Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

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F R O N T I E R S IN APPLIED MATHEMATICS The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.

EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo Chavez, Cornell University Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI) Pat Hagan, NumeriX, New York Matthias Heinkenschloss, Rice University Belinda King, Virginia Polytechnic Institute and State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph Smith, North Carolina State University Anna Tsao, Institute for Defense Analyses, Center for Computing Sciences

BOOKS PUBLISHED IN FRONTIERS IN A P P L I E D MATHEMATICS

Banks, H.T., Buksas, M.W., and Lin, T., Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles andTechniques of Algorithmic Differentiation Kelley, C. T., Iterative Methods for Optimization Greenbaum.Anne, Iterative Methods for Solving Linear Systems Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMG:A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Rude, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel, Sabine and Vandewalle, Joos, The Total Least Squares Problem: Computational Aspects and Analysis Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F., Multigrid Methods Buckmaster.John D., The Mathematics of Combustion Ewing, Richard E., The Mathematics of Reservoir Simulation

Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

H.T. Banks North Carolina State University Raleigh, North Carolina

M.W. Buksas Los Alamos National Laboratory Los Alamos, New Mexico

T.Lin Virginia Polytechnic Institute and State University Blacksburg, Virginia

Siam Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2000 by the Society for Industrial and Applied Mathematics. 1098765432 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication Data Banks, H.T. Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts / H.T. Banks, M.W. Buksas.T. Lin. p. cm.– (Frontiers in applied mathematics ;21) Includes bibliographical references and index. ISBN 0-89871-459-1 1.Acoustic emission testing. 2. Electromagnetic waves-Scattering. 3. Surfaces (Physics) I. Buksas, M.W. II. Lin.T. Ill.Title. IV. Series TA4I8.84.B362000 620.1'127-dc2l 00-032944

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trademark.

Contents Foreword

ix

Preface

xi

1 Introduction

1

2 Problem Formulation and Physical Modeling 2.1 Motivation 2.2 Estimation Methodology 2.3 Reduction to Specific Problems 2.3.1 Acoustic Reflectors

7 7 15 18 23

3 Well-Posedness 3.1 A Variational Formulation 3.2 A Semigroup Formulation 3.3 Enhanced Regularity of Solutions 3.4 Convergence of Finite Element Approximations 3.5 Inverse Problem Methodology

27 27 35 42 47 48

4 Computational Methods for Dielectrics with Supraconductive Backing 4.1 The Forward Problem for the Debye Polarization Model 4.2 The Inverse Problem for a Debye Medium 4.2.1 Sample Results 4.2.2 Reconstruction with Inferior Accuracy 4.2.3 Identification of Material Depth 4.2.4 Results from the Two-Step Algorithm 4.2.5 Estimating Depth and Physical Parameters 4.2.6 Results for the Simultaneous Estimation of Depth and Dielectric Parameters 4.3 The Forward Problem for the Lorentz Model 4.4 The Inverse Problem for a Lorentz Medium vii

57 57 63 65 71 72 74 76 77 79 85

viii

Contents

5 Computational Methods for General Polarization Models 89 5.1 The Forward Problem 89 5.1.1 Galerkin Methods for the History Approximation 91 5.1.2 Approximating the History ofe ( t ) 92 5.1.3 Implementing the Hysteresis Term 95 5.1.4 Specific Implementation: Constant Material Parameters 96 5.2 Results of Simulations with the General Model 97 5.2.1 Dielectric Response Function Representation of the Debye Model 97 5.2.2 DRF Representation of the Lorentz Model 99 5.3 The Inverse Problem with the General Polarization Model . . . . 101 5.3.1 Practical Conclusions 105 6 Computational Methods for Acoustically Backed Dielectrics 6.1 The Forward Problem with an Acoustically Backed Layer 6.1.1 Numerical Methods 6.1.2 Simulation Results 6.2 Identification of Material Depth 7 Concluding Summary and Remarks on Potential Applications 7.1 Summary of Mathematical and Computational Results 7.2 Limitations and Unanswered Questions 7.3 Future Technological Possibilities

107 107 108 109 119 129 129 131 133

Bibliography

137

Index

145

Foreword

The human extension of visual ability has been very important to technology, society, and culture. This extension of vision has been to the very small, the distant, and the internal. Microscopy—viewing the small—continues to open new doors into the structure of materials, including biological materials in their relation to human disease. Radar systems and other airborne sensor modalities enable environmental monitoring important to the health of our planet and essential to flying safety. Internal imaging is central to medical diagnosis, on one hand, and to the nondestructive assessment of the integrity of nonliving structures, on the other. This volume in the SIAM Frontiers in Applied Mathematics series is a milestone in the human endeavor to extend vision. It is concerned with imaging the interior of materials or objects and shows, in a systematic way, how internal reflecting surfaces can be exploited to aid the process of "seeing within." Convincing evidence is presented that highly reflective items, such as metal surfaces, can substantially aid characterization of materials interrogated by short electromagnetic pulses. Also, and perhaps most intriguing, the work herein suggests that slowly moving acoustic wave structures can also serve as useful reflecting surfaces and can be tracked by fast moving electromagnetic pulses to enhance internal visualization. In short, I believe that this volume can foster a significant increment in imaging technology using low energy electromagnetic radiation. The use of electromagnetic pulses interacting with specially placed reflective surfaces, whether solid or acoustic, is a new dimension that will substantively impact medical imaging, subsoil investigation, and structure evaluation. The text is accessible to the advanced undergraduate or early graduate engineering,bioengineering, geology, and mathematics or physics student. Its strengths include a clear discussion of materials properties from the electromagnetic point of view, a careful formulation of the imaging problems addressed, solid treatment of mathematical issues, and useful illustration of computational methods and results. While confined to internal vision in onedimensional settings, this volume will stimulate further developments in internal vision to include two- and three-dimensional interior assessments. It is an excellent and robust source of applied mathematics and engineering research challenges for the future.

ix

In the next quarter-century one can hope for the early diagnosis of cancer in such inaccessible organs as the ovary and the pancreas to enable curative action, and for improved, more convenient intestinal, heart, and breast surveillance. Imaging of structures in our cities, including bridges, buildings, road beds, and tunnel supports, must progress if structural aging processes are to be detected and arrested. Soil, lake, and river imaging is essential for pollution control and remedy. "Internal seeing" will be increasingly important in law enforcement in the search for concealed weapons, explosives, and biological or chemical threats. This volume materially addresses these current and emerging needs.

Richard Albanese U.S. Air Force Research Laboratory Brooks Air Force Base San Antonio, Texas

x

Preface This monograph contributes to the general area of electromagnetic imaging. Our treatment has several distinguishing features. First we explore with some care the possibility of using interfaces, either with supraconductive boundary materials or with acoustic wavefronts, as reflectors to determine geometry as well as dielectric material characteristics of a "target." Moreover, we use windowed microwave pulses from an antenna-like source to provide nonharmonic time dependent interrogating signals. These are inputs to a general time dependent form of Maxwell's equations containing Ohmic conductivity along with quite general material polarization constitutive laws to represent dispersiveness in dielectric media. Mathematically, we offer a rigorous treatment of the resulting integro-partial differential equation system with delta function inputs in a variational framework. We provide both theoretical and computational analysis in the context of inverse or parameter estimation problems. Our efforts are at the level of fundamental research on the basic question: Can supraconductive interfaces or acoustic waves be used as "electromagnetic mirrors" in imaging scenarios? While we believe we provide an affirmative answer to this question in the abstract, we also believe our specific examples provide computational validation that such an approach holds direct promise for several classes of interrogation applications (these are discussed in Chapter 7). The literature addressing electromagnetic imaging is extensive and the mass of relevant information can be viewed as even larger by including work on acoustic imaging. However, we are unaware of texts or research articles that systematically exploit interface phenomena, the electrodynamics of material responses, and time dependent interrogating signals in an integrated manner. We perceive that it is the variational framework of our effort that permits this integrated approach. We hope that this integrated approach will evolve into a standard for the field with particular extension to more complex targets. Electromagnetic theory offers fascination and challenge from both a physical and mathematical perspective. For the nonspecialist in electromagnetic theory (with which the authors of this monograph readily identify themselves) an effort in electromagnetic imaging can appear daunting. However, we have tried xi

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Preface

to make our presentation accessible to a wide audience (the mature upperclassman, and especially graduate students or postdoctoral students in mathematics, physics, and engineering). We do not derive the free space Maxwell's equations from first principles (i.e., Coulomb's law plus relativity theory), but we do summarize the needed background in Chapter 2, where we formulate the basic mathematics and physics associated with material responses and imaging. Our brief literature survey in Chapter 1 can be skipped on a first reading, but a reading of Chapter 2 is an essential prerequisite for the remainder of the monograph. For the theoretical foundations of Chapter 3 (the details of which can be skipped or read only cursorily), we assume a background of analysis usually taught in a first graduate course in functional analysis. The computational material which is the heart of Chapters 4, 5, and 6 should be accessible to anyone with a standard course in numerical analysis which includes some numerical optimization. Finally, an elementary understanding of electromagnetics and acoustics would be helpful but we feel it is not necessary if leaders are willing to either accept our summaries of the physics in Chapters 2 and 6 or to pursue extra reading (references are given throughout the presentation) on their own. The first author has (successfully he thinks!) covered the material in Chapter 2 with graduate students who had no special electromagnetic training. We believe that electromagnetic imaging provides exciting challenges to which modern mathematicians, engineers, and physicists with computational skills can contribute at the frontiers. The tremendous advances in mathematical and computational capabilities over the past several decades permit consideration of time domian problems that were completely out of reach for our predecessors. This monograph represents our attempt to share our enthusiasm for exciting scientific opportunities in electromagnetic imaging, and, hopefully, to stimulate others to join the pursuit. This monograph grew out of the efforts of the second author in his Ph.D. thesis of August 1998 at North Carolina State University and several related research projects of the first author. The major stimulation in all cases came from Dr. Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB, San Antonio, Texas. The authors are deeply indebted to Dr. Albanese for his continual encouragement and numerous specific technical discussions throughout the course of the research discussed in this monograph. His many suggestions and questions provided significant impetus for our research efforts. Moreover, he read with extreme care several earlier versions of the manuscript. His thoughtful challenges and comments always improved our presentations and in some cases provoked us to obtain new results. Finally, he made a major contribution to the speculative comments in Chapter 7. (Dick, it is indeed exciting and great fun to dream with you!) We would also like to thank Dr. Mac Hyman of Los Alamos National Laboratory for his support of the second author during the completion of this manuscript. His encouragement and guidance have been most helpful.

Preface

xiii

Our research has been strongly supported by the U.S. Air Force Office of Scientific Research under grants AFOSR F49620-95-1-0236, AFOSR F49620-981-0180, AFOSR F49620-95-1-0375; in part by the U.S. Department of Education under a GAANN Fellowship (MWB) through grant P200A40730; and the Department of Energy (MWB) under contract W-7405-ENG-36. The AFOSR grants were under the auspices of Dr. Marc Q. Jacobs, who as a Program Manager has unfailingly encouraged our (and many others') serious involvement with AFRL scientists and engineers through the years. For this we are most grateful. A number of postdocs and graduate students (particular thanks to Julie Raye) in the Center for Research in Scientific Computation made suggestions and corrected typos in earlier versions. Finally, we wish to acknowledge a debt to Dr. Yun Wang. It was through her enthusiasm for electromagnetic problems while working with Dr. Albanese at Brooks AFB that our collaboration with Dr. Albanese and his group began.

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Chapter 1 Introduction A survey of the mathematical literature reveals considerable interest in the identification of material parameters describing electromagnetic phenomenon. For our purposes, we categorize the materials and the models employed to describe them as either dispersive or nondispersive, where dispersive materials are denned as those in which planar electromagnetic waves propagate with phase velocities that depend on the frequency of the waves. Thus an incident transient pulse in a dispersive medium will spread and change shape, even in a homogeneous medium. When modeled in the frequency domain, this is manifested as parameters which depend explicitly on frequency. In time domain models, the same phenomenon can be captured with constitutive laws in which the electric and/or magnetic polarizations are expressed in terms of the convolution of the history of the electric and magnetic fields. The equivalence of the two in the case of electric polarization dispersion is shown by Jackson [Jac75, p. 306]. Alternatively, one can introduce dispersiveness through Ohmic conductivity, or more generally, by a conductivity denned via a convolution with the electric field [APM89]. Simple Ohmic conductivity results in a dissipation term in the electromagnetic equations. In identification of electromagnetic material parameters, the emphasis has been on one-dimensional scattering problems where planar electromagnetic waves impinge on dielectric slabs. In a series of four papers [KK86a, KK86b, KK87, KK89] Kristenson and Krueger examine this problem with a wave splitting technique and the derivation of scattering operators which satisfy imbedding equations. A nice summary of this general technique involving invariant imbedding coupled with wave splitting (discussed also in the earlier papers [CK83, CDK83]) can be found in [AMP94]. The reconstruction of the functions representing the physical parameters is carried out via the deconvolution of the imbedding equations which relate incident and reflected waves through a convolution kernel. The physical model in these papers includes dissipativeness via Ohmic conduc1

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Banks, Buksas, and Lin

tivity but no polarization-induced dispersiveness; it also covers stratified media, meaning that the material is inhomogeneous only in the direction of the propagation of the waves. Scattering techniques using wave splitting were applied to a general hyperbolic model by Weston [Wes72], who considered a dissipative wave equation equivalent to the problem of planar waves in stratified media. These results were extended by Krueger [Kru76, Kru78, Kru81] to cover media in multiple slabs, thus containing multiple discontunities in the material parameters. Corones and Sun [CS93] used the same method of wave splitting and invariant imbedding to reconstruct coefficients in a one dimensional wave equation with a smooth source term. In another paper, He and Strom [HS91] also employed wave splitting in a scattering problem for stratified materials illuminated with waves generated by a magnetic dipole. Some progress has also been made with more general geometries, largely due to the increasing sophistication of wave splitting techniques in higher dimensions. In a paper by Weston [Wes88], a decomposition of solutions of the dissipative wave equation in R3 is given and integro-differential equations are derived for the reflection operator. The reconstruction of the velocity and dissipation coefficients from the kernel of this operator are demonstrated. Inverse problems involving polarization-based dispersive materials follow a similar pattern of development in moving from one dimensional scattering with planar waves to more general settings. Beezley and Krueger [BK85] began investigation into these problems in one dimension by employing a method similar to that described above for conductivity-based dissipative wave equations. Imbedding equations are derived for the reflection operator relating the incident and scattered parts of the split wave solution. The dielectric response kernel is reconstructed from the imbedding equations derived for homogeneous semiinfinite and finite slabs. The reconstruction is also carried out numerically in the presence of noise applied to the reflection kernel. Well-posedness results for these problems involving wave splitting and imbedding equations are developed by Bui [Bui95] in which a time domain model involving both conductivity and electric polarization is considered. Under stringent smoothness (regularity) assumptions on the electric field and on the displacement and conductivity susceptibility kernels, theorems governing the existence, uniqueness, and continuous dependence of solutions (to the imbedding equations) in the context of both the forward and inverse problems are given. In [Sun92] Sun combined wave splitting/invariant imbedding techniques with Green's function methods to identify the source current embedded inside a dispersive material using a time-domain approach in systems similar to those of Beezley and Krueger. In a paper by Lerche [Ler86], a different integral equation is derived relating the dielectric response function to an operator representing the frequency domain absorption characteristics of a material. Wolfers-

