)l 2 dxdh .
Ihl~+2~
[a(x + h)OQ(x + h ) - a(x)OQ(x)l 2
R
I>t(Q)
"
_
R
ihln+28
I~(~ + h)! 2 I~Q(~ + h)
-
dxdh
~Q(~)I ~ dxdh
Ihi n+28
~
+2 fih I
[a(x + h)! +
I~(~)1-< CIl~llco l(Q) ~ ,
when x E supp ~Q and [hi _ l(Q). Hence, the first and the second terms in the above upper estimate are dominated by C[[a[[2,[Qi. Thus, by (4.1.6) and (4.1.8), we see that a fulfills condition (4.1.3). The converse implication is proved similarly. We turn to the proof of the equivalence between (4.1.3) and (4.1.4). It is not difficult in principle, though several details have to be checked. The first is the following. L e m m a 9. Condition (4.1.3) is equivalent to ll(~
-
~Q)FQliw~.. <_ CIQI ~/~
(4.1.9)
for every dyadic cube Q, where ~Q is given by
Let us assume that (4.1.3) holds, and let Q be a dyadic cube. There exists a constant C such that
(a
-
~Q)FQ
-
(a
-
~Q)FQCQ,
521
Analysis of Elliptic Operators with Q' - CQ. Thus we have (a - ~dQ) FQ - (a - aQ,)OQ, FQ + (aQ, - "~Q)FQOQ,.
(4.1.10)
Since a E (~s, we have
II(a- aQ')'~Q, Iloo _ C Z(O) s, and by construction we have llFQllw~,. - C l(Q) -'
IQ1112.
Together with (4.1.3), the two last relations imply
II(a - ~Q')~Q'FQIIw~,. ~ CIQI 1/~.
(4.1.11)
The second term in the right-hand side of (4.1.10) is easy: we have laQ, - ~dQI <_ C l(Q) 8 and
IIFQCQ'llw~,o -- C
t(Q) -~ IQI~/2;
hence, with (4.1.10) and (4.1.11), we obtain
II(a- a~)FQIIw~,o <_ C IQI1/2. Let us conversely assume (4.1.9), and let Q be a cube. There exists a finite number of dyadic cubes Qi, 1 <_ i <__io, with io independent of Q, such that io
~
FQ, (x) - 1
(4.1.12)
i=1
when x E supp CQ. We then write io
(a - aQ)~Q - E
io
(a - ~Q,)FQ, ~Q + E
i'-i
(aQ, - aQ)FQ, ~Q,
i--1
and argue as before. This gives (4.1.3) in a similar way. L e m m a 10. Let a be some measurable function. If a fulfills (4.1.4) and if (T~)~eI iS a family of vaguelettes, with envelope, regularity, and cancellation indices all larger than 1, then there exists a constant C such that
I< a, T~ > Q~cQ
12 4y(~)~ ~ CIQI
(4.1.13)
J. Angeletti et al.
522
for every cube Q. This implies that condition (4.1.4) is equivalent to
[ < a, Gx > [2 4j(x)" _
E
(4.1.14)
Q~CQ
for every cube Q.
To prove (4.1.13), we use the operator T such that T@x = T~ for all 6 I. Then T belongs to the Lemari4 algebra, and by (2.3.2), the index in estimates (2.3.1) on the matrix elements < T@~, ~x > is greater than 1. If a fulfills (4.1.4), we have ASa 6 BMO, by definition of A* and because B M O is characterized by (4.1.4) with s = 0. Moreover, we have .
(4.1.15)
AST*a = (A*T*A-S)(A*a) .
(4.1.16)
=
We write Now, the operator A - ' T A " belongs to s because its matrix elements are equal to
2 (j(~)-j('x))s < T ~ , , ~ , >,
and satisfy (2.3.1) with index ~ - s > 0. Theorem 3 implies A-STA" 6 s Since s is stable under taking the adjoint, AST*A -* also belongs to s Furthermore, since the operators in s are continuous on BMO, we have
AST*a 6 B M O by (4.1.16), which is equivalent to T*a 6 B M O ' . By (4.1.15), this shows (4.1.13). The equivalence between (4.1.4) and (4.1.14) is just a consequence of the preceding considerations, and is left to the reader. By Lemmas 9 and 10, it remains to prove that (4.1.9) and (4.1.14) are equivalent. Let a fulfilling (4.1.9), and let Q be a dyadic cube. Using (2.4.13), we have
I< (a- Q)FQ,
> I: 4
<_ CIQI.
(4.1.17)
,k6I
If Q = Qxo for some A0 6 I, and if j(1) < FQ , G~ > = 0. Hence we have
_> j(A0), we know that
I< a, FQG~ > [2 4j(~)* <_ CIQI, E j(~,)>jo
Analysis of Elliptic Operators
523
where jo = j(Ao). We now use again (4.1.12). The preceding inequality is valid for each Qi as well. By summation, it gives (4.1.14) as desired. Let us conversely assume that (4.1.14) holds, and let Q be the cube Qxo once more. We have to prove that (4.1.17) holds. If j(A) < jo, we use the fact that
/rtn (a - "SQ) FQ = O, and write
L< ( -aQ)FQ,a >l= C
j~~"tr(a(x) t,
-
"SQ)FQ,(x)[G~, (x)
-
a:~(Ao)) dx
~supp FQ
l(Q)S2Jn/22Jl(Q) dx - C2J(~+I)I(Q)I+~IQ I.
This inequality only holds if supp FQ N supp G x ~ 0, of course. Since, at each fixed level j(A), there are only a finite number of such G~, independently of j0 and j(A), we have
I< ( a - "SQ)FQ, Gx > 12 4j(x)8 j(A)<jo
~_ C
~
2J(n+2)4JSl(Q):(l+S)[Q[2~_C [QI.
(4.1.18)
j<jo
If now j (A) >_ j0, we use the orthogonality property
=O, which implies that the family (FQG)~)j(A)>_jo satisfies the vaguelettes estimates, with large envelope index, and regularity and cancellation indexes at least equal to 1. Hence, statement (4.1.13) in Lemma 10 implies
Z
I< a, FQG~, > 12 4j(x)~ <_ C IQt-
Q),CQ
With (4.1.18), this gives (4.1.17). Theorem 10 is completely proved, Using Remark 1, Theorem 10 admits the following corollary. Corollary 6. For 0 < s < 1, we have
BMO 8 -(-A)
-8/2 (BMO) - A-~(BMO) .
m
524
j. Angeletti et al.
This means that a E B M O s if and only if (-A)S/2a E B M O or A s a E B M O , and that [i(--A)S/2a[[BMO or [[ As a[[BMO define equivalent norms on B M O ~.
R e m a r k 3. Our definition of B M O s is not the standard one. To see that it is equivalent to the definition given in [19], just use Theorem 10 and the results of Frazier and J awerth [9]. Remark
4. In Lemma 9, we can choose F-
1[0,11-
1 This amounts to say that the Haar system provides an when s < ~. 1 unconditional basis of B M O s , for the weak-, topology, if s < 7" 4.2 T h e c o m m u t a t o r e s t i m a t e
One of the most celebrated results of Calderbn is his first commutator theorem, which states that whenever a is Lipschitz, the operator [x/:-A , Ma] is bounded on L 2. Since it has a Calderbn-Zygmund kernel, it is thus also bounded on L p, 1 < p < c~. The spaces B M O s are the good spaces to extend Calderbn's result. T h e o r e m 11. Let 0 < s < 1 and a E C s. Then, the c o m m u t a t o r s [ ( - A ) s/2, M~] and [As, Ma] are bounded on Lp, 1 < p < oo, if and only if a E B M O s.
We will briefly show the proof for [As, Ma] only, the treatment for [ ( - A ) s/2, Ma] being essentially the same. For each )k E I, we compute the image of ~x 9 [AS, Ma] ~
= A S ( a ~ ) - a As ( ~ )
=
(4.2.1) A
Let T and U be the two operators defined by T~
- (a- m),(a))2J(~)s~,
U~), - AS((a - m ~ ( a ) ) g ~ ) .
The operator T is the easier to analyse. We first notice that, since a E CS, the functions T ~ x , A E I, satisfy estimates (2.2.1-2.2.3), with large envelope index q, and regularity index r - ~ for example. The cancellation property is not fulfilled, nevertheless, since T~
= < a , ~ x > 2j('x)s
Analysis of Elliptic Operators
525
But it is apparent that a E B M O 8 if and only if the scalars fRn T ~ , I, satisfy the so-called Carleson condition, which reads
)~ E
2
/R n T ~
_ C IQI,
(4.2.2)
Q~cQ
for every cube Q. Hence, T is continuous on L p when a E B M O ~, by virtue of the following lemma. L e m m a 11. Let T be an operator such that the family (T@x)~eI satisfies the estimates (2.2.1-3). Then the kernel of T is of Calderbn- Zygmund type, and T itself is Calderbn-Zygmund (i.e. bounded on L 2) if and only if condition (4.2.2) is fultilled. This lemma is classical in Calderbn-Zygmund theory. It is essentially equivalent to the David and Journ6 theorem [7]. The reader is referred to [16] for further references. We now know that when a E B M O ~, T is bounded on L 2. By Calderbn-Zygmund theory, it extends to L p spaces when 1 < p < c~, and by a simple argument to the operator U also. Indeed, we will prove that U = T*, (4.2.3) provided a is real-valued, which we can assume. Equation (4.2.3) is a consequence of the orthogonality relation between wavelets, and the diagonal form of A (in the case of the commutator [ ( - A ) ~/2, Ma], this part of the argument has to be slightly modified). We have
< U~,~
> =< (a-m~(a))~,AS~ =
2
<
=
2
<
=< 9~,T9,
-
> >
-
m.
>
>,
thus proving (4.2.3) and the "if' part of Theorem 11. The converse part starts again with (4.2.1), which can be written as [As, Ma] : T* - T, always assuming a to be real-valued. By Lemma 11 we know that [A~, Ma] has a Calderbn-Zygmund kernel. We compute the image of 1 (see [7] for the precise meaning of this) which leads to
[AS,Ma](1) -- AS(a). If [A~, Ma] is continuous on L 2, it must map L ~ to B M O . then implies that a E B M O ~.
Corollary 6
526
J. A ngeletti et al.
4.3 T h e r e s o l v e n t o f M
Let us now come back to our operators M and L, and see what happens. We use again the notation of Subsection 3.1. T h e o r e m 12. If the entries of A are in B M O ' , for some s E (0, 1), then Sr is continuous on W 8'p if 1 < p < oc, with lim
liS;liw,.p = O.
(~oo
Corollary 7. W 2+s''p
(4.3.1)
If ~ is large enough, M + ~ is invertible from W s''p to
0 < 81 < 8 < 1, 1 < p < oc
Let us prove Theorem 12. Since we already know that lim
IISr
~--~oo
= O,
it will be enough to prove that S; is continuous on the homogeneous space IU ~,p, with lim ][S~[[r162 = 0. ~-+oo
We have seen (Remark 1) that A8 is an isomorphism from IU s,p to L p. It is thus equivalent to prove that ASScA -s is bounded on L p, with lim
{--+oo
[[ A8 S~ A -s lip = 0.