Introduction

3

dorf [Wol91] extends these results to finite and semi-infinite slabs and derives exact solutions to the integral equations for the dielectric response. A geometry different from the usual slab was considered by Kreider [Kre89], who posed the problem of reconstruction of the displacement susceptibility kernel from reflected data in a stratified cylinder. The problem is, however, rendered one-dimensional through angular symmetry of the cylinder and the fields. The dispersive material is permitted inhomogeneities in space, although the dielectric parameter is restricted to functions which are separable between the variables of space and frequency: e(x,w;) = e\(x)E 2 ((w]. Another treatment which allows for spatially inhomogeneous dispersive materials is considered in [HFL96]. In this case an optimization approach is applied to match simulations to experimental as well as synthetic data. The constitutive equations permit convolution terms (i.e., displacement and conductivity susceptibility kernels) in both the electric polarization and the conductivity. The limitations of simultaneous reconstruction of the spatial and time-varying parts of the kernel functions is considered when both transmission and reflection data are available. Another inhomogeneous two-dimensional problem is considered by Colton and Monk [CM94a, CM95] who use a frequency domain approach in modeling the interaction of electromagnetic waves with human tissue for the detection of leukemia in bone marrow. The geometry of the medium is presumed known and the two dimensional domain is further partitioned into subdomains of known geometry. The only unknowns are the constitutive parameters describing the bone marrow, and these are considered functions of space as well. The reconstruction of the bone marrow parameters is demonstrated and is shown to be robust in the presence of relative noise with magnitude as high as 1%. The methods in [CM94a, CM95] are versions of a class of dual space methods that have been proposed for solving inverse scattering problems in the frequency domain for an inhomogeneous body [CM94b, CK92]. A succinct summary of Herglotz kernel based versions of this approach can be found in [AMP94]. Among other fundamental efforts in the frequency domain are those of Oughstun and Sherman [SO81, OS88] who employed improved asymptotic analysis to extend the classical theory of Sommerfeld and Brillouin on propagation of electromagnetic pulses in dispersive Lorentz media. A problem directly motivated by an application is given in [RGKM97], where time domain models with polarization and conductivity operators were formulated for the penetration of radar waves in soil. The ground is represented by a stratified semi-infinite slab and electromagnetic signals are generated by a circular loop of current. Under stringent regularity conditions on parameters and kernels, unique determination of dielectric parameters is guaranteed for a sufficiently rich set of observations. Finally, we note that inverse scattering problems involving the the geometrical reconstruction of a body with known dielectric properties have been widely

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Banks, Buksas, and Lin

studied (e.g., see [BKP97, vdBCK95] and the references therein) using integral equation methods connecting incident and reflected wave fields. In this monograph we work exclusively in the time domain and formulate both forward and inverse problems for planar waves incident on layered strata. Using the full time varying Maxwell's equations, we incorporate very general polarization constitutive laws. These convolution formulations are sufficiently general to include all known or hypothetical polarization mechanisms as special cases. In our model, we employ a source term that can represent an antenna producing a point source spatial input consisting of pulsed microwaves. In a variational formulation suitable for both theoretical and computational analysis we investigate two types of inverse problems. In the first we assume that a layer of dielectric material with unknown dielectric parameters and geometry (thickness) has a back boundary of supraconductive (e.g., metal) material. The inverse problem consists of estimating dielectric properties and geometry from reflections of incident microwave pulses. In the second problem, the supraconductive back boundary is replaced by a second layer of known dielectric properties and an acoustic reflector or mirror. A chapter is devoted to a rather thorough theoretical treatment of the problems with supraconductive back boundary in a variational formulation. Existence, uniqueness, and continuous dependence results for the forward problems are given. A related semigroup formulation is discussed. For inverse problems we present theoretical foundations (including a convergence framework for approximation methods) which lead to existence and continuous dependence (on data) of estimates. We develop computational methods based on finite elements in space and finite differences in time, and we use these to demonstrate that microwave pulses from the antenna point source can produce Brillouin or Brillouin and Sommerfeld precursors in Debye or Lorentz media, respectively, similar to those found using asymptotic analysis or frequency domain techniques. In a series of computational examples, we investigate the feasibility of estimating dielectric parameters and geometry using observations of electric field reflections from dispersive layers. Emphasis is on estimating conductivity and polarization constitutive laws (Debye, Lorentz, or more general media represented by a general dielectric response function, i.e., a displacement susceptibility convolution kernel). We develop and test a two-step algorithm in which reflections from the first interface in a stratified media are used to obtain initial estimates of dielectric parameters. Reflections of the media propagated field from a supraconductive back interface are used to estimate layer thickness and to refine the initial estimates of the dielectric parameters for the layer. A modification of the inverse problem wherein one replaces the supraconductive back interface by a stationary standing acoustic wave is then investigated. It is demonstrated that this acoustic wave can be tuned to enhance reflections from the second interface to permit useful application of the two-step algorithm.

Introduction

5

In the closing chapter we discuss the importance of and potential applications (biomedical, civilian, and military) for the techniques developed in this monograph. In closing this introductory chapter, we note the differences between the physical models and the solution methods employed in some of the literature cited above and our discussions which follow. A number of these contributions differ from ours in that they deal only with dissipative materials. This eliminates the need to reconstruct parameters which are functions of frequency or convolution kernels which are functions of time. Some do consider materials which are inhomogeneous in the space variable, however, and while the model formulation is in the time domain, the forward and inverse problems in many are solved by wave-splitting and invariant imbedding techniques. A number of these papers do consider physical problems similar to ours, in which planar electromagnetic waves impinge normally on slabs of material. Among the references cited which employ dispersive models, many differ from ours by using a frequency domain approach which is best suited for physical problems involving time harmonic solutions. Among those treatments which use time domain models, most again use the wave splitting approach popular with the dissipative materials to formulate the forward and inverse problems. Only one contribution [HFL96] uses an optimization approach and formulates an inverse problem using the time domain data itself. While this paper and others consider inhomogeneous materials, the geometry of the material slab is considered to be known. This differs fundamentally from our problems in that we do not assume the thickness of the material slab is known a priori and attempt to identify this dimension along with the parameters describing the electromagnetic material properties.

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Chapter 2

Problem Formulation and Physical Modeling 2.1

Motivation

The ability to interrogate the interior of tissues and other materials has wide ranging applications to medical imaging, the early detection of anomalies, and other problems in nondestructive interrogation of sensitive materials. Noninvasive interrogating techniques are most valuable in determining substructure in biological tissues due to the fact that they usually result in much less discomfort in subjects. Low amplitude or low energy microwaves (electromagnetic waves in the frequency range of 3 to 300 GHz) can pass through many media without causing any known damage. On the other hand, chemical and physical changes in biological tissue can result in changes in its electromagnetic characteristics such as electric and magnetic polarization mechanisms and conductivity. Furthermore, it is hoped that the in vivo electromagnetic characteristics of tissues and organs can be correlated with metabolic functioning. Hence the accurate determination of these dielectric properties can be employed in the evaluation of the functional integrity of tissues and organs in subjects. Other applications in biology and the environmental sciences of the use of microwaves in noninvasive interrogation procedures can be found in a recently published review article [AMP94]. Additional potential applications for the interrogation ideas discussed below include nondestructive damage detection in aircraft, mine, ordinance and camouflage detection, and subsurface and atmospheric environmental modeling. Use of ultrasonic waves is another popular technique employed in noninvasive interrogation of media in both industrial and medical applications. It has been well known since 1922 [Bri22] that electromagnetic and sound waves can interact in a medium and influence each other's propagation. This interaction has been 7

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Figure 2.1: Schematic diagram of geometry. the subject of substantial investigation in acoustooptics [DD91, Kor97, XS92], and numerous acoustooptic devices have been developed in many applications in industry—neural nets, optical excision, and fiber optics, to name just a few. The goals of electromagnetic interrogation as presented here are the determination of both the geometry and dielectric properties of the materials under investigation. We consider the generalized problem depicted schematically in Figure 2.1. The domain £7 of the object under consideration has both a known and an unknown portion of the boundary. The unknown portion F(g) is presumed to be backed by a supraconductive material with an effectively infinite conductivity. On this boundary with outer normal n, we thus have E x n — 0 and B • n = 0. Note that the unknown nature of the boundary is represented by its dependence on a set of parameters q which are to be determined to establish the geometry of the object. Alternatively, we may suppose that the supraconductive backing material is replaced by an interface with an acoustically excited medium. Reflections from the interior of the domain £7 would then be created by changes in material properties induced by the acoustic wave. The location of these changes in the material properties would then be described in analogous fashion by the parameters q. The electric and magnetic fields inside i7 and exterior to 17 (this region will be denoted 17o) are governed by the macroscopic Maxwell's equations [Jac75, Bal89,

Problem Formulation and Physical Modeling

9

Str41, E1193]. To describe the electromagnetic behavior of complex materials, we express Maxwell's equations in a general form which includes terms for electric and magnetic polarization. We have

The vector-valued functions E and H represent the strengths of the electric and magnetic fields, respectively, while D and B are the electric and magnetic flux densities, respectively. The two current contributions are denoted by Jc, the conduction current density, and Js, a source current density. The electric and magnetic polarizations are represented by P and M, respectively. The scalar quantity p represents the density of free electric charges unaccounted for in the electric polarization. The three quantities M, P, and Jc embody the behavior of the material in response to the electromagnetic fields. Additional material dependent equations (constitutive laws) are required to determine their dependence on the components of the fields E and H. The dependence of these equations on the material is reflected both in the choice of a mathematical model and in the parameters (possibly operators) appearing in the model. Estimation of these parameters or operators are the goals in the inverse problems we formulate below. The region OQ external to the medium is treated as empty space and is devoid of conductivity or polarization effects, hence M = 0, P = 0, and Jc = 0 in Slo, and all of the necessary parameters for the determination of the fields are assumed known in this domain. The source current density term Js will also be nonzero only at points in QQ. The presence of any time varying vector valued current source will generate the electromagnetic waves in this domain which illuminate the target medium fi. We make certain assumptions about the material which are reflected in the constitutive relations in the domain Q. For the media of interest to us, we can neglect magnetic effects; we also assume that Ohm's law governs the electric conductivitv. Hence for x & £1

For dispersive media, it is generally recognized [AMP94, APM89, Bal89, Jac75, E1193] that frequency dependence of conductivity a as well as the dielectric permittivity e (in a displacement constitutive law D = eE to be dis-

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cussed momentarily) is important. This is often [APM89, Jac75, HFL96, BBOO] treated by assuming a nonlocality in time through introduction of conductive and electric susceptibility kernels in convolution relationships for Jc — JC[E] and P = P[E]. We will do precisely this below in our treatment of electric polarization. However, following arguments in [APM89], one can establish that introducing frequency dependence via susceptibility kernels in the polarization automatically results in frequency dependence of the conductivity even if one uses the simple form of Ohm's law given above (i.e., assuming instantaneous or local-in-time dependence of Jc on the electric field E). We therefore shall not, in our treatment here, assume a more complicated relationship for conductivity even though the ideas and methods presented in this monograph could readily be used to do so. This is not necessary to obtain the desired frequency dependence of conductivity and it would add to the computational burden in the inverse problems we treat here. Moreover, it is not at all clear that one could separately estimate frequency dependence of conductivity and polarization using observations of the macroscopic electric field (i.e., additional nonuniqueness of parameters in the inverse or parameter estimation problems would be introduced). Thus, with no real loss of generality and for simplicity, we restrict ourselves to the usual Ohmic conductivity in our discussion in this monograph. To describe the behavior of the media's macroscopic electric polarization P, we employ a general integral equation model in which the polarization explicitly depends on the past history of the electric field. This model is sufficiently general to include microscopic polarization mechanisms such as dipole or orientational polarization as well as ionic and electronic polarization [Bal89, Fr658, E1193] and other frequency dependent polarization mechanisms. The resulting constitutive law can be given in terms of a polarization or displacement susceptibility kernel g (which we shall sometimes refer to as the dielectric response function or DRF)

by

We note that this model presupposes that P(0, x] = 0. In the electromagnetic literature (e.g., see [Jac75, BK85, APM89, E1193]), the relationship is often expressed as

in the case of E(t,x] = 0 for t < 0, which is of interest here. These are both related to our formulation by the simple change of variables: s = t — £. We prefer our form of the equation since then any time derivatives of P are borne

Problem Formulation and Physical Modeling

11

by the kernel function g and not the variable E. Specifically, under (2.1) the term Q^-(t,x) — P(t,x), which will appear in subsequent equations, is given by

while the more traditional representation leads to

The presence of E under the integral term in (2.3) complicates the analysis and solution of the Maxwell's equations considerably. Although the first formulation leads to the additional terms g(0, x)E(t, x) and g(0, x)E(t, x), we shall see below that these terms cause no increase in the complexity of the problem analysis or computation. We note that an attempt to include a component of the polarization which depends on the instantaneous value of the electric field would add a delta function in the time variable to the DRF g(s, x). This introduces some mathematical complexities which, for simplicity, we avoid by treating instantaneous polarization when it arises in a different, but completely equivalent, manner. Allowing the instantaneous component of the polarization to be related to the electric field by a dielectric constant so that P;n = eoxE and denoting the remainder of the electric polarization with P, we find

where er — I + x > 1 is a relative permittivity. The parameter er can be treated as a spatially dependent parameter to allow for instantaneous effects on displacement in Q due to the electric field originating in QQThe constitutive law in (2.1) is also sufficiently general to include models based on differential equations and systems of differential equations or delay differential equations (see [BJ70]) whose solutions can be expressed through fundamental solutions (in general variation-of-parameters representations). For example, the choice of kernel function g(i] = e~t/T6o(es — £oo)/ r m ^ corresponds to the differential equation of the Debye model for orientational or dipolar polarization in fi given by

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Here, es is the static relative permittivity (sometimes denoted esr, e.g., see [Bal89, p. 50]). The presence of instantaneous polarization is accounted for in this case by the coefficient €00 in the electric flux equation. That is, er = e^ in fi, er = I in f^o- The remainder of the electric polarization is seen to be a decaying exponential, driven by the electric field, less the part included in the instantaneous polarization. This model was first proposed by Debye in [Deb29, vH54] to model the behavior of materials whose molecules possess permanent dipole moments. The magnitude of the polarization term P represents the degree of alignment of these individual moments. The choice of coefficients in (2.4) gives a physical interpretation toes and eoo as the relative permittivities of the medium in the limit of the static field and very high frequencies respectively. In the static case, we have P = 0, so P = eo(es ~~ eoo)E and D = ese0E. For very high frequencies, rP dominates P so P w 0 and D — e^vE. We will also consider the Lorentz model for electronic polarization which, in differential form, is represented with the second order equation:

The so-called plasma frequency is defined to be up = ujQ^/es — e^. A simple variation of constants solution yields the correct kernel function

where Z/Q = For more complex dielectric materials, a simple Debye or Lorentz polarization model is often not adequate to characterize the dispersive behavior of the material. One can then turn to a combination of multiples of Debye, Lorentz, or even more general nth order mechanisms. In the time domain, such an nth order model is given analytically by

which in the frequency domain takes the form

where q(s) is the polynomial q(s] = Y^=oajs^- ^o model multiple (e.g., N) Debye, Lorentz, or nth order polarization mechanisms one would then have (in

Problem Formulation and Physical Modeling

13

the frequency domain)

where the degree of Qnum is less than the degree of Qden. In the time domain one obtains the equivalent polarization law

where k i ( N ) > K2(N) corresponding to a stable medium. To see that (2.6) is included as a special case of the general polarization law (2.1), we first observe that in the time domain (2.6) becomes P — ^-j Pi, where

These equations can be written as first order n^ dimensional vector systems for the variables

where Mi = diag(l,..., 1,0:^.), [/^Q] = col(0,... ,0,/30), and Ai is the n; x n; matrix with 1's on the superdiagonal, (—a l 0 , — a | , . . . , —a^._ 1 ) in the n, row and zeros elsewhere. We note that this matrix representation requires some care in its interpretation, as the elements of the vector Pi and the quantity E are all vector-valued functions themselves. The matrices Mj and Ai should be viewed as the matrix representations of linear transformations acting on a vector of unknowns, in which the transformations are expressed through scalar-vector multiplication of the scalar coefficients of Mi and Ai and the vector elements of

A.