We now compute AsS; A-" ~x, for each A E I. We recall that -
=
which gives As& A - ' ~
-- AS{[aaf~- m~(aaf~)] 2 -j()')` aa af~ e~} = [As, aa~] 2-J(x)saaazox +
-
(4.3.2)
^'
The first term in the right-hand side of (4.3.2) is treated as follows. By Corollary 1, the functions 2-Js0a0~0x, when j(A) - j, satisfy vaguelettes estimates with size constant Cj, where 2J(2-s) C j < _ C 22j + r < _ C r Hence, the operator T1, defined by Tl~.X = [As, aa~](2-J(~)sOaOZ~x), is continuous on L p, by Theorems 1 and 11, with [[TI[[p <_ C r
(4.3.3)
To treat the second term, in the right-hand side of (4.3.2), we will use the following.
Analysis of Elliptic Operators
527
L e m m a 12. For every a , # e { 1 , . . . , n } the family (AS(2-J(x)sOaO~Ox)) satisfies the vaguelettes estimates, with a size constant Cj, when j(A) = j, satisifying 4J c . It follows from the vaguelettes estimates given by Corollary 1 that 4J(~)
Im ,,.I < C 4J(") -b ( 2 ~ t#(,x)-J(~)l 2-(n+e)d(),,~) for some e >_ 1 and for every A,p (see for example [11] or [13] for the calculations leading to such an estimate). Since we have
< As2-J(Z)so~oBO~, ~
> -- 2 (j(z)-j(~))s mz,,,
we obtain
I< As2-J(Z)so~o~O~,~z >I<__C
4#(,) ~ 4J(,) +
2 ~ I#(z)-J (~)12-(n+~- s)d(~,,l~)
By Theorem 3, this last estimate gives Lemma 12. If T2 is the operator defined by
T2 ~
- (aaz - m z ( a a # ) ) A s 2-J(z)sO~O~Ox,
we see that, thanks to the previous lemma, we can estimate its L p norm exactly as we did for Sr in the proof of Theorem 5. Moreover, since we know here that N2-~ (A) - O(2-Js), j -~ oc, inequality (3.1.14) gives us
IIT IIp < c r
(4.3.4)
The relation (4.3.2), meaning that AsSf A-S - TI + T2,
together with inequalities (4.3.3) and (4.4.4) imply Theorem 12, with the estimate
ll&llw,, valid for large (.
___c
528
J. Angeletti et al.
4.4 T h e r e s o l v e n t of L We only quote here the results that can be obtained along the lines we have developed in subsections 2.2 and 3.2. T h e o r e m 13. Let 0 < s < 1. When the entries of A are in B M O s, the operator (L+~) -1 is continuous from W -l+s''p to W l+s''p if ~ > O, [s'] <_ s and 1 < p < co. Moreover, formula (3.2.4) is valid on these spaces for large enough (depending on the space), and formula (3.2.15) is valid for each ~ > O.
Theorem 13 and Corollary 7 together give Theorem B. However, an important step in proving our results on the operator L has not yet been explained, namely the study of the Galerkin operators and Theorem C. This is the subject of the last section. w
T h e G a l e r k i n a p p r o x i m a t i o n of t h e r e s o l v e n t of L
5.1 T h e c o n v e r g e n c e r e s u l t s
Let us now consider again the Galerkin operators (L + ~)j = 7rj(L + r
when ~ > 0. Recalling the notation of Subsection 3.2.2, we have Fj,r - 7r~{(L + ~)j}-lTrj, and, if f is some test function, u and uj are the solutions of the equations (L + r
(L + ()u = f ,
= 7rjf .
This is equivalent to u=(L+~)-lf
,
uj = rj,r f .
From the Lax-Milgram lemma, we know that lim [[u- uj[Iwx,2 = 0.
j--+oo
(5.1.1)
We address here the problem of describing the various topologies for which the analog of (5.1.1) holds. Keep in mind that Vjo will be equipped with various topologies in the sequel, and accordingly completed in some cases, though we will always keep the same notation. When we will need to specify the topology, we will denote by (Vj, E) the space Vj equipped with the topology induced by the space E. We will prove the following theorems, thus recovering Theorem C.
Analysis of Elliptic Operators
529
T h e o r e m 14. /f the coefficients aaz are in V M O , then lim
j--+oa
!1- - ujllw,,, = 0
(5.1.2)
for every p E (1, oo ) . T h e o r e m 15. If the coefficients a~z are in B M O 8, for some s E (0, 1), then lira [ [ u - ujllw~+o,, = 0 (5.1.3) j-+oo
for every p E (1, oe).
Both are consequences of the following theorem, which is a stronger version of Theorem 8. T h e o r e m 16. I f a ~ E V M O , the operators Fj,; are uniformly bounded from W - 1,p t o W I ' p , 1 < p < oo. The same is true from W - lq-s,p t o W I+*,p i f a ~ z E B M O ~, 0 < s < 1. As we pointed out before, the uniform boundedness from W -1,2 to W 1'2 is the only easy case. Before proving the others, let us show how we deduce Theorems 14 and 15. We begin with the V M O case, and a fixed exponent p in (1, oe). We claim that there exists a constant C such that
V j e IN" V v e Vj
Ilvllw,,,, _< c II~,(L + Ovllw-,,,,.
(5.1.4)
Indeed, we may write v - Fj,(w, w 6 Vj, because Fj,( is invertible on Vj. Since, by definition, we have
=j(L + r162
-w,
inequality (5.1.4) is equivalent to V j e IN V w e Vj
ttrj,r
_< c Ilwilw-,,,,
which is exactly the content of Theorem 16. Let us now prove (5.1.2). We have II- - - j l l ~ , . ,
< II~-ltw,.,
+ II~j- - u j l l w , . , .
Inequality (5.1.4) implies
I[lrju - uj[Iw~,~ <_ C 117rj(L + r
- uj)llw-~,~
< C ll~j(n + ~)=ju - = j f l I w - , , ,
<_ C I[Trj(L + ~)TrJ-uI[w-l.~.
J. A ngeletti et al.
530 Hence we finally have
- u llw,,, <_ c II
J-ullw,,,,
and (5.1.2) is proved. The relation (5.1.3) is proved along the same lines. Let us remark that this last inequality also gives the convergence rate, as in the classical case p - 2. We now have to prove Theorem 16, to which the rest of this section is devoted. The proof follows the same strategy as for Theorem 7 : we will construct a parametrix of (L + ~)j, from which we will prove the desired estimates when r is large enough. Then, we will extend them to all ( > 0. The main idea here is to do all the required calculations in the space itself, equipped with suitable topologies. Thus, all the operators will be replaced by their Galerkin approximations in Vj, and all the functions to be constructed will live in Vj. 5.2
S t u d y of Fj,i for large r and V M O coefficients
If A C I and j(A) <_ j - 1, we define Oj,x E Vj by
7rj (L), + {)Oj,), = ~ , ,
(5.2.1)
and the operator Pj,;, on Vj by
Pj,r (,I, x) = Oj,~. They are Galerkin analogs of the (0x)Xel and P~ we used before, and play the same role. In particular, we will prove L e m m a 13. (i) The families (4J()~)Oj,~)j()~)<_j_1 and (O~O~Oj,x)j(x)<j_l satisfy vaguelettes estimates, with regularity index > 1 and size constants Cj, when j(A) = j, controlled by
Cj(~) <_ C
4J(),) 4j(~) + t:"
(ii) The operators Pj,i are uniformly bounded (with respect to j) from (Vj, W -I'p) to (Vj, WI'P), 1 < p < oc. Part (ii) is a consequence of part (i), using Theorem 1 and Remark 1. Part (i) is the new fact to be proved. Before explaining it, let us describe the rest of the construction, when r is large. We define the operator Rj,; on Vj by
(L + ~)jPj,r = Idyj
-
Rj,r
By Lemma 13 (ii), Rj,r is uniformly continuous on (Vj, W-I,P).
(5.2.2)
Analysis of Elliptic Operators
531
L e m m a 14. For every p E (1, oc), we have lim
(--~oo
llRj,r II
= 0,
uniformly in j. This lemma is proved like Theorem 6, once Lemma 13 is known. Hence, we can conclude at this stage that there exists, for each p E (1, co), a constant C and a real ((p) > 0 such that if ( _ ((p),
Fj,~ - 7r; Pj,r
- Rj,r
(5.2.3)
by (5.2.2), and
_< c
(5.2.4)
for every j E IN. Let us now prove part (i) of Lemma 13. The proof relies on assuming the factorization
(uj(Lx + ( ) ~ ) - 1 _ Mj,x,r
+r
where the operator Mj,~,r is defined by this formula. This means that
Oj,~ = Mj,~,r
(5.2.5)
We already know the family (OA)j(A)<_j-1 to satisfy the estimates described in Lemma 2 and Corollary 1. We will successively prove that the actions of zrj and of Mj,~,; preserve them. L e m m a 15. Let j E IN and A e I with j(A) _ j - 1. Let f~ be a function satisfying estimates (2.2.1)-(2.2.3), with size constant Co, envelope index q, and regularity index r. Then, its projections 7rj fx also satisfy these estimates, with size constant CCo, where C is an absolute constant, with the same envelope index, and min([r], r0) as regularity index. This is easy to prove, once we recall the Lemari~ commutation formula [12]. The size estimate (2.2.1) is elementary, and left to the reader. The only point to elucidate is the regularity estimate (2.2.2). But this is an immediate consequence of the following" for every multi-index a, with [a[ < r0, there exists a pair of biorthogonal multiresolution analyses (V(~) , V(~)) with associated projections (~rj " (~) ,~,j =(~) ),such that
O~ Trj - 7r~~ ) 0 ~ . This is Lemari6's formula, by which the estimate (2.2.2) on O~Trj fx is reduced . . to. the . elementary size estimate ( . . 2) 21 on 7r!~)j O~fj~._,provided ,lal<- r and [al _< ro. Hence, in formula (5.2.5), we know the family (TYjOA)j(A)<_j_1 to satisfy the same estimates a s (OA)j(A)<j-1. What remains is to examine the operators Mj,x,r
J. A ngeletti et al.
532
L e m m a 16. Let P(D) be a constant coet~cient elliptic operator
P(D) = -a,~c9aO~ + ao, with ao > O, and let Mj be the operator, acting on Vj, defined by the formula (TrjP(D)r~) -1 = MjrjP(D)-lr~. Then, there exists a sequence (mj,k )keZ, such that 3 C>0
C Imj,kl < 1 + ]k]n+2ro-l'
VkEZ ~ VjEIN
(5.2.6)
-
Mjf(x) - ~
VfE~
m j , k f ( x - k2-J).
(5.2.7)
k6_Z"
Moreover, the constant C in (5.2.6) depends only on the ellipticity constants of the operator P (D). When applied to P(D) = L~ +~ and f~ - 7rj0~, j(A) _< j - 1, formulae (5.2.6)-(5.2.7) give the desired estimates on 4J(~)0j,~ and on 0a0~t?j,~. We skip the details which can be found in [18]. It thus remains to prove Lemma 16. Again we only sketch the proof, which is also described in [18]. If f E Vj, and a(w) = a~w~w~ + ao, then we have 1 ^
[(1rJP(D)Tr~)-l fHw) = aj(w) f(w),
where
~j(~) = ~
o(~ + 2.k2J)l~(2-J~ + :.k)l ~,
kEZ"
being the scaling function associated to (Vj)jez. Similarly, we have A
[lrJP(D)-17r~ f ~ w ) - ( ) j (~) f(w), where
~ J (~) - keZ"
2"k2J) 1~(2-J~ + 2.k)l ~
Hence, if mj(w) is defined by the formula
1
~,j(~) = ~ ( ~ )
(
)
(~),
J
(5.2.s)
Analysis of Elliptic Operators
533
we see that
A -
Since mj is a 27r2J periodic symbol, we obtain formula (5.2.7), where mj,k is the k-th Fourier coefficient of mj. The estimate (5.2.6) on these coefficients comes from the integrability of Oamj, whenever lal _ n + 2r0 - 1. This 1 j last property is a consequence of (5.2.8), since !~ and (~) have the same Taylor expansion near zero, up to order n + 2r0. We let the reader check this property. Hence Lemmas 13, 14, and formulae (5.2.3) and (5.2.4) are completely proved. This gives Theorem 16 in the V M O case and when r is large enough. 5.3
S t u d y of Fj,r in t h e o t h e r c a s e s
The extension of estimate (5.2.4) to all r > 0 relies on an analog of Lemma 6: L e m m a 17. For every ~ > 0 and j E IN, the operator F j,r is continuous on L p, 1 <_ p <_ oo, uniformly in j and p. This lemma implies the boundedness of Fj,r from W -1,p to W 1,p is exactly the same as for deducing Theorem 7 from Lemma 7. Thus, the V M O case in Theorem 16 is complete up to Lemma 17, whose proof will be commented on in the next paragraph. Let us now turn to the case where the coefficients a ~ are in B M O ~ for some 0 < s < 1. We have L e m m a 18. If aa~ E B M O s, then Uj,r is continuous on W -1-ks,p, 1 < p < c~, and lim [[Rj,r = 0, (5.3.1) ff--~oo
uniformly in j. The proof mixes the ingredients we used for Theorem 6 and Theorem 12. We start from
Rj,r ~ ~, = 7rjO,~[a,~z - m), (ao,~)]OZOj,r , when j(A) _ j - 1. For each value of a, let Tj,a (where the subscript ~ is dropped) be defined by { [ a , z - mx(a~z)]OZOj,r
Tj,~ ~ )~ -
0
ifj(A) _ j - 1, otherwise.
We have to estimate the norm of Tj,~, acting from W -1-bs,p to W s,p. It is equivalent to the norm of Tj,~(I + A) on W 8'p. Now, Tj,~ (I + A) is
J. A ngeletti et al.
534
bounded on L p with a norm tending to zero as { tends to infinity: this is the content of Lemma 14, and can be proved as for Theorem 6. This is the W -1,p estimate. Now, to obtain the W -l+s,p estimate, it is enough to investigate the behaviour of
^'Tj,.(I + on L p, as for Theorem 12. We have here an analog of identity (4.2.5), which reads
A~Tj,o~(I + A) A -s ~;~ --[A~, a ~ ] 2-J(~)8(1 + 2J(:~))O~Oj,), +(a~o -m~,(a,~o))A ~ (2-J(:9~(1 + 2J()'))OZOj,~,) if j(A) _< j - 1. Since the estimates we have on the functions (1 + 2J(~))OzOj,~ are of the same nature as for the functions 0a0~0x in (4.2.5), uniformly in j, we can argue as when proving Theorem 12. Thus we obtain the boundedness of Rj,; on W -l+s,p, and relation (5.3.1). Using (5.2.3) and Lemma 51, we complete the proof of Theorem 16, up to L p estimates. 5.4
Lp
e s t i m a t e s o n Fj,i
We finally discuss Lemma 17, which asserts that Fj,~ is bounded on Vj and is equipped with the L p topology, 1 < p < c~, uniformly in j. Since the analogous result is valid for the resolvent (L + r (see Lemma 6) as soon as the coefficients are real-valued, it is natural to expect Lemma 17 to hold in such a case, independently of any regularity condition. However, we do not know at the present time whether this conjecture is true or not. We will instead present a proof which relies on the V M O hypothesis. It is very much inspired by [2]. This proof has several steps. The first is to prove that if p > 2, Fj,~ maps W -I'p N W -1'2 in W 1,p N W -1'2 continuously, for all r > 0, uniformly in j. This is proved iteratively, along the lines of Corollary 5, using the resolvent equation rj,~ - rj,~ + ( ~ - ~)rj,~r~,~. (5.4.1) Here, ~ has to be chosen so large as to ensure the boundedness of F j,~ from W -I'p to W 1,p. Hence, if Fj, i maps W -l'q ~ W -1'2 in W l'q N W -1'2, with q _< p, we deduce that Fj,~Fj,~ maps W -l'a A W -1'2 in W l'q N W -1'2 with 1
1
2
q
n
m
q
Analysis of Elliptic Operators
535
as long as ~ < p, by Sobolev embeddings. By (5.4.1), this implies that Fj,r
maps W - 1 , ~ n W -1'2 in W 1,~n W -1'2. In a finite number of steps we get the desired result. Step 2 shows that there exists a natural integer M such that, for all > 0, F~,~ maps L 2 in L ~ uniformly in j. This is a consequence of the first step 1. Since L p (resp. W I'p) is embedded in W -l'q (resp. L q) when
Fj,r maps L 2 n L p~' in L 2 n
1
1
1
q
p
n
where the exponents pk are defined by
L p~+t ,
Po - 2,
1
Pk+~
=
1
2
Pk
n
n as long as Pk < ~, or equivalently k <_ M - 1, for some integer M. Then, Fj,r maps L2O L pM-1 in L2N L c~, and the result follows. Since the class of operators F j,r is stable under taking the adjoint, we 2 M maps L 1 in L ~ uniformly in j. If Fj,r deduce that Fj,r 2M(x , y) denotes its kernel, we have proved the existence of a constant C = C({) such that, for every j E IN and x, y E IRn,
Y)I <-- C(C).
(5.4.2)
Step 3 is the most technical step, and will only be sketched. It is the Davies perturbation argument, which allows us to go from (5.4.2) to
<_ c(r
(5.4.3)
for some constant "yo > 0 independent of j and r
To prove this, we fix
xo,yo E IRn, and consider O, a compactly supported function from the Lipschitz class such that
9
- O(yo) -I
and
o - yol
IlV ll - 1.
(5.4.4)
We then prove (5.4.2) for the perturbed Galerkin operator
eT'~,, j,r
-
-
(eTCFj,r
TM
where e +~r stands for the operator of pointwise multiplication by e •162 with 7 a positive constant to be chosen. Once (5.4.2) is proved we obtain
lr 2,rM (xo,yo)l
< C(r
e -~[~(~~162176
J. A ngeletti et al.
536
which gives (5.4.3) provided C(r and "), do not depend on xo and yo, and ~' "- ~/ovf~ for some absolute constant ~'o > 0. -~r is the inverse of the Galerkin operator Now, the operator e~r associated to e-~VL e ~r and to e-~r e ~ r By this, we mean that if we replace L by L, with
L f - - d i v A V f - ~/(AVf) 9V r - ~/div(fAVr
- ~/2(AV(I) 9V(I))f,
if we replace Vj by Vj to be described later, and if we set
(L + ()j
j(L + r
where
~j
e-'r r162
=
is a nonorthogonal projection onto Vj, then we have - -~j. [(L + r - 1-~j,
e~r162
as in formula (3.2.6). Thus we have to prove that (5.4.2) remains valid in the more general setting where L is replaced by L and Vj by Vj, for any (I) satisfying (5.4.4), and for "i, - ~/o v~. The replacement of L by L is easy to handle, since the higher order terms are not modified. It is the replacement of Vj by Vj which is the main source of technicalities. Recalling that ~ is the scaling function attached to (Vj)jez, the spaces Vj are generated by the functions
~ k ( X ) -- 2in/2 e -~[r162
~ ( 2 J x - k),
k E Z n.
By construction, if f E Vj, we have
kEZ" where Cjk = 2in/2 fR" f ( x ) e ~[r
~(2Jx -- k) dx.
The spaces (Vj)jez do not form a standard multiresolution analysis anymore. They, however, share enough properties for the generalization of (5.4.2) to hold" they satisfy the embedding condition
vj c Vj+l,
Analysis of Elliptic Operators
537
and if Wj is generated by the functions ~ (x) = e -~[r162
r
(x)
where j(s - j, then it is a nonorthogonal supplementary subspace to Vj in Vj+I, which can replace the usual wavelet subspace Wj. The main point here is that, although the functions ~x do not have vanishing moments, they have small enough moments. We stop here the description of step 3, and show why the estimate (5.4.3) implies the L p estimate for Fj,;, which is the last step. It is a consequence of Taylor's formula. Once more, we choose ~ large enough, and write ,-v
2M-2
~.j,~: + ( 2 M - 1)
(Z -- ~)2M-2 F2,M dz.
k=O
We already know that Fj,~ is continuous on L p, and the estimate (5.4.3) implies the same result for Fj,z2M. Hence we obtain the L p boundedness of F j,(:, uniformly in j. Theorem 16 is now completely proved, and so are Theorems 14 and 15. w
Conclusion
All the preceding results might be extended in several directions. They probably can be localized, in the sense that if the coefficients a~,z have better regularity on some open set, it seems reasonable to conjecture that the same property will be shared by the solution u of an inhomogeneous problem, and that the convergence of the Galerkin approximations uj will be better on this open set. Our results can also be extended to boundary value problems on regular open sets. Using a classical trick (see [15]), it is not hard to see, for example, that Theorem A generalizes to the case of Dirichlet condition, thus recovering results of Chiarenza, Frasca, and Longo [4]. We are currently developing this line of research, and hope to publish some new results soon. Finally another set of questions seem interesting to investigate, regarding the convergence of Galerkin schemes. The first question we think of is that of describing the properties of the approximation spaces Vj which are really needed for our results to hold. The second question is about the nonregular coefficients case: when the cofficients have no regularity at all, it is known that our W 1,p results are only true when p is close to 2. However, it is also known that u = (L + ~ ) - l f is HSlder continuous if f
538
J. A ngeletti et al.
is in some reasonable space. This is the celebrated De Giorgi-Nash-Moser theory. We hence address the problem of proving the convergence of the Galerkin scheme in the appropriate HSlder space, for suitable spaces Vj. Though we are only dealing with linear operators, all the known proofs of De Giorgi-Nash-Moser estimates have a strong nonlinear structure. How to adapt this theory to the context of Galerkin approximations is the main difficulty to solve, a difficulty we find challenging.