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Banks, Buksas, and Lin

The variation of parameters representation can be used (assuming -Pi(O) = 0) with Gi(t] = exp(M~lAit] to obtain

so that the first component Pi of Pi has the representation

we are thus led to the expression

This has the same form as (2.1), where the susceptibility kernel is the sum of kernels deriving from impulse response solutions of general nth order polarization models (including first and second order Debye and Lorentz models, respectively). Using a standard variation-of-parameters representation for systems with delays (see [BJ70]), one can also argue that polarization models with intrinsic explicit delays of the form

are also contained as special cases of (2.1). Polarization effects are dependent on physical variables such as mass density, temperature, material inhomogeneity, etc. (This dependence on physical quantities is usually expressed through changes in the coefficients in the mathematical models of the previous discussions.) This fact can be used to produce "acoustic gratings" to act as electromagnetic reflecting interfaces. In particular, the introduction of an acoustic wave will change the density of the fluid. (Indeed, acoustic waves are simply pressure waves which involve density variations.) This in turn will affect electromagnetic properties, such as the refraction index, of the fluid. This is known as the acoustooptic effect. Consequently, any electromagnetic wave transmitted into a material with an acoustic field will be modulated by the field. At the same time, the material electrostriction caused by the electromagnetic waves will also affect the propagation of the pressure wave in the fluid [MI68]. This produces a fully coupled nonlinear model with equations for both the electromagnetic and acoustic pressure waves (see [BL] and [MI68, p. 825]). In our initial efforts, we focus on the effects of variations in

Problem Formulation and Physical Modeling

15

acoustic pressure as a reflector of electromagnetic waves. We ignore the effect of electromagnetic forces in the acoustic equation under the tacit assumption that the effect is weak. To demonstrate the effect of the acoustic wave on the electromagnetic wave, we begin with a common assumption [Kor97] that the electric susceptibility is an afhne function of the acoustic pressure p(t, z):

Then we have Note that we have returned to an instantaneous model of polarization, rather than a DRF model such as Debye. Since it plays an important role in the subsequent analysis and computations, we compute

More generally, we may assume that the fluid in the acoustically affected part of the domain obeys a generalized pressure dependent polarization rule [Dan67, Chapter 9]:

To further simplify this preliminary investigation, we take

which we note is not a special case of (2.7). However, we shall see below that for typical values of dielectric parameters, the contributions of the /o(p) and f i (p) terms are several orders of magnitude smaller than that of the /2 (p) term (and the other terms in (2.19) below). Hence a reasonable first approximation to (2.7) is to ignore the /o, /i terms and take as the polarization assumption in the acoustic region the approximation

2.2

Estimation Methodology

Adopting a rather standard approach, we propose to identify the unknown parameters in a given model of polarization and a geometric representation by attempting to minimize the difference between simulations and observations of time-domain data. The data are measurements of the electric field at points in the exterior domain Q,Q at discrete times. The simulation is a computed

16

Banks, Buksas, and Lin

solution to Maxwell's equations with the constitutive laws for polarization, using candidate values of the geometric and material parameters. The criterion for optimization is a least-squares measurement of the difference between the simulation and the observed data given by

The Ei are measurements of the electric field taken at specific locations and times. The E(ti,£i;q) are solutions evaluated at the same locations and times from the simulation using the full set of parameter values q. We note that use of least squares formulations in inverse problems in electromagnetic scattering as well as in geophysical (seismic reservoir) problems is well established (e.g., see [PSS84, GW65, Bor99, BK89b] and the numerous references contained therein). We note two nontrivial difficulties with this approach and propose solutions, the efficacy of which will be demonstrated in the particular implementation discussed in this monograph. The unknown location of part of the boundary creates computational challenges. During the course of an iterative optimization procedure, simulations will be repeated many times for different locations of the unknown part of the boundary of 0. That is, iterative-based methods generally will involve changing domains and hence changing discretization grids in the usual finite element or finite difference approximation schemes. Any associated computational scheme (with domain changing with each iterative step) will be prohibitive in effort and time. We address this difficulty by employing the "method of mappings" [BK89a, Pir83, BKW90] and transforming the problem on £1 (J fio with unknown geometry to one with known geometry (a reference domain fi) at the expense of introducing additional unknowns into the equations that must be solved on this new domain. This technique, developed earlier [Pir83] in connection with general problems in shape design and optimization, has been used successfully in thermal inverse problems [BK89a, BKW90] related to the determination of unknown thermal boundaries and interfaces defined by material defects. Conceptually, one transforms a parameter dependent domain T>(q) = QQ U Q(q) to a reference domain fi which is independent of the parameterization q of the unknown part of the boundary (see Figure 2.2) through a parameter dependent transformation of the independent variable. That is, one defines a map T(q} : ^oUO(g) i—> fi so that X = T(q)X, where X = (x,y,z) and X = (x,y, z) are the independent coordinates in V = OQ U £l(q) and fi, respectively. On the original domain fio U &(<}) one has a dynamical system (e.g., Maxwell's equations) with coefficients independent of the unknown boundary parameters g, but the system must be solved on an unknown domain that changes as one changes

Problem Formulation and Physical Modeling

17

Figure 2.2: Schematic diagram of the method of mappings. estimates of the parameter q. On the reference domain £7, which is independent of the boundary parameters q and is a known domain, one obtains, through the coordinate change, a dynamical system with coefficients which depend on the boundary parameters q. Estimation of q (through iterative methods), while still nontrivial, is much more tractable in this latter formulation. Generally, we cannot expect these mappings T(q] to be C2, or even C1, which precludes finding classical solutions of the transformed system. For this and other reasons discussed below, we consider a weak formulation of the problem. As we have noted, the effect of the mapping is seen in additional coefficients appearing in the equations. The values of these coefficients must be interpreted through the inverse map T~l(q) to determine the geometric parameters. Another difficulty arises from the oscillatory nature of the time domain data. Varying some parameters in the model has the effect of changing the time at which reflected signals are detected by varying the distance the propagated waves travel (to and from the unknown boundary F(g)) and their speed of propagation. This causes the simulated data to move in and out of phase with observations as these parameters are changed, producing in the cost function J an oscillatory character with respect to these parameters. Thus, local minima of J can (and do) arise at values for which the simulation moves partially back into phase with the data (this is illustrated in Chapter 4). We choose to avoid the resulting difficult optimization problem by employing a multistep approach which separates the identification of geometric and physical parameters. (Other approaches are possible, e.g., use of an optimization algorithm that obtains global minima even in the presence of multiple local minima.) The data set is truncated in time to a period which contains only partial reflections from initial penetrations of the interrogating signal on the surface of the material, eliminating the dependence on geometry. This generates an estimate of the dielectric parameters that is used in a second step which attempts to recover the geometry. To avoid the oscillatory nature of the objective function, a different optimization criterion is used in the second step. This criterion

18

Banks, Buksas, and Lin

compares the secondary return times (i.e., return times after the pulse has reflected back from the unknown supraconductive-backed part of the boundary) of pulsed signals through media of varying geometry with the observed data secondary return times. A global optimization for improved estimation of all parameters is then attempted in a third step using the estimates of the geometry and the prior estimate of the dielectric parameters as initial estimates.

2.3

Reduction to Specific Problems

The choice of an interrogating input signal (in our case, a windowed microwave pulse from an exterior antenna in QQ) has profound implications on both theoretical and computational aspects of the inverse problem for estimation of dielectric and geometric parameters. A very popular (and readily implemented) choice consists of a polarized planar wave. This produces a signal with the E and H fields possessing nontrivial components in only one dimension in fio- If the interrogated medium £) has some homogeneity (in planes parallel to that of the interrogating planar wave), a similar reduction of the E and H fields occurs in the body fi. To discuss our inverse problem ideas, we consider in this monograph the problem of interrogating an infinite slab of homogeneous (in the directions orthogonal to the direction of propagation of the plane wave) material by a polarized plane wave windowed microwave pulse. Specifically, as depicted in Figure 2.3, the interrogating signal is assumed to be a planar electromagnetic wave normally incident on an infinite slab of material contained in the interval \z\, z2] with faces parallel to the xy plane. The electric field is polarized with oscillations in the xz plane only. For the acoustooptic problem, we will amend this geometry slightly by adding a second homogeneous layer. Under these assumptions, it is easy to argue that the electric field is parallel to the ? axis at all points in fio (the region external to the slab) and that the magnetic field is always parallel to j. Furthermore, these fields are homogeneous in intensity in the x and y directions. Thus E(t, x) = iE(t, z), H(t, x) = jH(t, z) as shown in Figure 2.3. In our problem, the electric flux density D and polarization P inherit this uniform directional property from E and hence will be denoted hereafter by their scalar magnitudes D and P in the i direction. Since we have assumed that the material properties are homogeneous in the x and y variables, the propagating waves in SI are also reduced to one nontrivial component. Thus the problem's dependence on x and y disappears since the resulting fields are necessarily homogeneous in these variables. This makes it possible to represent the fields in 17 and £7o with the scalar functions E(t, z) and H(t, z). Under these assumptions, the differential operation V x A reduces to —i-of- + j^f- and the Maxwell's equations from Section 2.1 become

Problem Formulation and Physical Modeling

19

Figure 2.3: Geometry of physical problem.

We eliminate the magnetic field from the equations by taking the space derivative of (2.9) and the time derivative of (2.10) and using the equation for electric flux density D — eE + P, where e = eo(l + (er — l)-fa)j to obtain

Here and throughout, Is will denote the indicator function for a set S and This is the differential equation of concern for both our simulations and inverse problem calculations. We also define the domain of the computation (after the method of mappings has been applied) to be the interval 17 = [0,1] which contains f^o- An absorbing boundary condition is placed at the z = 0 boundary of the interval to prevent the reflection of waves. This can be expressed by

20

Banks, Buksas, and Lin

For our formulation we assume that the location of the boundary at z — z\ is known, while the location of the original back boundary at z — z% (equivalently, the depth of the slab) is unknown, i.e., F(g) = {x G R \z = z^}. For the first problem of interest, we have a supraconductive backing on the slab at z — z-2. The boundary conditions on this supraconductive reflector can now be determined explicitly, again assuming that the method of maps had been applied and therefore the back boundary of the material is now found at z = 1, the edge of the computational domain. (The precise map will be discussed below.) The vector normal to the surface is n = k and the condition B • n = 0 is satisfied automatically since B = Bj. The condition E x n = 0 becomes [Eyi — Exj]r — 0 and hence the condition Ex = Ey — 0 on the boundary. Sinc E = (E= iE, this is equivalent to the condition that E(t, 1) = 0. x, 0,0) Substituting the expression for P derived in (2.2) we obtain the strong form of the equation

where indicator functions /^ have been added to explicitly enforce the restriction of polarization and conductivity to the interior of the transformed medium 0 = [zi, 1] and er = e/e0 = I + (er — l)Ifi > I throughout QQ (J^Alternatively, we can apply integration by parts to the integral term and arrive at a different form of the equation

This form of the equation has the advantage of requiring less continuity in the dielectric response function g(s,z). Furthermore, one of its terms disappears when .5(0, z) — 0, which is the case of interest in the simulations. We shall use the weak form of this equation in our computational investigations in Section 5.1 below. Analysis of the well-posedness of (2.14) in weak form is carried out in [Buk98]. Due to the forms of the interrogating inputs, the dielectrically discontinuous medium interfaces, and the possible lack of smoothness in mapping the

Problem Formulation and Physical Modeling

21

original domain i!0 U^ = [0> z?\ t° ^ne reference domain fi = [0,1], one should not expect classical solutions to Maxwell's equations in strong form. For both theoretical and computational purposes, it is therefore desirable to write the system equations in weak or variational form. Using the spaces H = 1/2(0,1) and V = H^(Q, 1) = {0 6 Hl(0,1)|0(1) = 0} and the boundary conditions (2.12), we can write (2.13) in weak form as

with initial conditions

Ine coefficients in (2.15) are given by

and { - , - ) is the I/2 inner product (or equivalently, any appropriately chosen topologically equivalent inner product—the relevance of this remark will be clearer after our discussion below of the method of maps for this example). The functions a,/?, and 7 are dependent on parameters which must be identified. These are assumed to be in L°° but may lack any additional regularity. The source current is under our control, so we can and do choose its form precisely. For example, we may use

Here, a; is a specified angular frequency of the input signal (and the carrier frequency of the resulting planar wave) and 6(z) is the Dirac distribution which has infinite mass at z = 0. The signal is truncated at a finite time tf by the indicator function /( 0 ) t / )(t). We avoid the complications arising from a discontinuous (in i) input signal by choosing tf so that the sinusoid g s ( t ) is zero at tf. Hence cutf = HTT for some positive even integer n. This is equivalent to requiring that the end of the signal occur after an even integral number of half-periods. (We require an even integer since far-field radiated signals integrate to zero.) We note that this windowed signal can be equivalently chosen with additional smoothness in t by replacing the indicator /(o,t/) with a slightly smoother (continuous or even differentiable) truncating function. By this modification one obtains an input signal for which the theoretical results in Chapter 3 below are valid.

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Banks, Buksas, and Lin

This windowed pulse input signal is most helpful in identifying the physical and geometric parameters separately. Since it has a finite duration, the wave's reflection off the surface of the media and its subsequent reflection off the back surface will not necessarily overlap for a sufficiently short pulse. This makes it possible to split the resulting data and perform the two estimation steps described above separately. We turn to the details of the method of maps for this example. The application of the method of mappings is quite straightforward in this particular case. The value of z\ is presumed known, while the surface at 22 is inaccessible and therefore its location (i.e., the value of 22) is not known. We use a piecewise linear mapping which leaves the interval (0, z\) invariant and maps (zi, 22) to (21,1), thus mapping the original domain f2 0 (J^ = [O?^] to the reference domain & = [0, 1]. The new coordinate variable z in £1 = [0, 1] is defined by

We can express the function as

The parameter £ appearing the equations can be identified in lieu of the depth 22 — z\ which of course can be recovered from it. Wherever spatial derivatives appear in the equations, we must replace these with the derivatives with respect to the new variable z using the chain rule

Hence dz — f'(z)dz and the expressions for the inner products in the weak form are modified accordingly to

Problem Formulation and Physical Modeling

23

Here the functions and ip have been expressed in terms of the new variables 0(2) = 0(z),i/;(z) = ^(z). For convenience, we will drop the over tilde notation in subsequent discussion. Returning to (2.15) again, we see that in mapping the original Maxwell system on the domain fioU^ to the reference domain fi = [0,1], the {••,-} in (2.15) should be interpreted in the sense (2.17), (2.18) above. This will be done in all computational results presented in Section 4. Note that for the theoretical discussions of Section 3 we can without loss of generality use the usual inner products. This one-dimensional problem also permits a simple interpretation of the return time when identifying the geometry. Before the method of maps is applied, the unknown geometric quantity is the depth, 2-2 — z\, of the medium. After the mapping is applied, the actual depth of the medium is fixed, but the effects of changing geometry are reflected in the new coefficients (actually the weighting parameter £ in the inner products which appears in the equation). Since there is only one path for the pulse to pass through the medium and be reflected back, the first return of the transmitted signal is a singular event and simple to identify. To further clarify the identification of the return time, we restrict our attention to materials which are sufficiently thick and pulses which are sufficiently short so that the first reflection of the pulse off the surface at z\ is completed before the transmitted signal has returned from the back surface. Using c, the speed of light in a vacuum, as an upper limit on the speed of propagation in the material, this condition requires that tfC < 2(z^ — zi).

2.3.1

Acoustic Reflectors

The presence of the supraconductive boundary condition at z = 1 in the previous formulation of the problem makes sense only in applications where an appropriate conducting object can be introduced. While this situation will occur in a number of important applications (e.g., metal backed stealth surfaces

24

Banks, Buksas, and Lin

in the military setting) there are many applications for which no such backing is present. This motivates the use of acoustic waves as an artificially produced reflector for the interrogating electromagnetic pulse. To study this configuration, we consider a two-layered medium normalized to the intervals [21,22] and [22,1] in which the medium in [22,1] is modulated by the acoustic wave. Maxwell's equations under the same planar wave assumptions as before, together with the basic constitutive laws, yield the following equation in the domain 0 < 2 < 1:

In our investigation, the material in 2 € [0, 21], which is air, is assumed to have zero electric polarization and zero conductivity; hence er = 1 , a = 0 in this domain. The material in 2 6 [21, 22] is assumed to obey the Debye law described in the previous section. We consider electromagnetic wave propagation in the normalized interval 2 G [0, 1], assume that the solid slab occupies the space for 2 G [21, 1], and an acoustic wave is given in the material in 2 G [22,!] with 0 < 21 < 22 < 1. Figure 2.4 is a sketch of the geometry considered in the presentation below. For the purposes of our discussions, we suppose that the acoustic perturbations are restricted to the domain [22, 1] and that the material outside the space 2 G [0, 1] can absorb the electromagnetic wave completely. We also assume the initial conditions We return to the polarization relation (2.7) and consider the contributions of the various terms in the overall equation (2.19). We first note that in general, the terms E, E, and E" behave like u)2E,ujE, and ^-E, respectively, where u> is the carrier frequency and A is the wavelength of the propagating wave E. For the systems considered in the part of this monograph concerned with the acoustically backed layer (with carrier frequency TT x 1010 rad/s and wavelength roughly \ x 10"1 m), these terms thus are of the order 1021£, 1010E, and 104£, respectively. Thus for er w 1, a/eo « 107, and c2 = I/CQ^Q w 1017 we find the terms erE,(r/c0E and -^E" are order 1021E, W17E and 1021£, respectively. On the other hand, the polarization law (2.7) leads to

where fo(p) — KP, fi(p) = %KP and f^p] — Xo + ^P with xo representing the instantaneous polarization and p the standing acoustic wave. For the parameters used in Chapter 6 below (K — 0.9,p w l,p w u>p — 106,p w a;2 = 1012) we find /o(p),/i(p), and /2(p) are order 1012, 106, and 1, respectively. Thus we find fo(p)EJi(p)E,Rand h(p)E are order of 1012£, 1016E, and 1020£, respectively.

Problem Formulation and Physical Modeling

25

Figure 2.4: Geometry of acoustooptics problem.

Thus a reasonable first approximation might ignore the fo(p) and fi(p) terms and result in the polarization assumption (2.8). Using the scalar form of (2.8) in (2.19), we have the following partial differential equation for the electric field in the region disturbed by acoustic waves:

where er = 1 + Xo- This equation is very similar to the one given in [MI68] derived from thermodynamical considerations. The absorbing boundary conditions for the electric field are found to be

In the first domain [0,zi] the speed of propagation is exactly c, hence the absorbing boundary condition at z — 0 is exact. For the boundary condition at

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Banks, Buksas, and Lin

z = 1 we have made use of the approximate phase velocity for wave propagation in the medium described by (2.20), vph = /e C+K . We note that this is not exact for all frequencies because of the presence of dispersion caused by the nonzero conductivity a of the material. We ignore this difference on the grounds that the dispersive effect is very slight for the small conductivities of the materials in which we are interested. Furthermore, we will limit the duration of simulations to times before spurious reflections due to inexactness in the boundary condition at z = 1 can return to affect observations. For the material in the interval [21,22] we follow the previous section and eliminate the second order derivative of P in (2.19) by substituting the derivative of the Debye polarization law in (2.4). This results in a first order derivative of P along with a similar additional term for E in the basic Maxwell's equation. Thus, we use the following initial-boundary value problem to model the dynamics of the electromagnetic fields:

where

and /[ Xl)X2 ](x) is the usual characteristic function for the interval [xi,X2J.

Chapter 3

Well-Posedness 3.1

A Variational Formulation

In this section we consider well-posedness questions for the variational form (2.15) of Maxwell's equation with a general polarization term, absorbing left boundary conditions and supraconductive right boundary conditions as discussed and formulated in the previous section. Without loss of generality, we may take er = 1 for our discussions in this section. That is, we are concerned with existence, uniqueness, continuous dependence, and regularity of solutions to

We seek solutions t -* E(t) with E(t) in V = ##(0,1) = (0 <E #^(0,1) : 0(1) = 0} satisfying (3.2) and (3.1) for all 0 G V. Here we use standard Sobelev space notation, e.g., tf^O, 1) = {0 € L 2 (0,1) : 0' 6 L 2 (0,1)} found in [Wlo87, LioTl] for example. We shall do this in the context of a Gelfand triple setting V °-> H <—> V*, where H = L 2 (0,1). We assume throughout this section that the slab region fi = [zi, 22] has been mapped to [zi, 1] so that our domain of interest is Q = [0,1] with 7, /?, and a(t) bounded on [0,1] and vanishing outside [zi,l]. Thus, in (3.1) the inner products should be properly interpreted in the sense of (2.17), (2.18). However, as explained in the last section, one can equivalently treat the well-posedness questions of this section using the unweighted L2 inner product in (3.1), and we shall do this. From a general theory presented in [BIW95], [BSW96, Chapter 4], one sees that (3.1) differs from the usual lightly damped second order systems of [BSW96] only by the presence of the terms cE(t, 0)0(0) due to the absorbing left boundary 27

28

Banks, Buksas, and Lin

condition and { J0 a(t — s)E(s)ds, 0) resulting from the convolution representation for the polarization. (The (@E, 0} term is of no additional consequence in any well-posedness analysis since it is readily handled in developing a priori estimates.) From the general theory one might expect to seek solutions of (3.1) in the sense of L2(Q,T;V)* ~ L 2 (0,T;F*) with E G L 2 (0,T;\0,£ e L 2 (0,T;#) and E G L 2 (0,T; V*} for appropriate interpretation of the (-,-) in (3.1), i.e., the duality product {•,•)}/*, v which reduces to the H = L2 inner product in all terms of (3.1) except the first and last. We recall from Section 2.3 that the input J(t] results from a point source (antenna) at z — 0 and hence J(i) has the form g(t)6(z) for a windowed time signal g ( t ) . This motivates our desire for results allowing J(t) values in V* . In addition to differences one might encounter due to the polarization term, if one obtains as usual E G I/ 2 (0, T; H}, where H = Z/ 2 (0, 1), then questions arise concerning the interpretation of the boundary term cE(t, 0)0(0) which has the appearance of pointwise evaluation of an L 2 (0, 1) "function" at z = 0. Correct interpretation of this term will result from our arguments below. We follow the general approach using sesquilinear forms as in [BIW95, BSW96] which are standard in the research literature [LioTl, Wlo87]. We first rewrite (3.1) by adding a term (kE,(f))fj to both sides of the equation. The positive constant k is chosen so that (3 = k + /3 satisfies fi > e > 0 on [0, 1] for some constant e; this is possible since by assumption (3 G L°°(0, 1). We define a sesquilinear form a\ : V x V —> C by

Equation (3.1) can then be rewritten as

for all 0 G V, where it is readily seen that a\ is ^-continuous and F-elliptic. That is, there are positive constants Ci,c 2 such that

To establish existence of solutions to (3.2), (3.4), where $ G V, \& G H, we follow the ideas in [BSW96] and choose a subset {wi}^! spanning V. (Without loss of generality we may assume linear independence of these elements.) Let Vm = span{tui,... ,wm} and define Galerkin "approximates"

Well-Posedness

29

where the {e™ (t}}™^ are determined by substitution into (3.4) and requiring this system of ordinary differential equations to hold for 0 = Wi, i = 1, 2, . . . , m. This m-dimensional s s t e m is solved with initial conditions

where the 3>m, $m are chosen in Vm so that $m —> $ in V, ^m —> \I> in #. We thus find that Em(t) satisfies (3.4) with 0 = Em(t) G V so that for each £ we have the system

This system will allow us to obtain bounds for {Em}, {Em}, and {Em(-, 0)} that are independent of m. Using the fact that

we may rewrite (3.6) as

Integration along with use of the V ellipticity of a\ yields

where

30

Banks, Buksas, and Lin Assuming that a is bounded on [0,T] x [0, 1] we have for T £ [0, t]

Thus we find

We also have

Finally, to consider the term Ta, we use (assuming that

We obtain

Combining (3.7), (3.8), (3.9), and (3.10), we obtain

Well-Posedness

31

Assuming that J E Hl(Q,T; V*} and using the boundedness of {Em(Q)} in V and {Em(0}} in H (which follows from the convergences of {$m} and {^m}, respectively), we may employ Gronwall's inequality along with the inequality (3.11) to conclude that {Em} is bounded in C(0,T; H}, {Em} is bounded in C(0, T; V) and {Em(; 0)} is bounded in L 2 (0, T). Thus we find (extracting subsequences and reindexing as usual) there exist E E L2(0, T; V), .E E L2(0, T; #), and EL E L 2 (0,T) such that

The limit function E is a candidate for solution of (3.2), (3.4) and we must verify that E = E, EL = E(-,Q) in some sense and that we may pass to the limit in the version of (3.4) for Em to obtain (3.4) for the limit function. First we note that for each m

and

Passing to the limit (in the weak H sense in (3.12)) we obtain

We find that (3.14) holds in the H sense for each t € [0, T] and hence E — E while (3.15) yields that jE?(t,0) exists and is continuous in t. In actuality E(t,Q) is absolutely continuous with E(t,Q) — Ei(t) for almost every t. We note, in fact, that the same arguments used in [BGS97, Lemma 5.1(b)] can be used to establish that Em also converges weakly in C(0, T; H ) to E so that E E C(0,T; #) n L 2 (0,T; V). Thus we have that our candidate E for solution of (3.2), (3.4) satisfies

We must show that E satisfies (3.4). For this we follow directly the arguments of [BSW96, pp. 100-101]. Taking V e Cl[Q,T] with ^(T) = 0 and

32 choosing i^j(t) = i(j(t}wj where the {'Wj}°^1 fixing ji, that for all m> j, Em must satisfy

Banks, Buksas, and Lin 1areas

chosen before, we have,

Integrating by parts in the first term and then taking the limit as ra —* oo, with the convergences of (3.16)-(3.18) we obtain

It follows that for every Wj we have in the L 2 (0, T) sense (except in the first term which is in the distributional sense in t)

Since {wj}(^.l was chosen total in V we thus obtain that E G L2(0,T; V*) and that E satisfies (3.4). From (3.14) we know that E(0) = $ and the arguments that E = * follow exactly as those in [BSW96, p. 101]. Hence we find that E is a solution of (3.2), (3.4). Continuous dependence of solutions to (3.2), (3.4) on $,^, and J follow readily from the inequality (3.11) and some standard arguments. Noting that I • \H < (J\ • \v for some constant /^ and letting

we observe that (3.11) implies

Well-Posedness

33

for some positive constant v independent of ra. Using Gronwall's inequality again, we obtain

Recalling that E m (0) = $m -» $ in V and £ m (0) = ^ m -» $ in # so that from (3.19) we have limtf m < K, where A" = |*|^ + (c2 + l)|$|v + ^\^\2Hl(o T-V*)> we may use weak lower semicontinuity of norms, the convergences of (3.16) and (3.17), and (3.20) to conclude that

Since the mapping ($, ^, J) -> (E, E) is linear from F x F x Hl(0,T; V*} to L 2 (0,T;V) x L 2 (0,T;7J) we see that (3.21) yields continuous dependence of solutions (E,E) of (3.2), (3.4) on initial data ($,*) and input J. For uniqueness of solutions to (3.2), (3.4), we again follow the standard arguments given in [BSW96, pp. 102-103]. In this case the details are tedious but rather straightforward. As usual, it suffices to show that the only solution of (3.4) corresponding to zero initial data ($ = \I/ = 0 in (3.2)) and zero input (J = 0) is the trivial solution. Let E be a solution corresponding to $ = \I> = J — 0 and for arbitrary s in (0, T) define

so that i/>s(T) = 0 and i^s(t] 6 F for each t. We then find that

Hence, choosing (f) — ijjs-(t) in (3.4) and integrating over t from 0 to s, we have

Observing that

34

Banks, Buksas, and Lin

and

we may use (3.22) to obtain

It follows immediately that

From the definition of TJJS we have for each

so that

Using (3.25) and arguments exactly like those behind the estimate (3.8) for TI (T) we find for t < s

Well-Posedness

35

Using (3.26) and (3.27) we thus obtain

or

for arbitrary s E (0,Tj. Invoking the Gronwall inequality once again, we conclude that E(£) = 0 on (0,T) and solutions of (3.2), (3.4) are unique. Summarizing our discussions in this section, we see that we have proved the following result. Theorem 1. Suppose that J E Hl(0,T;V*), 7,/5 E L°°(0,l), 00 a E L°°(0,T;L (0,1)) with a, (3,7 vanishing outside [zi,l]. Then for $ E V = #^(0,1),^ E H = L 2 (0,1) ; we have that solutions to (3.2), (3.4) exist and are unique. These solutions satisfy E E L 2 (0, T; V) n C(0, T; H), E E I/ 2 (0, T; #), and E E L 2 (0,T;y*). Moreover, t —* E ( t , Q ) is absolutely continuous with E(-,Q) E L 2 (0,T). The solutions depend continuously on (3>,\I>, i 7) as maps /rom I/ x H x tf^O,!1; V*) to L 2 (0,T;1/) x L 2 (0,T;//).