Acknowledgments. It is a pleasure for us to acknowledge the influence of P. Auscher, who showed us the references [3] and [6], and of N. Lerner and G. M6tivier, especially regarding the results on nondivergence form operators. We also warmly thank M. Berg for her typing the manuscript, the referee for his useful comments, and the editors of this book for their patience and cooperation. References [1] Aronson, D., Bounds for fundamental solutions of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), 890-896. [2] Auscher, P., A. McIntosh, and P. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal., to appear. [3] Chiarenza, F., LV-regularity for systems of PDEs with coefficients in V M O , in Nonlinear Analysis, Function Spaces and Applications, vol.5, Krbec, Opic, Rakosnik (eds.), Prometheus Pub. House, 1994. [4] Chiarenza, F., M. Frasca, and P. Longo, W2,p-solvability of the Dirichlet problem for non-divergence elliptic equations with V M O coefficients, Trans. Amer. Math. Soc. 336 (1993), 841-853. [5] Chiarenza, F., M. Frasca, and P. Longo, Interior W 2,p estimates for non-divergence elliptic equations with discontinuous coefficients, Ricerche di Mat. 40 (1991), 149-168. [6] Coifman, R., R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611-635. [7] David, G., and J. L. Journ6, A boundedness criterion for generalized Calderbn-Zygmund operators, Ann. Math. 120 (1984), 371-387. [8] Escauriaza, L., Weak type (1, 1) estimates and regularity properties of adjoint and normalized adjoint solutions to linear non-divergence form operators with V M O coefficients, preprint. [9] Frazier, M. and B. Jawerth, A discrete transform and decomposition of function spaces, J. Funct. Anal. 93 (1990), 34-170. [lO] Frazier, M., B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS-AMS Regional Conf. Ser. in Math. 79 AMS, Providence, 1991.
Analysis of Elliptic Operators
[11]
[12] [13] [14]
[15] [16] [17]
[ls] [19]
539
Lemari~, P. G., Alg~bres d'op~rateurs et semi-groupes de Poisson sur un espace de nature homog~ne, Th~se, Orsay, 1984. Lemari~, P. G., Ondelettes vecteurs ~ divergence nuUe, Rev. Mat. Iberoamericana 8 (1992), 91-107. Meyer, Y., Ondelettes et Opdrateurs, vol. 1-3, Hermann, Paris, 1990. Sarason, D., Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. Taylor, M., Pseudodifferential Operators and Nonlinear PDE, Birkhguser, Basel, 1991. Tchamitchian, P., Wavelets and differential operators, in Different Perspectives on Wavelets, I. Daubechies (ed.), Proc. Sympos. Appl. Math. 47, AMS, Providence, 1993, pp. 77-88. Tchamitchian, P., Wavelets, functions and operators, in Wavelets. Theory and Applications, G. Erlebacher et al. (eds.), Oxford University Press, Oxford, 1996, pp. 83-181. Tchamitchian, P., Inversion de certains op~rateurs elliptiques ~ coefficients variables, SIAM J. Math. Anal., to appear. Triebel, H., Theory of Function Spaces II, Birkh/iuser, Basel, 1992.
J.M. Angeletti Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225 [email protected] S. Mazet Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225 [email protected] P. Tchamitchian Laboratoire de Math~matiques Fondamentales et Appliqu~es Facult~ des Sciences et Techniques de Saint-J~rSme 13397 Marseille Cedex 20, France et LATP, CNRS, URA 225 [email protected]
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Some Directional Elliptic Regularity for D o m a i n s w i t h C u s p s
Matthias Holschneider A b s t r a c t . In this paper we discuss how the position-scale half-space of wavelet analysis may be cut into different regions. We discuss conditions under which they are independent in the sense that the Toeplitz operators associated with their characteristic functions commute modulo smoothing operators. This is used to define microlocal classes of distributions having a well-defined behavior along the lines in wavelet spaces, and it allows us to describe singular and regular directions in distributions. As an application, we discuss elliptic regularity for these microlocal classes for domains with cusp-like singularities. An extended version of the present paper may be found in [5].
w
Introduction
The classical definition of local singular directions of a distribution rl is given by the wavefront set (see, e.g., [6]). At a given point x it is, roughly speaking, the cone of all directions in which the Fourier transform of the localized distribution Crl does not decay rapidly, where r is any smooth function t h a t is supported in some neighborhood of x. More precisely, for fixed r E C ~ (IRa), x E supp r a direction ~ E IRa\{0} is regular if in some conic neighborhood ~/~ ~ we have k E 7 =~ I(r
rl)^(k)l -< Cg(1 + ]kl) - y , N - 0, 1, 2, . . . .
The complement of the regular directions is denoted by Ex,r directions at x are then defined as
The singular
F~z -- NCEz,r Multiscale
Wavelet Methods
Wolfgang Dahmen,
541
for P D E s
A n d r e w J. K u r d i l a ,
a n d P e t e r O s w a l d ( e d s . ) , p p . 541-5{}5.
Copyright (~1997 by Academic Press, Inc. All r i g h t s of r e p r o d u c t i o n in a n y f o r m r e s e r v e d . ISBN 0-12-200675-5
M. Holschneider
542
where the intersection is over all r E II7~~ n) with x E supp r However this concept of singular and regular directions does not always fit with what one would intuitively call a regular direction in a distribution. W h a t we mean is best illustrated by an example in 1lt2. Consider the set K - {(x,y) E lR 2 9x >_ O, lYi <- x2} and let ;g be a function that is of very low regularity in the complement of the cusp K, whereas inside the cusp it is smooth. It is easy to see that there is no direction in which the Fourier transform of the localized function CX decays rapidly and so all directions are singular. This, however, is in contradiction with our intuition, in the sense that if we approach the singularity along any path contained in the set K, no irregularity is to be noticed and one would like to call the direction of the cusp regular. This is clearly only a very vague statement. To give a more precise definition we have to introduce the wavelet transform. We shall be very brief and we refer to the literature for a more detailed discussion (e.g. [3, 2, 7, 1, 8, 4]). Let g E 8(IRa), the class of Schwartz of rapidly decaying functions. In addition, suppose that g has all moments vanishing
xm g(x) dx - 0 for all multi-indices m. Then the wavelet transform of s E LP(IR n) with respect to g is defined by the following convolution:
Wgs(b,a) - W[g , s](b,a) - ((Ta * s)(b)
=
/ -~'~ ( x - b ) a
s(x) dx,
(1.1)
with a > 0 and b E IRn. Here we have introduced the following notations, that we shall use in the sequel:
~(t) = -~(-t), go = g(.la)l a', gb,a = ga("- b). The wavelet transform thus maps functions over the real line to functions over the open half-space ]Hn = {(b, a) :b E IRn, a > 0}. From the definition, it is clear that the wavelet transform is a sort of mathematical microscope whose position is fixed by b and whose enlargement is given by lla. To put it differently, )4;[g,s](b,a) is obtained by "looking at s at position b and at scale a." As a general statement, one can say that local regularity of s is mirrored in a certain speed of decay of )/Ygs. For instance, a uniform (in b) decay of O(aoo) as a --+ 0 of the wavelet coefficients is equivalent to the C ~ regularity of s. More quantitative information is available. So a uniform decay of O(a <~) with c~ E (0, 1) is equivalent to s E A a, the space of HSlder continuous functions of exponent c~. A tentative definition of a regular direction at x is therefore any direction ( for which the wavelet transform decays faster than any power of a, if the microscope approaches the singularity along a path that is tangent
Directional Elliptic Regularity
543
to r in x in such a way that it looks at a scale that is small compared to the distance to x. That is we say, vaguely speaking, a direction is regular if along a parabolic line we have rapid decay of the wavelet coefficients =
0).
This idea will be made more precise in Section 5. In particular, the definition will be modified in such a way that it becomes independent of the choice of the wavelet g. w
T h e basic formulas of c o n t i n u o u s wavelet analysis
For the convenience of the reader, we shall list here the basic formulas of wavelet analysis. We limit ourselves to formal expressions. They actually have a precise meaning when we consider the wavelet analysis in So(lR n) or S~(lR n) (see below). Let f be a complex valued function over ]Rn. Let g be another such function. The wavelet transform of f with respect to the analyzing wavelet g is defined through (we write dx for n-dimensional Lebesgue measure)
Wg f (b' a) - /rtn l f f ( x - b ) f (x) dx
b E ]Rn a > O"
We also write W[g, f](b, a) instead of Wgf(b, a). Here b 6 IRn is a position parameter and a 6 IR+ is a scale parameter. The wavelet transform of a function over Rn is thus a function over the position-scale half-space IHn = IRn • IR+. If we introduce the dilation (Da) and translation operators (Tb) by
Das(x) = s(x/a)/a n,
Tbs(x) = s ( x - b),
then we may also write the wavelet transform as a family of scalar products )/Ygf (b, a) -
(gb,a, f ) ,
gb,a --
TbDag,
or as a family of convolutions indexed by a scale parameter
Wg f (b, a) = (ga * f)(b),
ga = Dag,
g(x) = "ff(-x).
The convolution product is defined as usual, by
(s 9r)(x) =/rtn s(x - y)r(y) dy = (r 9s)(x) . If we introduce the Fourier transform 1
~
eikx.g(k)d k
M. Holschneider
544
1/.
then the wavelet transform may also be written as
Wgf(b,a)-
(27r)n
The wavelet synthesis A4 maps functions over the position-scale halfspace to functions over IRn. Let r = r(b, a) be a complex valued function over I[-In and h a function over lRn. Then the wavelet synthesis of r with respect to the synthesizing wavelet h is defined as
l(x-b) r(b, a) -~ h a 2.1
dbda a
R e l a t i o n b e t w e e n }4; and M
We now list some relations between ~V and ~/[. The wavelet synthesis is the adjoint of the wavelet transform--both with respect to the same wavelet--l/Y~ = A4g
/IH VPgs(b'a)r(b' a) dbda - /Ft ~(~) M ~ ( ~ ) d ~ . n
a
n
The combination A4hl/Yg reads in Fourier space
M h W , . ~(k) ~ m,,h(k)~(k),
~,.h(k) -
~(ak) h(~k) e~.
fo
^
a
Note that the Fourier multiplier mg,h depends only on the direction of k, mg,h = mg,h (k/lkl). This is because the measure da/a is scaling invariant. In case that g and h are such that
fo
^
g
Cg,h
with 0 < I%,hl < oc, we say that g, h are an analysis reconstruction pair, or that h is a reconstruction wavelet for g. In this case we have
MhWg = %,h11. We say that g is strictly admissible if g is its own reconstruction wavelet, or (what is the same) if Vk e IRa\{0}
9
i~(ak)12
da
- -
--
Cg.
a
A sufficient condition for g to have a reconstruction wavelet r is that for some c > 1 we have
~ - 1 < fo ~ I~(ak) 12 da < c.
Directional Elliptic Regularity
545
If this condition holds, we call g admissible. In this case, the following function r will be a reconstruction wavelet for g:
~(k) - ~ ( k ) /
i~(ak)! 2 d__aa ,
(2.1)
a
If g and h are an analysis reconstruction pair, then the following formula holds
/ ~ Wgs(b, a)Whr(b, a) dbda a
Cg,h fR . S(X)r(x) dx.
--"
In particular if g is strictly admissible, then we have conservation of energy
(~ = ~,~)
/
IW~(b,~)l= dbd~ n
cg
/R
a
Is(x) n
I~ dx
.
Let H and r be functions over the half-space ]I-In. We define a (noncommutative) convolution for functions over the half-space via
(n 9 r)(b, ~)
-/~.
)
r[ ~b- b' a r(b' a') a t
~ a t
~
dbldal a t
9
(2.2)
Formally, we can write
(Wg,~4hs)(b, a) - Hg,h * s(b, a),
Hg,h -- Wgh.
In the case where g and h are an analysis reconstruction pair with
Cg,h = 1, we clearly have (Wgj~4h) 2 -- ~Yg.~4h and hence l-Ig,h 9IIg,h
-
-
l-lg,h.
The mapping r ~ IIg,h 9 r is a projector into the range of the wavelet transform )/Yg. In case t h a t g is strictly admissible with cg,g - 1, we have t h a t Hg,g is an orthogonal projector. The most i m p o r t a n t formula for our work is the following. Consider a function s over ]Rn. Suppose t h a t g is admissible. It thus has a reconstruction wavelet r. An explicit formula has been given in (2.1). Then, since ~ u - - ( ~ ) h . A ~ r ) ~ / V g , the wavelet transform of s with respect to g and the one with respect to h are related via the so-called cross kernel equation" Whs = IIg_~h 9 Wgs,
w
IIg_+h = Whr.
(2.3)
Some function spaces
All formulas t h a t we have given so far have a well-defined meaning if the wavelets are taken in some subset of the class of Schwartz, as we will recall nOW.
M. Holschneider
546
3.1
The analysis of 80(R n)
Let S(IR n) denote the class of Schwartz consisting of those functions that, together with their derivatives, decay faster than any polynomial and such that the norms Ilsll~,,, = sup
tER n
Jt"O~s(t) ! < oo,
are all finite for all multi-indices a and ~. They generate a locally convex topology which makes S(lR n) a Fr~chet space. We denoted by 80 (]Rn) the closed set of functions in S(lR n) for which all moments vanish:
Va e IN n, /
s(x)x • dx - 0 ,,t-'->, Vm > O, ~(k) = o(k m)
(Ikl
-~ 0).
The Schwartz space of functions over the half-plane lI-In shall be denoted by S(IHn). It consists of those functions r for which the following norms are all finite
JJrlJ~,l,m,n'-
sup
(b,~)e~ ~
J(a + 1/a)k(1 +
Ibl)~O~'O2r(b,a)J
< oo.
Note that this means that r, together with all its derivatives, decays rapidly for large b and for large or small a. It can be shown that w:
So(~ ~)
x
s0(~ ~) -+ s ( ~ ) ,
(g, ~) ~ wg~
is continuous. The same holds for the wavelet synthesis defined through
r(b'a) l~-h n (X -a b
dbdaa
and we have that
./~4 : 8o ( IRn) x S ( lH '~) ~ So (lR ) ,
( h , r) ~ ~[ h r
is continuous, too. However, in this paper we will not discuss topologies on the microlocal classes that we will define below. This can be done in an obvious way and we want to streamline the discussion. We note here the following important fact. For admissible g E 80(IR n) and arbitrary h E $0(]Rn), the cross kernel (2.3) is a function in 3(]Hn). It is thus very well localized.
3.2
Wavelet analysis of S~(R n)
We denote the space of linear continuous functionals 77 : 8 0 ( ~ n) -+ C by S~(IRn). We consider it together with its natural weak-, topology.
Directional Elliptic Regularity
547
The space $~(IR n) can canonically be identified with S'(IR)/P(IRn), where P(IR n) is the space of polynomials in n variables. The wavelet transform of 7/E $~ (IRn) can now be defined pointwise as Wgrl(b, a) = rl(gb,a). This is a smooth function (see, e.g., [4]) that satisfies
IWgv(b, a)i <_ c(1 + Ibl)m(a + a-l) m
(3.1)
for some m > 0. By duality we have that the mapping (this time for the fixed wavelet g e So (IRn))
is continuous. Here S' (]H~) is the dual of S(]H n) together with the weak-, topology. On the other hand, consider any r E $'(]Hn). Then we set for s E
~0(~ n)
=
Clearly, for h E 5:0 (lR~) Mh
:
r
Mhr
is again continuous. In the case of a locally integrable function r of at most polynomial growth we have
,~4hr(s) - - / ~ n r(b, a) W-Ks(b , a) dbda
3.3
M o r e g e n e r a l spaces
Many function spaces can be characterized in terms of wavelet coefficients. As a rule, the faster the wavelet coefficients decay, the more the analyzed function is regular. We come to the details. For this, consider a vector space B(IH n) of locally integrable functions
S(~t ~) C B(~t ~) C S'(~tn). Suppose in addition that B (IHn) is invariant under convolutions with highly localized kernels
r e S(]I-In), s e B(]H n)
==~ r , s, s , r ~_ B(]Hn).
The convolution of two functions over IHn is defined by (2.2). It then makes sense to pull back B(lH n) to a vector space of distributions over ]Rn. We shall denote this space of distributions by B(]Rn). It is defined through the following theorem.
M. Holschneider
548
T h e o r e m 1. For a distribution ~1 E S~(IRn), the following are equivalent. 9 There is a wavelet g E So(IR n) which is admissible in that it satisfies 12 da ,~ 1,
f0 ~176
for which we have
Wgrl e B(lHn).
9 For all h E So (IRn) we have
who
e
Proof: The passage from Wgr/to Whr/is given by the highly localized cross kernel (2.3). By definition, this operation leaves B(lI-I n) invariant.II Therefore it makes sense to speak of the space B(lFt n) associated with B (]I-In). It is precisely the space of distributions for which ~Ygrl E B (]Hn), where g is as given in Theorem 1. We still shall need an additional technical assumption on the spaces B(lI-In). Their multiplier algebra should contain the bounded functions
m e L~176
s e B(lH n) =~ m . s e B(]Hn).
This allows us to define the Toeplitz operators
.MhmWg :
B(IR n) --+ B(IRn).
For the rest of the paper we refer to the spaces B ( ~ n ) satisfying all stated properties and their pulled back counterparts over lFtn, B(I[:tn), as admissible local regularity spaces. We end this section with a remark concerning topology. Suppose that in addition B(]I-In) is a Banach lattice with norm
II
liB(w,)= llSIIB(W,).
Suppose further that for fixed I I E S(IH n) we have that
r~II,r is continuous. Then we can define a norm on BOR n) which makes it a Banach space by setting
IlalIB(R )- IlWgSIIB( ) 9
Directional Elliptic Regularity
549
This is reasonable, since for different wavelets satisfying the hypothesis of Theorem 1, we obtain equivalent norms. It is easy to verify the sufficient condition for B(IH n) to be stable under convolution with localized kernels in the case that B(lH n) is a Banach space. It is sufficient to find K and c such that for all s E B(IH ~) we can estimate
I]~(~" +~, ~')lls(.-I-) <-II~IIB(IH-)~('~
+
1/a)K( 1 +
Ibl)K.
Indeed, by a simple change of variables, we may write
II1-[. ~ilB(~,,) <-II~lls(~I-) c /IH" II(-~,
1/a)(o~ + 1/a)K(1 + I/~[)Kd/3dO~
The last integral is a finite constant. For the sake of simplicity, we shall not give a detailed discussion of possible topologies on the microlocal classes which we are going to define in the next chapter. 3.4
S o m e e x a m p l e s of local r e g u l a r i t y spaces
Many functions spaces of day-to-day functional analysis can be characterized by this easy concept. Most of them are contained in the following two scales of spaces. Let r :]R+ -+ ]R+ and a :IR n -+ ]R+ be two tempered weight functions over IR+ and ]Rn, respectively. By this we understand that they satisfy the following: r
- O((a + 1/a) n r
a(b + b') = O((1 + [b[)m a(b')),
for some m > 0. Then the following expressions define norms for functions over the half-space for all 1 _ p, q <_ c~:
I[rllp =
. Ir(b, a)lq a(b)db
or
II~ll~-
~
• r
r
do
1 (i .b b.) m
~ '---~ a
ir(b" a')lq
a'
db'da' ] p/q a' ~(b) db.
The associated Banach spaces are stable under convolution with highly regular kernels and, thus, they may be pulled back to ]Rn, giving rise to two scales of spaces. The first scale of spaces contains the Besov spaces, whereas the second scale contains the LP-spaces and Sobolev spaces (see, e.g., [7, 1, 4, 9] for more details).
550
M. Holschneider
w
M o r e g e n e r a l microlocal classes
In this section we shall be concerned with the problem of constructing new local regularity spaces out of old ones. The idea is easily explained. Consider an arbitrary set f~ C lHn. Eventually we want to consider lines f~ -- {(b - , ~ , a = ,~)}, in order to define regularity in the direction E lRn. But for the moment we restrict ourselves to generality, since it makes our discussion easier to understand. Fix, in addition to f~, an admissible, local regularity space B (lRn). It is characterized by the fact that the wavelet transforms of its members are in some vector space B(lHn). The regularity classes we want to construct are, roughly speaking, distributions whose wavelet coefficients have, on f~, a growth behavior governed by the class B (]FIn). A slightly more general concept is obtained if we take two local regularity spaces B1 (JRn) and B2 (lRn), both of the type considered in the previous section. We now want to cut the half-space ]Hn into two parts, say f~ and its complement f~c. In some sense, to be precise, we consider classes of distributions whose wavelet coefficients behave inside f~ like the wavelet coefficients of functions in B1 (~n), whereas in fF, these distributions have regularity governed by B2 (IW~). A naive approach might be to require that the restriction of }4;g7/ to f~ satisfies }4~grl E B1 (IHn), whereas its restriction to the complement of f~ should correspond (via A4h) to a function in B2(]Rn). However this definition might depend on the wavelets we use, and thus it is not useful. To get around this difficulty we first construct a suitable family of neighborhoods for f~. With these neighborhoods, it turns out that we can define vector spaces that are independent of the wavelets 9 and h. 4.1
A non-Euclidian distance
The first step of the construction is to introduce a non-Euclidian distance function adapted to the geometry of the half-space. A suitable choice is given by dist((b, a), (b', a')) =
+ I(b -
+ I ( b ' - b)/ l 9
This clearly is not a distance in the usual sense, since dist((b, a), (b, a)) = 1. However, a kind of multiplicative triangule inequality holds (see Lemma 1 below), so that log dist((b, a), (b', a')) actually is a distance. Note, however, that the distance function is symmetric, dist((b, a), (b', a')) = dist((b', a'), (b, a)).