3.2

A Semigroup Formulation

While the variational formulation of the previous section provides adequate wellposedness results for our subsequent discussions, it is of some mathematical interest to determine whether the integro-partial differential system (3.1), (3.2) has a semigroup based formulation. For the sake of completeness, we present such a formulation next while noting that for this particular problem, improved regularity results over our theorem in Section 3.1 will not be obtained using semigroup methods. For this section we assume that as before 7, j3 6 L°°(0,1) while a E L°°((0, T) x (0,1)) and a,/3,7 vanish outside Q. We further assume that t t—> a(t, •) is positive in O and and monotone decreasing to zero so that a(t, •) < 0 whenever a E Hl(Q,T). This monotonicity assumption is typical of the usual assumptions in displacement susceptibility kernels (e.g., see [BloSl, p. 102] or [Hop77]). We further assume for this section that a(t, •) is constant in £1 so that a(t,z) = In(z)a(t} for some monotone decreasing function a. We note that a(t,z) = —I^(z)g(i) for the Debye and Lorentz polarization laws for g given in Section 2.1 satisfies this separation assumption. Moreover the Debye DRF also satisfies the required monotonicity assumption. Unfortunately, due to its oscillatory nature, the Lorentz DRF does not satisify the monotonicity condition.

36

Banks, Buksas, and Lin We consider the term (tacitly assuming E(s) = 0 for s < 0)

from (3.1) and note that it can be equivalently written

where G(f ) = <*(-£)• We denote G(£) = a(-£) so that The approximation is valid for r sufficiently large (r = oo is permitted) so that a(t) « 0 for t > r. We observe at this point that G(£) > 0 with G(f ) > 0 on (— r, 0]. As in the previous section, we take V = -H^(0, 1), H ~ L2(0, 1) and assume that fi = [zi , 1] C [0, 1] is the region of interest. We shall have use of H = L2(17) and shall denote the restriction of functions (f) in L 2 (0, 1) to J7 again by 0 and write G L 2 (fi) whenever no confusion will result. Motivated by the "strong" form of (3.1), i.e., see (2.13) with er = 1, and using the above definitions and approximating, we may write (3.1) as

where, of course, the derivatives must be interpreted in a weak or distributional sense. Following [BFW88, BFW89, FI90] and [BMZ96], we define an auxiliary variable w(t) in W = L2G(-r, 0; H) by w(t](9] = E(t] - E(t + 9), -r < 9 < 0. Since G(9,z) = G(9,z] > 0 for 9 6 (— r, 0],z G £1 and G is constant in fi, we may take as an inner product for W the weighted L2 inner product

under which W is a Hilbert space. We note that by our notational convention explained above, we have w(t) G W for any E(t,z) with E(-, •) G LQ(— r, Q;H). Using a standard shift notation, we may write w(t) = E(t) — E(t + 6) = E(t) — £*(0), where El(0) = E(t + 0) for -r < d < 0. Adding and subtracting appropriate terms in (3.30), we find

Well-Posedness

37

or, equivalently,

where Gn = f_r G(t)d£ and w(t}(£) = E(t) - £?'(£).

For our semigroup formulation, we consider (3.31) in the state space Z = V x H x W = tfjj(0,l) x £ 2 (0,1) x L^(-r,0,H) with states (,^,ry) = (E(t),E(t),w(t)} = (E(t),E(t),E(t) - £'(•))• To define an infinitesimal generator, we begin by defining a fundamental set of component operators. Let A&C(V,V*) be denned by

where ^o is the Dirac operator SQIJJ = ^(0). Then we find

so that it is readily seen that a\ : V x V \—> C defined by

is symmetric, V continuous, and V coercive (i.e., ) > ci||y — AQ|<^|^ for constants A0 and c\ > 0). We also define operators B <E L(V, V] and K e £(W, F) by

so that

Since G(£, z] — 0 for z e (0, l)\fl, we abuse notation and write this as

even though, strictly speaking, TJ(^, 2) is only defined for z e £2. With these definitions and notations, (3.31) can then be written as

38

Banks, Buksas, and Lin

or

We rewrite (3.38) as a first order system in the state £(£) = (-B(t), £"(£), w(i)), where w(t) = E(t) — El. To aid in this we introduce another operator

defined on dom D =

We then observe that w(t) — E(i) — El satisfies

Thus we may formally rewrite (3.38) as a first order system and adjoin to it the equation We then obtain the first order system for £(£) given by

where A given by

is defined on

that is, ^.$ = (,A + BV' + A'ry, ^ + -Dry) for $ = (0,^,77). The forcing function ^" in (3.40) is given by T = 001(0,^7,0). To argue that A is the infinitesimal generator of a Co-semigroup, we actually consider the system (3.40) on an equivlaent space Z\ — V\ x H x W, where Vj is the space V with equivalent inner product {^i,^}^ = <5"i(^i,2), where <TI is the sesquilinear form given in (3.34). Recall that <TI is symmetric, V continuous and V coercive so that it is topologically equivalent to the V inner product. Under the assumptions on a, 0, 7 listed in the beginning of this section, we can establish the following. Theorem 2. Suppose that 7,^ e £°°(0, l),a e L°°((0,l) x (0,1)) with a,/?, 7 vanishing outside Q. We further assume that a can be written a(t, z) = In(z)a(t),

Well-Posedness

39

where a € Hl(Q,T) with d(t) > 0, a(t] < 0, so that a is constant in z on£l and positive nonincreasing in t. Then the operator A is the infinitesimal generator of a Co-semigroup on Z\ and hence on the equivalent space Z. To prove this theorem, we use the Lumer-Phillips theorem [Paz83, p. 14]. Since Z\ is a Hilbert space, it suffices to argue that for some AQ, A — \Q! is dissipative in Z\ and H(\I — A] = Z\ for some A > 0, where 7£(AJ — A} is the range of XI — A. We first argue dissipativeness. Let $ = (0, ip, 77) e dom A. Then

We consider estimates for the last two terms in (3.41) separately. From (3.37) we have

Moreover,

Finally, since G(9) > 0, G > 0, and r; € dom D requires ry(0) = 0, we may argue

40

Banks, Buksas, and Lin

Combining these estimates with (3.41), we obtain for $ 6 dom A

which yields the desired dissipativeness in Z\. To establish the range statement, we must argue there exists some A > 0 such that for any given ty = (//, v, £) in Z, there exists $ in dom A satisfying

In view of the definition of A, (3.42) is equivalent to the system

for ((/>, i/;, 77) e dom ^l, (/^, i/, ^) e Z = V x H x VF. The first equation is the same as if) = A^> — /x while the third can be written as 77 = (A — D}~1(£ + tp) = (A — D)~l(£ + A0 — /z). These two equations can be substituted in the second to obtain an equation for . If this equation can be solved for (f> £ V, then the first and third can be solved for -0 and 77, respectively. The equation for (j) tha must be solved is given by

or

If we can invert (3.44) for 0 e V, then i/j = X(f> — /j, is in V, 77 = (A — A0 — /z] is in dom D C W, and

is in H so that (0, ^,77) is in dom A and solves (3.43).

Well-Posedness

41

Thus the range statement reduces to solving (3.44) for <j> £ V. This in turn reduces to invertability of the operator A We first observe th while rj(9] — ^—j^ — [£ + X(f> — p] satisfies r/(0) = 0 and hence is in dom Thus, for 0 G H , K(X — D)~1X satisfies

and

Hence for A sufficiently large we have

Thus if we define the sesquilinear form

we see that for A sufficiently large, a\ is V coercive and hence, by the LaxMilgram lemma [Wlo87], it is invertible. It follows immediately that (3.44) is invertible for 0 £ V. This completes the arguments to prove Theorem 2. Let S(t) denote the semigroup generated by A so that solutions to (3.40) are given by

Solutions are clearly continuously dependent on initial data £o and the nonhomogeneous perturbation J-. The first component of C,(t} is a solution E(i] of (3.30). To argue that the solution agrees with that obtained in Section 3.1, one can now use the arguments in Chapter 4.4 of [BSW96]. In summary, one argues equivalence for sufficiently regular initial data and nonhomogeneous perturbation. Then density along with continuous dependence is used to extend the equivalence to more general data (see [BSW96] for details).

42

3.3

Banks, Buksas, and Lin

Enhanced Regularity of Solutions

As we noted at the beginning of the previous section, additional regularity of solutions to (3.2), (3.4) over those presented in Section 3.1 are not readly obtained using semigroup methods. It will be useful, however, in our subsequent theoretical discussions of finite element approximations and a general approximation result in inverse problems to have additional regularity in solutions. One can prove, under the assumptions and results presented in Section 3.1, a weak convergence of finite element approximations (along the lines discussed in [BSW96] and [BZ99]) to (3.2), (3.4). However, these results are not adequate for development of an inverse problem approximation methodology similar to that of [BSW96, Chapter 5], where a type of strong convergence is needed. In this section we give arguments for additional regularity of t —> E(t,-) which will prove adequate for subsequent theoretical discussions. This will be made possible by additional assumptions of smoothness in t for the source and coefficient functions in (3.2), (3.4). Due to the essential singularities and discontinuities in space in the source and dielectric functions, additional smoothness of solutions in the spatial variables is not readily obtained. Using traditional arguments (given in [BZ99]) we are able to obtain the following regularity theorem. Before stating this theorem, we recall the additonal notation for j > I : Theorem 3. Let J(t,z) = g(t)6(z) with g € #2(0,T) and g(0) = g(0} = 0, (3, 7 e L°°(0,1) and a 6 H2(0,T; L°°(0,1)) with a,/3 and 7 vanishing outside fi. Further assume that $ e #"^(0,1), ^ 6 -##(0,1), and
Well-Posedness

43

To establish the desired regularity for E, we first smooth all the given data and then solve the smoothed problem to obtain solutions with some added regularity. We then pass to the limit by letting the smoothing parameter tend to zero. Let 0 < A < zl be given. Since g G # 2 (0,T), $ G #1(0,1) and # G #£(0,1), we can construct #A G # 3 (0,T), $A G #£(0,1) and ^ A E #£(0,1) such that

and

We next define an approximating source function

It is easily seen that where the norm is in # 2 (0, T; V*). We then consider the smoothed problem: # x (0, T; #) n L2 (0, T; I/) with

Find E& G # 2 (0,T; V"*)n

and for a.e. t G (0,T), £'A satisfies

Here and below we use the notation a * E&(t, z) = JQ a(t — s, z)E&(s, z)dz. An application of Theorem 1 guarantees the existence of a unique solution £'A to the problem (3.48)-(3.49). We proceed to argue that this solution has the additional regularity

First, we formally differentiate with respect to t both sides of (3.49) and introduce a new variable U A for E& to obtain

44

Banks, Buksas, and Lin

With (3.51) we take the initial conditions where ZA = -(7*A + /?$A - c2 $ A ) (note that JA(O) =0). It is easy to verify that \!>A £ V and ZA E #, so we can again apply the arguments for Theorem 1 to establish the existence of a unique solution WA to the problem (3.51)-(3.52) so that We next verify that WA = -^A- To assist in this, we define

It is clear that Integrating (3.51) over (0, £) and using (3.52), we obtain

Using (3.52), integration by parts and the consistency condition, we can readily argue that the expression in brackets in (3.54) vanishes. Comparing equations (3.53)-(3.54) with equations (3.48)-(3.49) and using the uniqueness of solutions to equations (3.48)-(3.49) we then find that Thus the desired regularity expressed in (3.50) follows from that of WAWe next formally differentiate both sides of (3.51) with respect to t and introduce a variable w& for WA to obtain

with

Well-Posedness

45

and

We combine (3.55) with the initial conditions

where YA = -(a(0)$A + /^A + 7^A - c2\&A). We tnen argue as abt the solution w& of (3.55) is the same as WA, and

In these arguments we use the conditions 7(0) = cr(0) = (3(0) = 0 as well as the consistency condition ^'(0) = c$"(Q). We next provide uniform (in A) bounds on EA- Recall UA = £A- Integrating (3.51) over (0, t) and then choosing 0 = EA (this is possible due to the regularity (3.56)), we obtain

After an integration by parts the second term in the right side of (3.57) can be written as

Using this in (3.57) along with the Cauchy-Schwarz inequality, we obtain directly

46

Banks, Buksas, and Lin

where we have used &(0) = 0. Here the constant C depends only on k, T, and the bounds 7 = |7|L°°, & — max{|a|L°°, H£,°O), and j3c = min{c2,/3}. Using the bounds for ^ A ,
for sufficiently small A. Applying Gronwall's inequality, we find

where the constant C is independent of A. In a similar manner, beginning with (3.49) we obtain the bounds for the first order derivative of E&

From these bounds, we conclude there exists a subsequence (which we again denote by E&) such that as A tends to zero,

Using the formula

with ty = E& and JBA, one can easily argue that

Finally letting A —> 0 in (3.49), we readily find that £^i must be the unique solution E to the problem (3.2), (3.4). The desired regularity of Theorem 3 follows from that for E\ .

Well-Posedness

3.4

47

Convergence of Finite Element Approximations

In this section we introduce approximation systems for (3.1) and (3.2). More precisely, we will consider (2.15) and will not make the tacit assumption that er = 1 as we did in the sections above on existence, uniqueness and regularity. In our discussions below, we shall be focusing on estimation of dielectric parameters (including, in general er) using partially reflected electromagnetic pulses and hence we cannot assume without loss of generality that er = I . In fact, we consider semidiscretization (discretization in space) of the equivalent system (see (3.4))

where a\ is defined as in (3.3). We seek approximations of the form EN(t, z) = 1 ^Lj=o ^(^}(f)jf (z) wnere the basis elements satisfy • =^) {0^} In particular, we shall be interested in the standard piecewise linear spline elements corresponding to a partition of the interval [0,1] by points {z^}jLQ = {j/N}jLQ. The piecewise linear elements 0^ are completely determine the usual requirement that 0^(z^) = 6ij. We define the corresponding quasiL2(0,l) projection operator (see [NV88]) PN : V* ^

where ((•,•)}N is defined by

with IN the nodal value linear interpolation operator associated with VN. It can be shown [NV88] that PN is a well-defined operator which satisfies

48

Banks, Buksas, and Lin

where With these definitions, our approximating systems for (3.59), (3.60) are given by

Since it is readily seen that (3.62), (3.63) is equivalent to a system of N ordinary differential equations with initial data, one can use standard results to guarantee, under the conditions of Theorem 3 for example, that a unique solution EN 6 H2(Q,T;VN) exists for each N. Moreover, under the regularity assumptions on parameters and initial data given in Theorem 3, one can prove the strong finite element convergences:

Detailed arguments for these convergence statements are given in [BZ99] which we invite the reader to consult. We shall not give these arguments here since these results will be special cases of the convergence statements proved in the next section in the context of convergence of approximations in parameter estimation problems.