Directional Elliptic Regularity
551
Clearly the upper half-space carries a natural group structure. It is given by the following composition law
(b, a) o (b', a') - 5aTb(b', a') -- (ab' + b, aa'), where ~'b and 5a stand for the translation and dilation as left actions of IR+ and ]Rn on IHn via
~ " (b, a) ~ (ab, c~a),
Tz " (b, a) ~ (b + ~, a').
(4.1)
The inverse element reads (b, a) -1 = (-b/a, 1/a), and the neutral element clearly is (0, 1). Note that we could have taken the logarithm of the hyperbolic distance instead of the distance above. However, in order to simplify certain proofs, we have opted for the distance function above. If we denote the distance of a point (b, a) from the point (0, 1) by A((b, a))
-
dist((b, a), (0, 1)) - exp(llog al) + Ibl (1 + 1/a)
=
max{a, 1/a} + ibl (1 + 1/a),
then we have the following relation: dist((b, a), (b', a')) = A((b, a) -1 (Dr, at)). Note also the identity
A((b, a) -1) - A((b, a)). Moreover, the function A is a tempered weight function
~((b + aZ, ~a)) < (~ + 1/~)(1 + lZl)~((b, a)). The next lemma shows that a kind of triangule inequality holds. L e m m a 1. We have the following triangule inequalities
max(A((b, a))/A((b', a')), A((b', ~'))/A((D, a))} < ~((b, a)(b', a')) < ~((b, a))~((b', a')).
M. Holschneider
552
Proof: To prove this inequality, note that an elementary direct computation shows that
aa'A((b,a)-l(b',a'))
= =
aa'A(((b' - b)/a,a'/a)) max(a 2, a '2} + a ib - b'] + a' Ib - b'[
<_ max{a 2, a '2} + a Ibl + a Ib'l + a' ]bi + a' Ib'l. On the other hand
aa'A((b,a))A((b',a')) - (max(1, a2}+]bl+a [b])(max{1,a'2}+tb'i+a ' [b'l). Upon neglecting positive terms, we have aa'A((b, a))A((b', a')) > max{l, a 2} max{l, a '2 } + alb I + max{l, a '2 } ibl + a' Ib'l + max(l, a 2} Ib'l. Therefore the difference between the last and the previous expression is majorized by max{l, a 2 } ma.x{1, a '2 } - max{a2, a '2 } + Ibl (max{ 1, a '2 } - a') + Ib'l (max{ 1, a 2} - a) _> 0. The proof of the rightmost inequality follows now from the identity
A((b,a)-~)=A((b,a)). The remaining inequality follows as usual from the previous one, namely A((b, a)) < A((b,a)(b',a'))A((b',a') -1) = A((b,a)(b',a'))A((b',a')). m
This immediately implies the following relation for the distance function dist((b, a), (b', a')) = A((b, a) -1 (b', a')) < A((b, a))A((b', a')), and dist((b, a), (b', a')) > A((b, a))/A((b', a')). Thus the following triangule inequality holds dist((b, a), (b", a")) <_ dist((b, a), (b', a'))dist((b', a'), (b", a")), and dist((b, a), (b", a")) >__dist((b, a), (b', a'))/dist((b', a'), (b", a")).
Directional Elliptic Regularity 4.2
A f a m i l y of
553
neighborhoods
Let us introduce the closed non-Euclidian balls
U((b,a),r) = {(b',a') E lHn: dist((b,a), (b', a')) _< r}. Note that they all are obtained by dilations and translations of the balls around the point (0,1). More precisely, since the distance function satisfies dist((-yb +/~, ~/a), (Tb' + t3, 7a')) = dist((b, a), (b', a')),
-7 > 0,/3 E IRa,
we have
U((b, a), r) = Tb6aV((O, 1), r). An equivalent system of neighborhoods U' is obtained by translating and dilating the family of balls defined via the following inequality: (a - [1 + r + 1/(1 + r)]/2) 2 +
Ibl < (1 + r + 1/(1 + r))2/4.
They are Euclidian balls with the "south pole" at the point (b = 0,a 1/(1 + r)) and the "north pole" at the point (b - 0, a - 1 + r). The equivalence is expressed by the fact that for some constant c > 1 we have
U'((O, 1),r/c) C U((O, 1 ) , r ) C U'((O, 1),cr). We may leave the elementary calculations to the reader. We are interested in when it makes sense to speak of a certain regularity in one set and another regularity in another set of the half-space. Consider therefore two arbitrary subsets i2, E C IHa. We say that f~ and E are well separated if the following holds. For (b, a) E f~, consider a non-Euclidian ball U((b,a), r) with center (b,a) and radius r. Choose r small enough so that U does not meet E. Then "well separated" means for us that for some e > 0 we may choose r such that the following estimate holds true for small a~
r > a -e. In other terms, we define more formally and slightly more generally: Definition 1. We say that two sets f~ and E are well separated if for some e > 0 we have (b, a) E f~ =V dist((b, a), E) > A((b, a)) ~. Here the distance between a point and a set f~ C ~ a is defined as usual, dist ((b, a), f~) =
inf dist ((b, a), (b', a')). (b',a')Ef~
M. Holschneider
554
Note that although the non-Euclidian distance diverges at small scale, the Euclidian distance might tend to 0 as a -+ 0. As a somewhat typical example, consider in ]H 2 the sets ={a
E={a>lb[ ~}M{a
They are well separated iff ~ >/3. However, their Euclidian distance always tends to 0 as a -~ 0. We have still another useful characterization of two sets ~ and ~ being well separated. For this, consider the sets
61/aT-b~,
(b,a) E ~.
Now both sets are well separated iff each of these sets is contained in the complement of a non-Euclidian ball U((0, 1), r(b, a)), with r(b, a) >
(fl/~T-b~ C U((O, 1), A((b, a))~) c. 4.3
(4.1)
More about well-separated sets
Since the distance function is continuous in the Euclidian topology, it is clear that the distance of a point and a set and its Euclidian closure are the same. Therefore a set is well separated from another if and only if its closures are. The notion of "well separated" is inherited by subsets. If D C ~ and ~E is well separated from E, then ~2 is well separated from E, too. In addition, the notion of "well separated" is symmetric. L e m m a 2. If Q is well separated from ~ then ~ is well separated from ~. Proof: By hypothesis, (b, a) E ~ and (b', a') E ~ implies that dist((b, a), (b', a')) > A((b, a)) ~.
(4.1)
We claim that this implies that dist((b, a), (b', a')) > A((b', a')) ~' , for e' = e/(1 + e). On the contrary, suppose that for some points we have dist((b, a), (b', a')) - A((b, a) -1 (b', a')) <_ A((b', a')) g . This implies, via the second triangule inequality, in particular, that
A((b',
<
555
Directional Elliptic Regularity
and therefore by the choice of s that A((b',a')) d <_ A((b,a)) e. This implies dist((b, a), (b', a')) _ A((b, a)) ~, which is in contradiction with (4.1). m For every s > 0 let us introduce the following non-Euclidian neighborhoods of a set fl C lHn:
U U((b,~),
~((b, ~))~).
The system of such neighborhoods constitutes a fundamental family of neighborhoods in the following sense. We have that F~ (f~) c is well separated from f~. In addition, if E is well separated from f~, then for some s > 0 we have that
z n r~(~) =0.
Thanks to the triangule inequalities we have the following associativity for the s neighborhoods. L e m m a 3. For any set f~ C ]I-tn the following holds true. For s > O, s > 0 and s >_ s -F s + s we have
r~ (r~i (n)) c r~ (~). On the other hand for el > O, 0 < s enough such that s
__ el/(1 + s
and e3 > 0 small
l+e3 . .s . +. e2i . . . e2 < 0 ,
we have
r ~ (r~, (~)c)c D r ~ (~). Proofi We show the first part. If (b",a") E Fe2(Fe 1(fl)), then for some points (b', a') E ]H n and (b, a) E f~ we have dist((b", a"), (b', a')) <_ A((b', a')) ~2,
dist((b', a'), (b, a)) _< A((b, a)) ~1.
Therefore by the triangule inequality we have dist((b", a"), (b,a)) <_ A((b',a'))~2A((b,a)) ~. Now, as before, by the reverse triangule inequality we have
~((b', ~'))/Z~((b, ~)) < Z~((b, ~))~,
M. Holschneider
556 and therefore finally as claimed
dist((b",a"), (b,a)) <_ A((b,a)) el+e2(l+el) To show the second statement we have to prove that F, a (fl) n F, 2 (r,, ( ~ ) ~ ) - 0. Suppose on the contrary, that there are (b, a) such that 3(b', a') e ~
9 dist((b, a), (b', a')) <_ A((b', a')) ~3,
(4.2)
and simultaneously some (b", a") E F•2 (~r~)c satisfying V(/3, a) E fl
9 dist((/~, a), (b", a")) > A((/~, a)) '1 ,
and for which we have dist((b, a), (b", a")) <_ A((b", a")) '2 .
(4.3)
In particular we have dist((b', a'), (b", a")) > A((b', a')) '1 . Then we have by the triangule inequality
~X((b', a'))'l
< dist((b', a'), (b", a")) _ dist((b', a'), (b, a)) dist((b, a), (b", a")) < ~((b', ~'))'~((b", ~"))'~.
Now, relation (4.2) implies
m((b, a)) ~ A((D', at)) l+e3 Moreover, (4.3) implies
A((D", art)) l-e2 ~ z~((b, a)). It follows that Therefore
l+e~}
A((bn~ all)) < A((b', al))l--e2 l+e 3
1 < A((b', ~,)),3-,1+,, 1-,~
which is impossible by the choice of e3, since A((b', a')) >_ 1. m This immediately implies the following lemma which we shall use in the next section.
557
Directional Elliptic Regularity
Lemma 4. Let E D f~ be such that E c is well separated from fl. Then there is a set E, E D E D f~ such that E is well separated from E c and f~ is well separated from E c. Proof: Some F~(f~) with e small enough will do. m
4.4
C u t t i n g the half-space
Let us come back to our original goal of dividing the half-space into two sets of different regularity. As we already said, it is not possible to speak of regularity B(IH ~) inside a given set f~ C ]H~, since this notion is not independent under highly regular Calder6n-Zygmund operators, or to put it more simply, it might depend on the given wavelet used for the definition. However, if we require regularity B(]H n) in a region which is slightly larger than f~, it then follows that the same regularity holds true in f~ for any wavelet. By abuse of notation, let E (respectively, f~) denote the operator that restricts functions over IHn to the set E (respectively, to f~). That is, we have E : r ~-4 xEr, where ) ~ is the characteristic function of E. Theorem 2. Consider two sets E and f~ and suppose that ~ D f~ in such a way that Ec and ~ are well separated. Let g, g', h, h' E So(IR n) satisfy for some c > 1 (s = g, g', h, h'), c- 1 <
[~(ak)i2 da <:c.