3.5

Inverse Problem Methodology

We return to the parameter estimation problems introduced in Chapter 2.2 above. These problems are special cases of very general parameter estimation or inverse problems as developed in [BK89b] and [BSW96] where a fairly complete computational framework is developed. Here we shall present only a brief discussion along with specific approximation results for the systems and problems of direct interest to us. We focus on the estimation of dielectric (essentially polarization and conductivity) parameters and those geometric parameters that can be included (via the method of maps) in the coefficients e r , 7, /?, a of (2.15). We shall assume throughout that these functions and parameters have been parameterized by some set of a finite number of constant parameters (e.g., the

Well-Posedness

49

coefficients in a polynomial or spline representation for the function g or the parameters in a specific polarization model such as the Debye or Lorentz models of (2.4) and (2.5), respectively). As in Chapter 2.2 we shall denote these parameters (either e r,7,/3, a or their equivalent parameterizations) by the fin dimensional vector q (we drop the overbar notation of Chapter 2.2 on the vector parameter q with the tacit understanding that it is some Euclidean space — of fixed dimension — vector throughout) and denote explicitly the dependence of solutions to (2.15) or (3.59), (3.60) on these parameters by E(q) or E(t, z; q}. The estimation problem then consists of minimizing over some set Q of admissible parameters the least squares functional

subject to (3.59), (3.60), where we suppose that measurements {E^} of the electric field at t — tj, i — 1 , 2 , . . . , S", and z = 0 are available as sampled data. In practice, one carries out the computations for the minimization problem using an approximate system such as (3.62), (3.63) or a corresponding fully discrete approximate system for (3.59), (3.60). Thus, one actually attempts to solve the minimization problem for

for some value (or values) of A/", presumably sufficiently large so that solutions of (3.62), (3.63) are "close" to those of (3.59), (3.60) in an appropriate manner. It is then hoped that the minimizing parameters q* (possibly not unique!) for (3.65) somehow approximate the minimizing parameters q* (again, possibly not unique!) for (3.64). A framework that guarantees that all is as should be is precisely the focus of theoretical inverse problem investigations. We briefly outline the questions and certain arguments that are usually made in such investigations, referring the reader to [BK89b] or Chapter 5 of [BSW96] for some more precise details as well as extensive bibliographies on inverse problems. Questions of importance usually include existence and uniqueness of minimizers q* and q*, and their continuous dependence on the the observation data {Ei} (often called "stability of the inverse problem"). Moreover, one wishes to know if minimizers q* converge in some sense to q*. This is a delicate and often difficult question in the case of nonuniqueness where the minimizers q* and g* are actually sets (e.g., see [BSW96, Section 5.2] or [BK89b]). In one approach (in its most elementary form, which incidentally is adequate for the problems of interest here), one assumes that (or formulates the problem so that) Q is compact, argues lower semicontinuity of the maps q i—> J(q] and q H-> JN(q) (this is guaranteed by continuity of q H-> E1^, 0; g), q i—> EN(ti, 0; q))

50

Banks, Buksas, and Lin

and thus obtains existence of minimizers for J and J^ in the usual manner. To obtain convergence of at least a subsequence {q*Nk} of any sequence of minimizers {q* } to some minimize! q* for J one can use the following steps: (a) Prove that for any arbitrary sequence {qN} in Q with qN —> q, we have EN(ti,Q;qN) -> E(ti,Q;q). (Note that by choosing qN = q for all N in this statement, we obtain immediately the finite element approximation convergence EN(t,Q;q) —» E(t,Q;q).) (b) Use compactness of Q and lower semicontinuity of J and JN to argue as follows: Let {q* } be a convergent subsequence of any (sequentially compact) set {q*N} of minimizers in Q with q* k —> q*. Then by the minimizing properties of q* k we have

Using (a) and taking limits in the above inequality we obtain

or g* is a minimizer for J over Q. Note that this sequence of steps actually proves existence of minimizers of J, assuming that one has minimizers for J^ (which involve finite dimensional side constraints (3.62), (3.63)). This latter is usually rather easily established. For the problems treated in this monograph, all of the ideas above can be made precise and a rigorous theory including existence and continuous dependence on data (even under approximations — see [BK89b]), and convergence of approximation methods can be derived. Since step (a) is usually by far the most difficult hurdle, we shall address only this statement before proceeding to the computational focus of our efforts in the next chapter. We wish to prove that {qN} arbitrary in Q with qN —» q implies

We shall assume that the parameterizations of (e^,~fN,/3N,aN) are such that qN —> q in Q implies e^ —> €. r,1N —* 7,p —* @,otN —* a un or (t, z) as appropriate. This strong convergence assumption can be weakened considerably (e.g., by considering the piecewise slab-like structure of the spatial domain) but this adds some tedium to the arguments below and hence we shall not do so here. Moreover, the assumption does hold for the parameterized Debye, Lorentz, and general DRF examples (which all involve finite dimensional parameterizations) treated in this monograph. We note that this assumption

Well- Posed ness

51

immediately provides uniform boundedness of (e^, jN,/3N,aN). In addition to the standing assumptions of Theorem 3 on all parameters and initial data (we use the additional regularity of solutions provided by Theorem 3 at several critical steps in the arguments below), we also assume that k in (3.59) is chosen so that J3N = k + (3N satisfies (3N > 6* > 0 uniformly, where 6* is some fixed lower bound. We recall that we may assume by definition that e^ > I uniformly in TV.

We observe that

and since {E(t;q)}t^0tT^ is a compact subset of V, we may use (3.61) with standard arguments to conclude that \PNE(t;q) uniformly E(t;q}\ in v —+ 0— t on [0, T]. Thus, to reach our desired conclusion of (3.66), it suffices to argue that in 1^, uniformly in t on [0, T]. We note here from (3.68), (3.63), and the definition of A^ that A^O) = 0, A^O) = 0. Choosing 0 = A"^ and subtracting (3.59) from (3.67) we obtain (again using the convolution notation a * E defined in Section 3.3)

Adding and subtracting PN E and its time derivative at appropriate places we thus find

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Banks, Buksas, and Lin

Recognizing that

we may integrate (3.71) over (0,£) to obtain

where the T^ are defined as twice the respective six terms on the right side of (3.71). It then follows that

where (see the definition of cr\} 0C = min{c2, J3}. We proceed to derive estimates for the terms in the right side of (3.72). We have

where 7 is a uniform bound for 7,7^. Moreover,

Well- Posedness where we have used

along with a as a uniform bound for aN . Continuing, we find

and

where we have used E 6 # 2 (0, T; /f) and e Next we argue

53

Banks, Buksas, and Lin

54

To estimate the last term, we first integrate by parts with respect to t to obtain (here we use A"(0) = A]V(0) = 0)

Finally, we incorporate these estimates into inequality (3.72) to obtain

where

, and Fconstants various for appropriately chosen 4 which depend on Fi,r upper bounds but not on N. Recalling that Theorem 3 guarantees that E € H2(Q,T;H)ftHl(Q,T]V), we may use (3.61) to conclude that \PNE(r) - E(r}\H ,\PN E(r] - E(r)\v, and \PN E(r] — E(T}\H converge to zero as N —> oo and that this convergence is dominated. Moreover, since {-E"(*)}te[o,T] is compact in V, we actually have \E(t) — PN E(t}\y —> 0 uniformly in t. Thus the uniform convergences N (e?,«/N,PN,aN) ) -* 0 uniformly (e r ,7,/3,a) allow us to conclude that 6 ( tto

Well-Posedness

55

in t. A simple application of Gronwall's inequality in (3.73) then yields

We remark that the above arguments and results along with (3.69) actually allow us to conclude that

and, moreover,

This latter convergence uses the fact that E 6 //"^O, T; //) C C(0, T; H) so that {^(Oltelo.T] ig compact in H and hence the convergence \PNE(t) — E(t}\u is uniform in t. As noted earlier, choosing qN = q for all N in (3.74), (3.75) yields relatively strong convergence properties for the finite element scheme of Section 3.4 based on piece wise linear splines.

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Chapter 4

Computational Methods for Dielectrics with Supraconductive Backing 4.1

The Forward Problem for the Debye Polarization Model

In this chapter we present computational results for both forward problem sim ulations and inverse problems based on the general formulation for the onedimensional geometry with supraconductive backing given in Section 2.3. We are concerned here with numerical results for the special case of a Debye medium £) with e r (z) = 600 denning the instantaneous polarization in Q. First, however, we formulate a Galerkin finite element approximation scheme for the system with general polarization. We return to the differential equation (2.11) and express it in weak form by

where the polarization P is for the moment of the form given in (2.1) and the mapping to fi = [0,1] has already been carried out. The term ^E(t, 0)0(0) is part of the weak form of the absorbing boundary condition. To facilitate the computations, we scale the time variable by a factor of c — 1/^/eo/^o and polarization P by a factor of 1/eo (i.e., t = ct,P = P/CO}Furthermore, we assume that the electric permittivity and magnetic permeability of the medium Q are constant. The new equation in the scaled variables 57

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(where we have dropped the overtildes on the scaled variables) is

where r(z) =e1 + /n(-2)(eoo — 1) is the relative electric permittivity so that ere0 = eand the impedance of free space is defined 770 = \//W^o ~ 376. Moreover, the {•, •} are the weighted inner products discussed in Section 2.3. We employ a first order Galerkin finite element approximation to discretize the problem in the space variable, yielding piecewise linear approximations for E(-,z) and P(-,z). We partition the interval [0,1] uniformly at the points z^ = ih, where h = l/N and i = 0, . . . , N and construct the standard piecewise linear spline functions ^(z) such that ^ (z^} = dij for i,j = 0, . . . ,N. We omit 0$ in constructing our finite dimensional approximating subspaces VN = spanj^, 0^, . . . , 0jv-i) so tnat f°r a^ the Dasis functions we have the essential boundary conditions 0f^(l) = 0. The computations detailed here are also simplified by the further requirement that the material boundaries of the slab Q — [zi, 1] coincide with grid points. We denote the index of the left boundary z\ of fi by j = L, i.e., z£ = z\. Since the right edge of the material ha been mapped to z = 1, this corresponds to the grid point z$. We seek an approximate solution of (4.1) in the space VN C V = ##(0, 1). Let EN and PN denote the the approximations of E and P in this space so that

By allowing both the space of solutions and space of test functions in (4.1) to be VN in the weak form of the equation, we obtain in the usual way the Galerkin finite dimensional system of equations given by

for e = (et?,e?,...,eyf_1) and p = (p(f,p?, . . . ,pj

The elements of the resulting N x N finite element matrices are computed in the usual manner (for i,j = 1, 2, . . . , N) by

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59

while

where the integrals are expressed in the scaled variables described in Section 2.3. We note that the variables e, p as well as the coefficient matrices should carry the index N, i.e., pN ,eN , MN ,KN , etc. But since this is well understood, we shall reduce the notation to that given in (4.4) and (4.5) in our subsequent discussions, reminding the reader that as usual all these quantities depend on the spatial discretization index N in the obvious ways. We have not yet imposed a particular constitutive law to govern polarization in the above formulation. We now restrict our consideration to the Debye model given in (2.4). Applying the same scaling in time and to P as above, (i.e., P = P/eo,i = ct) we obtain the scaled Debye polarization law

where d = ese — €00 and A = I/CT. Since this equation only holds inside the material domain, we can equivalently multiply the entire equation by /Q(Z); then the Galerkin approximation results in the system of equations

The matrix MQ is singular (the first L — 1 rows and the first L — 1 columns vanish identically), so in actual computations we solve this equation for the nontrivial variables pi,i = L, L + 1, . . . JV (i.e., po —Pi = • • • = PL-I — 0). This is equivalent to considering only the Lth through ATth elements of each vector (p, e) in the Galerkin approximation equation for P. With this tacit understanding, we may write the entire system of equations (4.4), (4.7) as

By substituting (4.9) and its derivative into (4.8) we obtain an equivalent system of equations

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This can be written as a first order system in the composite variable x = (e,p,e) as or

where

and ILR is the N x N identity matrix where the ones have been replaced with zeros in rows 1 through L — l. (We note that explicit dependence of the matrices Mi on the dielectric parameters are suggestive of the possibility of identifiability of these parameters.) We compute an approximate solution to this differential system with the standard Crank-Nicholson scheme, which is a member of a single parameter (9) family of schemes. Briefly, this can be summarized as follows. For a given value of 9 and a step size fc, the family of schemes applied to the differential equation x = f(t,x) yields the sequence of iterates xn w x(tn) = x(nk), where

and

This family includes the Euler scheme when 9 = 0, the Crank-Nicholson scheme when 9 — 1/2, and the implicit Euler when 9 = 1. Since xn+i appears on both sides of (4.12), the method is implicit unless 9 — 0. Since our system is linear, it can be solved directly for the value of x n +i even in the case 9^0. Applying sthis to our matrix system, we obtain another matrix problem for the iterations

where and XQ = 0, since the material is assumed to be initially electrically inactive. Equation (4.13) is reduced through block-Gaussian elimination to a block uppertriangular system of size 3N — L +1, in which only a single block of size N needs

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Figure 4.1: Debye model simulation (t = 0.7 ns) with distance in meters, electric field in volts/meter. to be factored. The LU factorization of this block can be computed once and used throughout the computation. Computational experiments with different values of 9 indicated a loss of stability in the numerical algorithm for values smaller than 0.5 and excessive dissipation for values of 9 which were substantially larger. Hence in the numerical results described below all calculations were carried out with the standard Crank-Nicholson time stepping scheme. We report graphically a sample of results from our forward simulations for the model with Debye polarization. Figures 4.1 through 4.4 depict (through time snapshots at t = 0.7, 5.0, 7.0,10.0 ns) the propagation of an electromagnetic wave through a material slab lying in z £ (.33, .89) meters. The material parameters (chosen here simply to test the computational algorithm) are a = 1.0 x 10~2 Ohm -1 , r = 1.0 x 10~n seconds, es = 35, e^ = 5, and the input has the form (2.16) with angular carrier frequency cj = 27r x 1.8 x 109 rad/sec. The numerical method is as described above, with the results depicted in the unsealed (i.e., in the original scales) spatial (z) and time (t) axes. In Figure 4.1 the incoming wave generated by the current source at z = 0 has yet to reach the left edge of the material at z\ = 1/3. In Figure 4.2 the signal has subsequently been partially reflected and partially transmitted. The reflected part of the field is the first part to be measured in the inverse problems discussed in the next section. In the transmitted part we see the formation of the signal precursor (the Brillouin precursors — see [APM89, Bri60]), which becomes more pronounced in Figures 4.3 and 4.4. These simulations were performed on the

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Figure 4.2: Debye model simulation (t = 5.0 ns) with distance in meters, electric field in volts/meter.

Figure 4.3: Debye model simulation (t = 7.0 ns) with distance in meters, electric field in volts/meter. reference domain Q = [0,1] with N = 450 and the time step was k = 1 x 10~4. Comparison of these (and other simulations) with independently generated solutions (finite difference and Fourier series) demonstrated the general accuracy

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63

Figure 4.4: Debye model simulation (t = 10.0 ns) with distance in meters, electric field in volts/meter.

and efficiency of the piecewise linear/Crank-Nicholson approximation methods in forward simulations. Moreover, such comparisons were convincing evidence that the dispersion in the solutions was due to the dielectric models and not due to any dispersion introduced by approximation techniques. We thus turned next to the use of these ideas in inverse problem techniques.

4.2

The Inverse Problem for a Debye Medium

The objective of the inverse problem is the reconstruction of the values of the parameters in the polarization model and the boundary geometry using information obtained through a scattering experiment of the type described in Section 2.1. Observations from the experiment are limited to sampled values of the electric field at selected points outside the material domain f£. The estimation problem is to minimize a suitable measure of the difference between the simulated prediction and a set of data taken from experiments as formulated in Section 3.5 of Chapter 3 above. The goal of our investigation here is to test the feasibility of this approach for the identification of dielectric and geometric parameters. We use the same physical configuration as in the forward problem described in Section 2.3; this involves a homogeneous slab obeying the Debye model of polarization and a planar electromagnetic interrogating signal. Therefore our

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observations and data consist of scalar values representing the i component of the electric field. The experimental observations consist of the value of the electric field at z = 0 at uniform intervals in time fi = iAT. Let EI denote the data we seek to reconstruct and E(t, z; q) be the electric field arising from a scattering experiment with dielectric and geometric parameters q. The inverse problem is performed by minimizing the /2 difference between the data and the simulation results. That is, we solve

where S is the number of sample data points. The set of admissible parameters Q is chosen to enforce limitations on the parameters that arise from physical or geometric considerations. The formulation of electromagnetic problems generally requires that physical parameters being estimated (e.g., dielectric parameters in polarization models such as those of Debye or Lorentz) be nonnegative. Geometric bounds will be determined according to how the boundary is represented by the parameters and will reflect both mathematical constraints and physical limitations on the boundary. This set can be made compact by providing both upper and lower bounds for each member. We remark that in actuality, we solve the approximate inverse problem depending on the Galerkin index N as discussed in Chapter 3.5. Thus, the criterion J and all solutions E should carry this index (which will be fixed unless otherwise stated in the results described below). Consistent with the convention descrbed earlier in this chapter, we shall, for notational convenience, suppress the explicit dependence on the index N. To test the feasibility of the estimation approach we produce synthetic data for the observations Ei by adding random noise to the results of the simulation with a known set of parameters. The absolute magnitude of the noise is relative to the size of the signal, reflecting the relative nature of uncertainties in measurements. Letting Ei be the data sampled from the solution with the true parameters, i.e., Ei = E(ti,0;q^}, we define Ei — Ei(l + vrji), where the rji are independent normally distributed random variables with mean zero and variance one. The coefficient v is chosen to adjust the relative magnitude of the noise. We express the magnitude of the noise as a percentage of the size of the signal by taking two times the standard deviation as the size of the random variable. Hence v = 0.05 corresponds to 10% noise and v = 0.025 to 5% noise. The feasibility of the inverse problem is measured by how successful it is at recovering the original values q* and the sensitivity of the results with respect to the magnitude of the noise v. In the absence of noise, an exact match for the parameter values has an error of zero (or roughly machine precision, allowing for nonessential differences in the method of computation). Minimizing J(q) is performed though an l^ trust region adaptation of Newton's method, using

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65

a Broyden-Fletcher-Goldfarb-Shanno (BFGS) secant update (see [Kel99]) for the approximation of the Hessian of the objective function d^d J(q)The parameters arising in the Debye model of polarization are the conductivity cr, the infinite and static limits of the dielectric permittivity €00, es and the relaxation time r. (Strictly speaking, the conductivity a is not part of the polarization model, but for convenience we will always group it with the other parameters which describe the dielectric properties of the material.) We will directly identify the related set of parameters
4.2.1

Sample Results

We present results from the inverse problem of identifying the dielectric parameters in the Debye model. All computations (electric field solutions and data generation) were performed with N = 100 and time step k — 1.0 x 10~3 ns. The data to be identified are generated with the Debye model using the following ("true") parameter values:

These parameters are considered reasonable for modeling the polarization behavior of water [BF95, p. 1505]. The carrier frequency of the interrogating signal is 1.2 GHz (hence the angular frequency is u = IK x 1.2 x 109 = 7.54 x 109 rad/s) and the duration of the window is if — 1.67 ns, which is two complete periods. The data consist of 100 measurements of the electric field taken at 2 = 0 every 0.06 ns from t = 0.06 ns to t = 6 ns. The width of the slab is fixed at 0.0375 meters, a quantity sufficient to ensure that the reflection from the surface at z = z\ is independent or temporally separated from the reflection from the metallic backing at z = z 2 . Eight different attempts are made at the inverse problem, each with a different level of random noise added to the data. The initial values of parameters used in all of the inverse problems are
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Table 4.1: Estimated parameters in Debye inverse problem: Test 1. Test % Noise True values: 1 0.0 2 0.5 3 1.0 4 1.5 5 2.0 6 2.5 7 3.0 8 5.0

a 1.0 x 10~5 9. 96 x 10 -6 0.00 0.00 1. 14 x 10 -2 0.00 7.34 x 10 -3 0.00 5. 56 x 10 -2

r 8.1 x 10--12 8.10 x 10 -12 1.23 x 10 -11 9.18 x 10-12 2.89 x 10-11 6.49 x 10 -12 1.93 x 10 -11 1.30 x 10-11 3.77 x 10-12

es 80 .1 80. 10 80. 20 80. 15 81. 18 80.46 80. 04 79. 95 78. 82

Coo

Residual

5.5 5.51 29.74 13.89 60.02 1.00 48.68 31.10 28.07

8.152 x 10~10 2.180 x 10~2 1.284 x 10"1 2.631 x 10-1 4.748 x ID"1 5.752 x ID"1 8.347 x 10-1 2.034

the true values and are typical of values used in testing algorithms [BK89b] in inverse problems.) The results are summarized in Table 4.1. We note that only the parameter es is consistently recovered in this inverse problem. While some of the values of r are close, the converged values of a and 600 are substantially off at all levels of noise. A partial explanation for the differing sensitivities can be found by reexpressing in an equivalent manner the equations which give rise to the wave equation and polarization equation. Consider the polarization equation,

The small magnitude of r suggests that P w eo(e« ~ eoo)E, f°r useful practical cases. We numerically verified P ~ eo(es — e00)£l by computing the ratio P/E during the pulse in the example given above and throughout the material domain. We observed that P/E ~ eo(e« ~ eoo) to within 10% for nearly all time points considered within the pulse. Therefore, we define Ps — eo(es — e^E so the total polarization may be considered as the sum of a static (nondispersive) and transient (dispersive) component. That is, P = Ps -f PT. The remaining polarization PT is easily seen to satisfy the differential equation

Note that PT satisfies the same equation as the original polarization P except that the driving function is different. We use the partitioning of the total polarization response into a transient and static term to rewrite the expression for the electric flux density and find

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Figure 4.5: Plot of

When the wave equation is derived from this new expression for D, es replaces €00 as the coefficient of the E term (i.e., the coefficient of E becomes er = 1 + -fo( 2 )( € s — !))• Additionally, the parameter €00 now appears only in the differential equation for PT. A simple calculation based on the magnitudes of the driving terms in the differential equations for P and PT indicates that the magnitude of PT is more than an order of magnitude less than P and Ps. (This was also verified by computing these values in the example given above.) This raises the question of whether e^ is effectively separable from e s , or in fact whether or not it is effectively estimable. We illustrate this idea graphically by plotting the objective function J(q) as a function of 600 alone. The plots of the objective functions in Figures 4.5 and 4.6 are normalized by subtracting the value of «/( = 2-rr x 1.2 rad/s the minimizing value of e^ has apparently slipped below the minimum value of 6^ = 1, the lowest value of e^ allowable on physical grounds. For comparison, we also show the dependence of the objective function on the parameters es in Figure 4.7 and a in Figure 4.8. Even in a comparatively small range of values for 6S, we see that the dependence is much stronger and that the minimum value is not perturbed by noise as far as e^.

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figure 4.6: Plot oof

Table 4.2: «/(
Frequency (GHz) 3.6 1.2 0.772 5.30 3.01 21.20

With respect to the conductivity paramater cr, the dependence of the objective function is moderate for a range of values which is several orders of magnitude larger than the value of a itself. Hence, we expect that cr is a difficult parameter to recover accurately, when accuracy is measured relative to the size of the parameter. This has been observed by others using different techniques [Alb97, KK86b]. The new polarization equation (4.15) also suggests that it may be difficult to independently estimate the parameters r and 600. Repeating the analysis of the original polarization equation on (4.15) for PT indicates that PT ~ — T6o(es — €00} E, with the remaining polarization effects being comparatively small in magnitide. If we assume that PT — —T€0(es — e^E, we have eliminated all other appearences of both r and 600 from the equations. (The parameter es, which is also present in this equation, also appears in the new version of the wave equation.) We will see in the examples below that it is possible to estimate the size of the driving term's coefficient —reo(e s — COQ), but there is

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Figure 4.7: Plot of J(q] — J(
Figure 4.8: Plot of

no additional means to separate the individual contributions of r and e^. This observation is borne out by results from another inverse problem in which the values a and e^ are held fixed. The conductivity a is fixed at zero while €00 is set to various values. (The interrogating frequency for this example is 3.6

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Table 4.3: Parameters resulting from incorrect e^. Fixed £00 1.0 5.5 10.0

Estimated r 7.62 x 10~12 8.10 x 10~12 8.64 x 10-12

Estimated es 79.96 80.10 80.26

Residual 3.94 x 10~4 6.36 x 10~n 1.61 x 10~4

r(es - GOO)

6.02 x 10~10 6.04 x 10-10 6.07 x 10~10

Table 4.4: Estimated parameters in Debye inverse problem: Test 2. Test % Noise True values: 1 0.0 0.5 2 3 1.0 1.5 4 5 2.0 2.5 6 7 3.0 5.0 8

a (fixed) 1.0 x 10~5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

r es
Residual 3.942 x 10-3 0.208 0.834 1.877 3.338 5.217 7.513 20.87

GHz.) Performing the inverse problem without added noise, we obtain different values of r according to the fixed value of e^. These results are summarized in Table 4.3. The residual column of Table 4.3 indicates that it is possible to get an acceptably low residual value with parameters far from the "true" values which generated the data. The last column shows that the coefficient of the driving term in the polarization equation (4.15) is being accurately reconstructed each time, even though the particular values of r and e^ are wrong. This supports the idea that the difficulty surrounding the identification of the parameters e^ and r arises because these two parameters are strongly coupled together in their influence on the objective function. The relatively low sensitivity of the objective function to these parameters (compared to e s ) further suggests that parameter values obtained by arbitrarily fixing one of the parameters and identifying a corresponding value for the other will yield parameters which capture the dynamics of the problem accurately (i.e., yield a small residual). To test this approach we perform the optimization problem with the parameter e^ fixed at the value 1.0 and the conductivity a fixed at zero. The results are summarized in Table 4.4. In the event that the exact value of e^ is known, we can fix the parameter at this value in the optimizations. The results of performing the inverse problem with CQO fixed at its correct value of 5.5 are shown in Table 4.5. We notice

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Table 4.5: Estimated parameters in Debye inverse problem: Test 3. Test % Noise True values: 1 0.0 2 0.5 3 1.0 4 1.5 5 2.0 6 2.5 7 3.0 8 5.0

a (fixed) 1.0 x 10~5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

T

600 (fixed) 80.1 5.5 80.10 5.5 80.5 5.5 80.93 5.5 81.35 5.5 81.77 5.5 82.06 5.5 5.5 82.63 84.39 5.5 ts

12

8.1 x 10~ 8.10 x 10~12 8.11 x 10~12 8.12 x ID"12 8.13 x ID"12 8.14 x 1Q-12 7.68 x 10-12 8.15 x 10-12 8.19 x 10~12

Residual 6.364 x ID"10 0.209 0.835 1.879 3.341 5.220 7.517 20.88

that, except in the case of zero noise, the residuals are slightly higher when the correct value of €00 is used than when €00 = 1.0 is used. This is explained by our observation in Figures 4.5 and 4.6 that the presence of noise uniformly lowers the optimal value of e^.

4.2.2

Reconstruction with Inferior Accuracy

As the data used in the above inverse problems are generated with the same numerical scheme that is used in the reconstruction process (a practice that is generally unacceptable in testing inverse problem methodology), we must be aware of the possibility that this affects in an unfair manner the tests of reliability of the reconstruction, and that, indeed, it might facilitate easier identification. Although we expect that the addition of random noise to the generated data would mask its fine details well enough to prevent unusually good convergence (and thus prevent baised conclusions about our identification algorithms), we nevertheless test for the possibility by performing the reconstruction using approximate solutions that are different in accuracy than the one used to generate the data. This is done by generating the data in a simulation with larger values of JV, smaller values of k or both. The results are summarized in Table 4.6 and demonstrate that concerns in this regard are unfounded. The second row of Table 4.6 contains the result in which the parameters in the reconstruction match those in the generation of data (N = 100, k = 3 x 10~ 4 ), resulting in a low residual and accurate reconstruction of parameter values because no noise has been added to the data. The remaining rows indicate the results when different values of N and k were used in the generation of the data while the same values of N = 100 and k = 3 x 10~4 were used in the reconstruction. The results obtained are similar to adding a moderate amount of noise to the data, as seen by comparison with Table 4.1.

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Table 4.6: Estimated parameters in Debye inverse problem with inferior accuracy. N k True values: 100 3 x 10~4 400 3 x 10-4 100 1 x 10~5 400 1 x 10~5

4.2.3

a 1.0 x 10~5 9.96 x 10~6 0.00 0.00 0.00

T

8.1 x 10-12 8.10 x ID"12 9.29 x ID"12 1.07 x ID"11 9.89 x 10-12

es 80.1 80.10 80.21 80.23 80.28

COG

Residual

5.5 5.51 14.98 22.88 17.67

8.152 x ID"10 1.530 x 10-3 4.206 x ID"2 4.430 x 10~2

Identification of Material Depth

Identifying the depth of the material with this method proved to be more difficult than identifying other parameters. We add the scaling parameter to the list of parameters to be identified, so q = (er, e^,^, / \ ? £) ? and we consider a slightly different inverse problem. In this case, the material is sufficiently thin and the duration of the data collection sufficiently long so that the part of the interrogating signal transmitted through the medium had returned after being reflected off the metallic or supraconductive back boundary. This is clearly necessary in any attempt to recover the depth of the sample. We see in Figure 4.9 the complicated dependence of J(q,d} on d, owing to the oscillatory nature of the time domain data. (The results are presented in terms of the actual thickness d instead of the scaling parameter (,.} Here, J(q*,d) is plotted for various values of d with the other parameters held fixed at their true values. The true depth of the material is d — 0.05 m, corresponding to the global minimum of J ( q ) . (This example uses the parameter values for water given above.) Figure 4.10 depicts one cause of the multiple local minima apparent in Figure 4.9. With the incorrect value of d, the return of the transmitted and subsequently reflected part of the interrogating signal is delayed by two full periods of the input signal. This brings it back into partial synchronization with the data, giving it a smaller error than nearby values where the data is out of phase. This produces a local minima as seen in Figure 4.9. Optimization results using the function J ( q ) , where
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Figure 4.9: Error as a functions of material depth. Depth is in meters.

Figure 4.10: Illustration of local minima at d = 0.07. Depth is in meters. terial) parameters and q — (
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Let tr(q, d) be the time at which the signal which penetrates ft first returns to z = 0 after being transmitted through the medium and reflected off its back surface, expressed as a function of the material parameters and the depth of the sample. Let A and 6 be positive constants with values to be determined and t1* be the duration of the interrogation signal. We approximate the return time by tr(q, d) w fr(q, d) = t^ where i is the first index such that

Equation (4.16) is satisfied when the finite difference approximation to Et(t, 0) is greater than the constant A. The constraint in (4.17) prevents the test from detecting the first reflection off the surface of fi, which is over at t — 1z\ -f-18*. (Note that the time is expressed in the scaled variable and hence the speed of light is unity.) The parameter 6 is chosen to delay detection further in order to avoid detecting numerical noise or the decaying electric field which trails behind the reflection when the value of the conductivity a is large. Typical values of these parameters which give good results are A = 20.0 and 6 — 0.05. We use the estimated return time f(q, d) to identify the true material depth d*. Let t(q*,d*) be the estimated return time (satisfying equations (4.16) and (4.17)) for the true data E1;, generated with the true parameter values q* and material depth d*. We then look for a root of the equation ft(d) = t(q*,d} — £(g*,d*), which represents the difference in estimated return times between the simulation and the data. The function ft(d) corresponding to the example given in detail below is plotted in Figure 4.11. The plateau of values for large d occurs because the deep reflection no longer returns before the end of the simulation and the detection algorithm returns tf, yielding ft(d} = t f — tr(q*,d*). Because ft(d) is a discontinuous function of a single variable, we employ Brent's method [PTVF94, p. 352] to approximate the root, since it does not rely on derivative information. The piecewise constant character of ft(d) also limits the accuracy available in the estimate of d. Therefore, we adopt the following two-step approach to finding the final value of this parameter. (1) Use Brent's method to find an estimate d\ of the root of ft(d). (2) Use (j = (1 — zi)/di as an initial value for the continuous optimization of J(q*,d) over the single parameter £.