~0~176
a
Suppose that ~1 E $' (IR n) satisfies
Then
Mh,
e
Therefore it makes sense to separate IHn into regions of different regularity, provided the regions are well separated. The proof is based on the following lemma. It estimates the influence of a function which is nasty inside some region E, under convolution operators over the half-space on another region E ~, when both are well separated.
M. Holschneider
558
L e m m a 5. Suppose .=' and .=. are well separated. Let r be a locally integrable function over lI'In that is equal to 0 except on ~', where it satisfies for some M > 0 and some c > 0
(b, a) e ~'
~
It(b, a)l <_ c A((b, a)) M.
Then, for P 6 8(]H n) we have that ~ - P 9r satisfies e
le(b,-)l <
a)) -k
for a/1 k ~ IN. Proof: We have to estimate the localization of P 9 r(b, a) for (b, a) E -By definition P . r(b, a) equals , -a7 ~ P
- - a- 7 - ' a'
a'
"
Using the action (4.1) of dilation and translation on lI-In, we may also write the following
P ( - b ' / a ' 1/a') r(ab' + b, aa') ~db'da' ~ l/aT--b,-,
Now by hypothesis on r, we may write with some K and some c > O,
Plugging this estimate into the previous expression, we have to estimate for (b, a) e F.,
A((b,a)) g f~
IP'(b',a')l db'da' a---7--,
(4.1)
I / a T _ b .'~j
with P'(b',a') = A ( ( b ' , a ' ) ) g P ( - b ' / a ' , l / a ' ) . Together with P, we have that P~ is highly localized. For A ___0, let us look at the following integral running over the complement of a non-Euclidian ball centered at (0, 1)"
/~ ((v,~,))>~
IP'(b',a')l
dbldal
a'"
Thanks to the high localization of P~, this function is decreasing faster than any power of 1/A as )~ ~ c~. Now, since E and F.~ are well separated we may use characterization (4.1) to conclude that the integral in (4.1) is estimated by F(A((b,a)) ~) for some e > 0. But this function is again rapidly decreasing as A((b, a)) ~ c~ and the proof is finished. II
Directional Elliptic Regularity
559
Note that the lemma we want to prove may be rephrased as follows: for all P E S(]Hn), we have that s ~ .=.' (P 9 (Es)) is infinitely smoothing in the sense that it maps functions of polynomial growth into rapidly decreasing functions over the half-space. P r o o f of T h e o r e m 2: The previous considerations imply the following: if we have P~l/Ygrl E B(]Hn), with g admissible, then for all g' E So(lR n) we have f~Wg, r/ E B(]I-In). To show this, note that the transition from l/Ygrl to Wg,~ is done by convoluting with a highly localized kernel P. Now, we may write aWg, rl = f l ( P 9 (EWgr/)) + fl(P 9 (ECWgrl)). Since by hypothesis B(]I-I '~) is invariant under multiplications with bounded functions and convolutions with P, the first term is again in B(IHn), whereas the second term is arbitrarily smooth. A slightly more complicated situation occurs in our theorem, since we can not conclude from fl4hr E B(]R "~) that r E B(]nn), since the wavelet synthesis is not injective. We can find a set E between E and f~,
such that E is well separated from the complement of E and f~ is well separated from the complement of .=.. This follows from Lemma 4. We may conclude that Z W , Mhr W,
= =(P1 *
e
where P1 = W f h for any admissible f E 80(lRn). In particular, we may choose f to be a reconstruction wavelet for g and thus it follows that P1 * l/Yg~ = Wg~. Now writing (as characteristic functions!) P~ = 1 - Nc, the last expression equals e(ei
9
- =(el
9
The set E is well separated from E c and thus the second term has rapid decay as A((b, a)) gets large. Let us call this function u. Then, since u is well localized we have r , u e 8(ln ), for all r E 8(]Hn). We therefore obtain, up to a function of rapid decay,
M. Holschneider
560
Now }4;g,~ = P 9 )/Yg~ for some P E $(]Hn). Therefore, since f~ C E is well separated from the complement of E we have at the beginning of the proof that fB/Yg,~ E B(IH ~) up to the well localized function f~u. But then clearly Mh, f~Wg, r/E B(]Rn). m Let ~ D f~ be open and let again f~ be well separated from the complement of E as before. Consider the two Toeplitz operators Tr~ = MhY, W~,
T, = Mh~Vg.
We then have proved the following
Corollary 1. We have that [
, T, ] =
T, - T,
is infinitely smoothing in the sense that it maps the tempered distributions in S~ (IRn) into smooth functions in So(IR~). 4.5
Microlocal classes
The theorems of the previous section may be used to define some very general microlocal classes. Suppose we are given two regularity spaces A(IR n) and B(IRn), and suppose in addition that B(]R n) C A(]Rn). Consider a set f~ C ]I-I'~. Since we are only interested in local properties we may suppose that f~ is bounded in the Euclidian norm. In order to avoid technicalities, we suppose that ~ is closed. The first type of local regularity classes corresponds to the idea that globally a distribution has a regularity described by A(IH n) whereas locally, in f~, we have some higher regularity of type B(]I-In). A dual idea would be to have the wavelet coefficients concentrated on the subset f~. That is, outside of f~, the wavelet coefficients are small, hence correspond to the higher regularity B(IH~), whereas inside f~, the coefficients are in A(]I-In). We want to make these statements more precise. Consider first the case of higher local regularity. Suppose that there is a sequence of closed sets {f~k), k = 1 , 2 , . . . with ~1 C . . . f ~ k C f~k+X"" C f~. We suppose that f~k converges to f~ in the sense that f~ -- Uf~k. k
Directional Elliptic Regularity
561
Suppose that ftk and ft~+ 1 are well separated for each k. Then clearly f~k and ~t~ are well separated for k < I. We then say that ~/E $~(lR n) belongs to the microlocal class A(f~; A,B) iff for some admissible g and all k we have E A(IRn) and Mgf~kV1]g~E B(lRn). By the results of the previous theorem it is clear that the definition does not depend on the specific wavelets nor on the family of approximating sets f~k. Indeed, by Lemma 3 we may take the family F1/k(ft) as universal family of approximating sets. Note however, that for arbitrary ft, the previous class might coincide with A(IRn). Indeed, in order to have an approximating sequence from the interior, of mutually well separated sets the "smoother" region can not be arbitrary thin. It must contain at least some non-Euclidian neighborhood of some set. Frequently one takes A = S~(]Rn) in which case one is only interested in the behavior of the wavelet coefficients around ft. In the next section we shall use this kind of classes to define directional regularity in distributions. Consider now the dual approach, where we want to formalize the idea of wavelet coefficients concentrated on ft. Suppose now that a sequence of open sets {~tk}, k = 1, 2 , . . . with
converges to gt in the sense that f~-- Nf~k. k
Again we require that (f~k)c and ft k+l are well separated for each k. We then say that r/E $~(IR n) belongs to the microlocal class E(~t; A, B) iff for some admissible g and all k we have
e A(IRn) and Mg(f~k)cwgT1 e B(]Rn). Note that in the case where B(~ n) = $o(lRn), this corresponds to the idea of having the wavelet coefficients concentrated on the set gt, where they satisfy the less restrictive regularity estimate given by A(]I-In). w
S o m e directional microlocal classes
We now propose to look at more specific examples of regularity classes, in particular, to those we mentioned in the beginning of the paper, that is, to classes related to the notion of singular or regular directions in distributions. Particular useful examples arise when we consider parabolic regions
M. Holschneider
562
or lines in wavelet space. As measure of regularity it is useful to consider the HSlder-Zygmund scale A s of spaces defined in wavelet space via tlslla -
sup
la-as(b,a)l.
Fix a vector ~ E IRn, I~I > 0 and consider the set E = E(~, 7) - { ( b - ~ , a -
) ~ ) " 1/2 > ~ > 0)
for some 7 > 1. We now say that r/ E Aa(IR ~) is locally of type ( a , ~ , 7 ) if it belongs to the microlocal class ~A,B with Ft - F~(E(~,7)) for some e > 0, A - S(IR ~) and B - A~(IR~). More explicitly, this means that the wavelet transform of s satisfies for some ~ > 0,
(b,a) E F~(S(~,-),))
=~ I W g u ( b , a ) l <_ ca ~
and (b, a) ~ F~(.--.(~, 7)) =v
i)4;g~(b, a)] <_ c(a + 1/a)k(1 + ib])k.
This corresponds to looking at the behavior of the wavelet coefficients under the following nonhomogeneous dilations: W~(~,~),
~ > 0.
Here c depends only on e. Actually, we may choose ~ such that I~i - 1. Indeed suppose ~' - ~ with/~ > 0, and denote by .=/the corresponding line. Then if (b,a) E E it follows that (b, fl-aa) E .=.'. Therefore the non-Euclidian distance between the two points is uniformly bounded by ~a +/~-a. Therefore, they define the same microlocal classes. Let us explain in which sense these classes are linked to singular and regular directions. To this end, replace for the moment the wavelet at position b and scale a by the characteristic function of the Euclidian ball centered at b and of radius a. As (b,a) tends to (0, 0), while always (b,a) E (gt(~, 7)), the support of the wavelets is contained in a cusp-like region, around the line in direction ~. This shows that the microlocal class (a, ~, 7) quantifies the regularity of s in direction ~.
5.1
Some elliptic regularity
We now want to apply the classes introduced above to a problem of elliptic regularity. For the sake of simplicity, we only discuss the Laplace equation and leave more general elliptic operators for subsequent papers. We say that an open domain ~ C IRn satisfies the cusp condition of degree 5 > 0 at x E 0Ft in direction ~, ~ E lRn - {0}, if there is some c > 0 such that
{yly-
1 <
ly-
1 <
c
563
Directional Elliptic Regularity
T h e o r e m 3. Let f~ satisfy a cusp condition at 0 of type 5 in direction ~. Suppose that 71 is a tempered distribution that satisfies inside f~ at At/- f for some distribution f supported b y e . If f is of type (~c,7, a), with 7 > ~, then it follows that 71 is of type (~, -),, a + 2). This theorem is a special case of a more general theorem. Let B(IR n) be a local regularity space of the type we have considered before, with B(IH n) the associated space of functions over IHn. It is straightforward to see that together with B(IHn), the space of functions which consists of the functions aVr(b, a) with r E B(IH n) is again an admissible regularity space. It will be denoted by (aVB)(IH n) (respectively, by (aVB)(lRn)). For a set f~ C IRn we consider the set
U K(b),
bEf~
where K(b) C ]I-In is the cone of opening angle 1 with top in b: g(b)-
{(/3, a ) . ]j3- b] <_ a).
We call this set the influence region of gt in the upper half-space. general theorem can now be stated as follows.
The
T h e o r e m 4. Let f~ C lR n be open. Suppose that .=. c ]Hn is well separated from the influence region of ft. Suppose that E' C .= such that =' and Ec are well separated. Suppose that 71 is a tempered distribution that satisfies inside f~ at
~-
f
for some distribution f E $(IR n) supported by-~. If f satisfies at
Mg~w~f e B ( ~ ) , with some admissible wavelet g E So(lRn), then it follows that A4gE'Wgri e (a2B)(lRn) . Proofi W e m a y suppose that g E $0(]R n) is spherically symmetric, g g([xl). Then with h - Ag we may write
wgA~ - -a-2Wh~. Now g and h are both admissible in the sense that
/o
l~(ak)[2 da --~ a
/o
o0
h(ak)
!
2 da --~1. a
564
M. Holschneider
From this it follows immediately that s E ,9~(IRa), and As satisfying
implies that s satisfies
The theorem is therefore proved if we can show that any distribution f ' that coincides with f inside fl satisfies again Mg~_WgAf e BORn). But this is follows from the next lemma, which justifies the name "influence region of ft." L e m m a 6. Let E C ]Hn be well separated from the influence region of f~. Then for all p E ,9 ~(IRn) with support in fl we have that
Mg-=w~p e So(~t~). Proof: By hypothesis there is an e > 0 such that for (b, a) E re(E) there is an n-dimensional Euclidian ball of radius > A((b,a)) ~ around b that is contained in the complement of the influence region of ft. Denote by r aC ~ function which is identically 1 on the complement of the unit ball of radius 1 and which is supported on the complement of a slightly smaller ball. Then denote by Cb,a the family of translates and dilates of r Therefore if p is supported by fl, it follows that
(b, ~) e ~ ~ W~p(b, ~) = W~(r
~) =
But now for every e > 0 we have that gb,aCb,a((b,a))~ tends to 0 as A((b, a)) ~ cc in S(lRn), in such a way that for all semi-norms in S0(lR n) and all M > 0 we have
I[gb,oOb,~((b,o))" [l,,~ <-- ~,,~,~A((b, ~))-~. This finishes the proofs of the lemma and the theorem, m References [1] Coifman, R. and Y. Meyer, Ondelettes et Op~rateurs III, Hermann, Paris, 1990.
565
Directional Elliptic Regularity
[2] Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [3] Grossmann, A. and J. Morlet, Decomposition of Hardy functions into square integrable Wavelets of Constant Shape, SIAM J. Math. Anal. 15 (1984), 723-736. [4] Holschneider, M., Press, 1995.
Wavelets: An Analysis Tool,
Oxford University
[5] Holschneider, M., Wavelet Analysis of Partial Differential Operators, in SchrSdinger Operators, Markov Semigroups, Wavelet Analysis, Operator Algebras, M. Demuth, E. Schrohe, B.-W. Schulze, and J. SjSstrand (eds.), Akademie-Verlag, Berlin, 1996, pp. 218-347.
[6] HSrmander, L.,
The Analysis of Linear Partial Differential Operators I, Springer Verlag, Berlin, 1983.
[7] Meyer, Y., Ondelettes et Op~rateurs ISzII. Hermann, 1990. [8] Torr~sani, B., Analyse Continue par Ondelettes, CNRS Editions, 1995. [9] Triebel, H., Theory of Function Spaces II, Birkh/iuser, Basel, 1992. Matthias Holschneider CNRS CPT Luminy Case 907 F-13288 Marseille [email protected]
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Subject Index A A posteriori error 52, 113, 122 Adaptive algorithms 138, 141, 142, 144, 158, 194, 238, 260, 414 Adaptive approximation 237, 261 Adaptive refinement 36, 47, 49, 276, 279 Additive Schwarz 8, 16, 24, 33, 35, 43, 348, 353, 357, 371,377, 419 Asymptotically optimal 4, 383, 388, 399, 404, 413 B
Bandwidth 399 Basis Haar 147 orthonormal 64, 73, 145,201,208 299-301,365, 477, 499, 519 Riesz 7, 40, 60, 65-67, 115, 145, 243-245, 370, 390, 421 wavelet packet 204 Bilinear form 7, 14, 43, 65-71, 351,419 Biorthogonal multiresolution analysis 390 wavelet 6, 246-248, 403 BMO 501,506-509, 528, 533 Boundary condition Dirichlet 41, 68, 208, 291,427, 537 essential 41 Neumann 24, 208, 291-292, 298, 421,428, 434 Boundary differential operator 291 567
Boundary integral equation 241,249, 253, 287, 292, 294-296, 384 Boundary manifold 287-288, 296 Bounded linear functional 24 C Cholesky decomposition 369-371 Coherence 474, 476 Collocation 111-113, 132, 224, 226 Commutator 508, 525 Compact support 243, 365, 415,448 Compactly supported wavelet 146, 162, 415, 519 Complement 60, 74, 110, 201,273, 300, 350, 364, 390, 516, 541,560 Compression 4, 46, 149, 183, 206, 271,287-289, 372, 413, 473 Computational complexity 287, 324, 369 Condition number 5, 33, 63-67, 254, 302, 402, 414, 433-435 Conditioning 61, 67, 71,429 Convergence analysis 99, 369 rate 9, 46, 67, 263, 287, 336, 350, 361,530 Convolution 152, 366, 542 Correlation 442, 465 D Daubechies wavelet 365, 373, 422 Decomposition algorithm 156, 244
568
Subject Index
Differential operator 94, 118, 138, 241,291,363, 495, 517 Dimension theoretical 474, 484-489 Domain decomposition 7, 98, 251 Double layer potential 240, 287, 295 Dual basis 273 function 243 wavelet 256 Duality 421,547 Dyadic cube 242, 260, 416, 499, 522 interval 477 E
Eigenvalue 9, 88, 178, 350, 505 Eigenvector 395 Elliptic operator 86, 237, 337, 384, 498, 562 Enstrophy 480, 484, 486 Entropy 200, 212-214, 231,474, 478 Error estimators 404 indicators 50, 52 Evolution equation 123, 140, 193, 449, 478, 485 F Fast wavelet transform 111 i 288 Filter quadrature mirror 146, 176, 203 Finite element 4, 41, 59-61, 92, 117, 141,238, 383, 415, 435 Function Haar 114, 147
orthonormal scaling 115 G Galerkin adaptive schemes 132 Galerkin method 62, 120, 254, 287, 414, 482, 516 Galerkin scheme 110, 132, 255, 290, 324, 334, 497, 516, 537 Generator 208, 391,401 H
Haar 114, 288, 364, 524 Hierarchical basis 9, 74-77, 122 Hilbert transform 139, 159, 162
Integral equation 4, 142, 193, 287, 349, 402 operator 139, 288-291,362, 498 Interpolation operator 76, 111, 119 property 114 Inverse Fourier transform 161,455 K
Kernel 152, 241, 293, 362, 524, 546 559 L Lacunary 199 Lagrange interpolation 111 Local linear independence 17, 44, 415 Localization 233, 444, 558 Lossy compression 473
569
Subject Index M Mask coefficients 16, 391-393, 422 Modulus of continuity 418 Multigrid 4, 125, 414, 427 Multilevel 4, 52, 122, 238, 348-349, 356, 358, 376, 412, 422, 435 Multiplicative algorithm 10, 61,348, 377 Multiresolution 5, 60, 138, 151,201, 289, 383, 422, 497, 536 Multiscale decomposition 111, 115, 390 Multivariate 242, 385, 427 Multiwavelet 40, 287, 288-290, 336 N Nested refinement 6, 28 Neumann problem 291,298, 434 Nodal basis 5, 14, 60, 86, 111,420, 433 Nodal value 18 Norm estimate 22, 38
358, 406, 420, 433 Preconditioning additive 101 Principal value 298, 326 Projection 50, 74-78, 110, 158, 206, 206, 299, 348, 360, 418, 448, 489, 499, 536 Prolongation 403, 406
Q Quadrature formula 121,142, 309, 414 R
Reconstruction algorithm 156, 170, 201 Refinable function 389, 393-396, 407 integral 395, 405 Relation 18, 60, 115, 146, 204, 255, 355, 462, 499, 552, 556 Riesz basis 7, 16, 65-67, 84, 115, 145, 243, 370, 390, 421
O Operator convolution 152, 160, 557 Optimal approximation 23, 46, 413 P Parametrix 497, 510, 530 Partitions 16, 73, 256, 316, 518 Phase atoms 473, 484, 485, 489 Positive definite 8, 32, 68, 292, 351,384, 502 Power spectral density 445 Preconditioner 4, 45, 87, 100,
Sampling 211 Scaling function 5, 40, 110, 145, 363-367, 414-416, 428, 433, 519, 532, 536 Schwarz operator 8, 35, 43 Sequence 5, 46, 72, 116, 145, 204, 474, 502, 532, 560, 561 Shannon entropy 206, 212-214 Shift invariant 152, 243 Shocks 139, 164, 188 Single layer 241,294 Singular integral 289, 309, 324, 337 Singularity 109, 126, 322, 479, 542
570
Subject Index
Sobolev estimate 513 exponent 421 norm 23, 248, 301 Spectral density 445 theory 352 norm 255, 403 Spline wavelet 211,365-368, 448 Square integrable 297 Stable splitting 4, 15 Stiffness matrix 34, 59, 70, 120, 224, 255, 288, 316, 387, 420 Subdivision interpolatory 115 scheme 115 Subspace splitting 7, 36, 362, 419 Support 19, 53, 94, 171,217, 275, 349, 390, 415, 448, 497, 562 Symbol 76, 292, 394, 502, 533 T Thresholding 142 Tikhonov regularization 347 Transform Fourier 140, 164, 216, 245, 446, 460, 505, 541 Hilbert 139, 162 wavelet 111, 122, 149, 288, 505, 542, 562 windowed Fourier 454 Truncation strategy 303 Turbulent flow 200, 233, 480, 486
U Unconditional basis 499, 524 V Vaguelette 497 Vanishing moment 146, 157, 176, 244, 275, 289, 302, 366, 480, 537 VMO 496-498, 509, 530, 534 Vorticity 480, 487 W Waveform 475, 489 Wavelength 446, 460 Wavelet decomposition 4, 111,152, 176, 238, 444, 452, 485 expansion 138, 156, 175, 237, 384, 488 representation 141, 153, 174, 185 transform 111, 149, 288, 505, 543, 562 Wavelet packet basis 204 orthogonal 477, 486 Wavelets biorthogonal 242, 246 Daubechies 365,, 422 Haar 288 orthogonal 5 periodic 444
W A V E L E T A N A L Y S I S A N D ITS A P P L I C A T I O N S CHARLES K. CHUI, SERIES EDITOR
1. Charles K. Chui, An Introduction to Wavelets 2. Charles K. Chui, ed., Wavelets: A Tutorial in Theory and Appfications 3. Larry L. Schumaker and Glenn Webb, eds., Recent Advances in Wavelet Analysis 4. Eft Foufoula-Georgiou and Praveen Kumar, eds., Wavelets in Geophysics 5. Charles K. Chui, Laura Montefusco, and Luigia Puccio, eds., Wavelets: Theory, Algorithms, and Applications 6. Wolfgang Dahmen, Andrew J. Kurdila, and Peter Oswald, eds., Multirate Wavelet Methods for PDEs
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