4.2.4

Results from the Two-Step Algorithm

The two step approach described above is extremely effective. For a test problem we use the physical parameters for water (a = 1.0 x 10~5,es = 80.1, €00 = 5.5,r = 8.1 x 10~12) and a material thickness of d* = 0.05 m. Note that only the thickness of the material will be estimated in this inverse problem. The

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Figure 4.11: ft(d) for example problem. Depth d is in meters.

leading edge of the material is at z\ — 0.25 and the simulation is run until tf = 7.33 ns, which is sufficient time for the transmitted part of the signal to return to z = 0 after subsequent reflections off the far boundary. The duration of the input signal is tsj = 1.66 ns and its frequency is LJ = IK x 1.8 x 109 rad/sec, yielding three full oscillations. The data is sampled 110 times over the duration of the simulation, hence AT = 0.0667 ns. The resulting data collected at z = 0 is plotted in Figure 4.12. The transmitted pulse can be seen passing through z = 0 over the time span 4.3 ns to 6.7 ns. In these simulations, the parameters in the signal return detection are A = 20.0,6 = 0.05 and we found that a wide variety of initial values of d in the range (0,1.0) lead to convergence to the correct value. The identification of the material depth is also very insensitive to noise in the data. We tested the sensitivity of the problem by adding normally distributed random noise of various magnitudes to the data set as in Section 4.2 and repeated the inverse problems using the initial value d0 — 0.02, with the results obtained shown in Table 4.7. Here we can see that noise as large as the amplitude of the original signal does not affect the depth estimation very much. This robustness of the depth estimate is due in part to the nature of the noise. Since it is relative noise, it has little effect on the data points between the surface reflection and the return of the signal which penetrated the material since these values are all nearly zero. Thus the return time of the signal is not affected, since the point at which the data has its first substantial jump is the same. Furthermore, the minimum of

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Figure 4.12: EI for inverse problem. Time in seconds.

Table 4.7: Results of depth estimation. Noise level % 5.0 10.0 20.0 30.0 40.0 100.0

Converged to 0.0499999128 0.0499998393 0.0499996808 0.0499994705 0.0499993745 0.0499983235

J(q) is still attained by the value of d for which the penetrating part of the signal returns at the correct time, causing the zero and nonzero parts of the data to match.

4.2.5

Estimating Depth and Physical Parameters

The method described above for estimating the depth d is limited by the need to know the exact values of the physical parameters q* to form the function ft(d] = t(q*,d] — f(q*,d*). We eliminate this requirement by modifying the algorithm and obtaining the following three step process.

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77

(1) Estimate the physical parameters by minimizing J(q) over the physical parameters q only. Do this over a data set sufficiently short so that it only contains the exterior surface (i.e., z = z\] reflection. Choose (o = (1 — z i ) / d o such that the penetrating signal does not return from the back boundary in any of the simulations. To do this, choose t'f < tr(q*, d*} and t'f < tr(q,do) for a wide range of values of q and do. Estimate qi only using the data Ei, where ti < t'f. (2) Using the estimate q\ from step 1, find the root of f l ( d } = i(q\,d) — f(q*,d*). Call this estimate d\. (3) Use the estimated value (j = (1 — z\)/d\ from step 2 and the estimate q\ from step (1) as an initial estimate for minimizing J ( q ) . To accomplish step (1), we use the physical property that the speed of propagation of waves in the media cannot be any faster than is it in vacuum. Since the deep reflection must transverse the distance z\ twice at speed one and the depth of the material twice at no more than speed one, tr(q,d) > 1(z\ + d) for all parameter values q. We need to choose the reduced test time t'f so that it is after the end of the first reflection and before the beginning of the deep reflection. Hence,

Using this conservative estimate, we can guarantee the existence of a suitable test time t'f, where d > tsff2 and where t** G [2zi +t{,2zi +2d]. In practice, we can relax the constraint on d because the typical speed on propagation in these materials is much less than 1. This is the case in the example given in the next section.

4.2.6

Results for the Simultaneous Estimation of Depth and Dielectric Parameters

As in the previous example, we use the physical parameters describing water for the generation of data: a* = 1.5 x 1(T5, T* = 8.1 x 10~12, e* = 80.1, e^ = 5.5, and the same material depth: d* = 0.05. All of these parameters will now be estimated in the inverse problem. The geometry and other parameters of the problem are identical to the previous example. Notice that the right inequality in (4.18) is not satisfied by this value of d*. We can see from Figure 4.12 that the slow propagation of the signal through the medium nevertheless results in a gap between the end of the first reflection and the beginning of the transmitted signal. The first stage of the optimization is carried out over half of the data up to the time t'f — 3.4 ns. The data was generated numerically using 600 basis

Banks, Buksas, and Lin

78

Table 4.8: Simultaneous estimation of Debye parameters and depth: Test 1. % Noise True values: Initial values: 0.0 0.5 1.0 1.5 2.0 2.5

a 5 1. O x io1.Ox 10~5 5 2 . O x io3 1.3x io0. 0 3 70 3. x io0. 0 3 4. 23 x io-

T

S .1 X 108 .2 x 10--12 8 07 x 10 -12 7 86 x 10 -12 7 76 x 10 -12 7 99 x 10 -12 7 76 x 10 -12 7 52 x 10 -12 -12

6S

Coo

80.1

5.5 5.3 5. 19 3. 33 2. 23 4. 37 2. 38 0 .0

81.0

80.09 79.84 79.85 78.94 80.31 77.82

d 0.05 0.055 0.0500 0.0501 0.0501 0.0503 0.0499 0.0503

Table 4.9: Simultaneous estimation: Test 2. % Noise True values: Initial values: 0.0 0.5 1.0 1.5 2.0 2.5

a

T

1.0 x IO-5 1.0 x ID"3 8.2 x IO-4 1.3 x IO- 2 0.0 3.70 x IO-2 8.00 x 10~2 4.23 x IO-5

8.1 x IO-12 10.0 x IO-12 8.11 x IO-12 7.88 x IO-12 7.77 x 10~12 7.99 x ID"12 12.36 x IO-12 7.52 x 10~12

e« 80.1 90.0 80.10 79.84 79.85 78.94 80.60 78.82

£00 5.5 3.0 5.58 3.48 2.29 4.33 41.72 0.0

d 0.05 0.1 0.0500 0.0501 0.0501 0.0504 0.119 0.0504

elements (i.e., N = 600) in the spatial (finite element) approximation. As before, random noise is added to the results of the forward problem to generate synthetic data for the inverse problem. Various initial values are used to test the range of convergence for the optimization problem. The results of one such test are given in Table 4.8. As before, we see that the estimates of a and e^ are not very good, while es and T are comparatively robust with respect to noise in the data. Further, the estimated depth of the sample is largely insensitive to noise. Performing the experiment again with a different initial value yields similar results as given in Table 4.9. Aside from the unreasonable values (underlined in Table 4.9) which appear in the case of 2.0% noise, these results resemble those in the previous trial, especially in the values of es and T. As before, the depth measurement is quite accurate. The results of the 2.0% trial indicate that the method as presented here is not perfectly robust, and some caution interpreting results is called for.

Computational Methods for Dielectrics with Supraconductive Backing

4.3

79

The Forward Problem for the Lorentz Model

The underlying numerical formulation for simulations using the Lorentz model is the same as that for the Debye model up to and including the the finite dimensional system of differential equations in (4.4) and the matrix definition in (4.5). We depart from the previous formulation in the choice of constitutive law to govern polarization. We use the Lorentz equation discussed in Section 2.1. Applying the same scaling as for the Debye law in previous sections, we obtain the scaled Lorentz polarization law

where u>0 = UJO/C,LJP = up/c and A = l/2cr. The derivation of the finite element equations for the polarization precedes exactly as in the Debye case, with the same provision that components of the variables p and p which are identically zero are dropped. (These are the ones corresponding to the exterior of the material domain fi.) The resulting system of equations is

By substituting (4.21) into (4.20), we obtain an equivalent system of equations

This can also be written as a first order system in the composite variable x — (e,p,e,p) as where

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and the submatrices are

In the implementation of the above algorithm, we avoid the construction of the matrices M, K, and F by writing a subroutine which computes the derivative of the system state x = M~1(F — Kx) when provided with the state x. This routine is then passed as an argument to another routine which performs the quadrature. The derivative computation is performed using only the smaller and banded matrices Mr and Ki and a more efficient version reuses the computed value of p where it appears in (4.20). Difficulties arise in the computation of solutions to the above equation when using physically realistic values of the material parameters, as the resulting linear system of equations is exceedingly stiff. In one example, we use the parameter values es = 2. 25,6^ = 1.0, u0 = 4.00 x 1016,r = 3.57 x 10~16 and the corresponding plasma frequency is UJP = 4.472 x 1016. These are typical values in the study of physical optics [BF95] . We perform the discretization described above on an approximation corresponding to N = 100 with the left edge of the material located at z = 0.01 meters and the right edge at z = 0.1 meters. We computed the matrices M and K using Matlab and from this found the condition number of the linear system matrix n(M~lK) = 3.23 x 1019, making this an extremely stiff system to be integrated. In the fourth order Runga-Kutta methods applied to this equation, the largest step sizes which did not lead to the blowup of the system are near k = 3 x 10~8 ns. To generate solutions to this system of equations, we must be content with very short simulations which observe the pulse propagating through a very short distance. With the restrictions described in the previous paragraph in mind, we graphically illustrate the results of the simulation with a different set of parameters. The following results are generated from a simulation where the material occupies the interval (1.0 x 10~6, 1.0 x 10~5) meters, and with material parameters (see [BF95]) es = 2.25, e^ = 1.0, o;0 = 1.779xl016 rad/sec, r - 7.14xlO~16 se The frequency of the interrogation signal is taken to be 8 x 105 GHz and the duration of the signal is equal to 12 of its periods. (The high frequency is necessary in order to see the formation of the Brillouin and Sommerfeld precursors associated with the Lorentz model.) The resulting pulse train thus has 12 complete cycles. Solutions from the simulation are displayed graphically in Figures 4.13, 4.14, 4.15, and 4.16. In Figure 4.13 the pulse is traveling to the right after having just entered the material. In Figure 4.14 it has undergone two reflections, first off the supraconducting back boundary followed by a partial reflection off the surface at z\ .

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81

z

Figure 4.13: Lorentz model simulation (t = 3.33x 10~5 ns) with depth in meters, electric field in volts/meter.

Figure 4.14: Lorentz model simulation (t = 1.33x 10~4 ns) with depth in meters, electric field in volts/meter. Figure 4.15 is two reflections later again, and in Figure 4.16 the pulse is traveling to the left after another reflection off the back boundary. Here we can see that the pulse has resolved into a part which resembles the Brillouin precursors en-

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Figure 4.15: Lorentz model simulation (t = 2.2 x 10~4 ns) with depth in meters, electric field in volts/meter.

Figure 4.16: Lorentz model simulation (t = 2.67 x 10 electric field in volts/meter.

4

ns) with depth in meters,

countered in simulations of the Debye model, and another part associated with the Sommerfeld precursors. These precursors are examined in detail in [BF95] using a frequency domain approach.

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Figure 4.17: Lorentz model simulation (t = 6.67x 10~6 ns) with depth in meters, electric field in volts/meter. These plots were made from a simulation using 1,000 basis elements (N — 1000) in the discretization of the space variable and with a step size of k = 3.0 x 10~9 in the scaled time variable. This represents an actual time step of 1.0 x 10~8 ns and over 26,000 steps are required to reach the state shown in Figure 4.16. Another illustrative example is obtained by setting the input frequency to the resonance frequency of the Lorentz oscillator. The resonance frequency for a second order oscillator as described in (2.5) is

Using the same values of the physical parameters as before, es = 2.25, €00 = 1.0,o;0 = 1-779 x 1016,r = 7.14 x 10~16 the damping in the model is very small and the resonance frequency is nearly identical to the natural frequency of the material, i.e., wr « UJQ. In this set of simulations, we take the driving frequency of the input signal to be the resonance frequency uj — ujr = 1.779 x 107 x 109 rad/sec. Because of the higher frequency, even finer discretizations of the space and time variables are required, and hence we use N — 20000 and dt = 1.0 x 10~10. Also, the material occupies the interval (4 x 10~7,1 x 10~5). The results of the simulation are given in Figures 4.17, 4.18, 4.19, and 4.20. Figure 4.17 shows the pulse at t = 6.67 x 10~6 ns, shortly after its entry into the medium. The closeup in Figure 4.18 shows some distortion in the shape of the originally square-rectified pulse. Figures 4.19 and 4.20 show the pulse at

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84

Figure 4.18: Lorentz model simulation (t = 6.67 x 10 electric field in volts/meter.

6

ns) with depth in meters,

Figure 4.19: Lorentz model simulation (t = 2.67 x 10 electric field in volts/meter.

5

ns) with depth in meters,

time t = 2.67 x 10~5 ns. Here the pulse has taken on a shape characteristic of the Sommerfeld precursors visible in other simulations. (See, for example, [BF95].) Also in Figure 4.20 we can see the separate Brillouin precursor, which has separated from the Sommerfeld signal because of its lower velocity.

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Figure 4.20: Lorentz model simulation (t = 2.67x 10~5 ns) with depth in meters, electric field in volts/meter.

4.4

The Inverse Problem for a Lorentz Medium

Because of the long execution times of the forward simulations using the Lorentz model, we restrict our attention to the identification of parameters using the reflection of the interrogating signal off the surface of the material and do not attempt to identify the material's depth. Based on our experience with the Debye equation, we also do not attempt to identify the conductivity parameter a. However, unlike the Debye problem, this problem is adequately sensitive to the parameter €00. Our attempts at estimating the parameters in the Lorentz model are carried out in the context of a test problem with a slightly different set of physical parameters from those used in the example above (since those parameters were chosen to accent the precursor development for demonstration, we now choose parameters more representative of real materials [BF95]). Here we use es = 2.25,6^ = 1.0, wo = 4.0 x 1016,r = 3.57 x 10~16. The interrogating signal is given a frequency of 1.2 x 1014 Hz and the signal is stopped after four complete periods of the input (| x 10~13 sec). The data for the inverse problem is depicted graphically in Figure 4.21. The first trials of the inverse problem attempt to establish a rough radius of the acceptable initial values for the parameters to be successfully estimated. This optimization is attempted first over the two parameters LJQ and up and then over three parameters WQ, u;p and r. Note that the optimization over T is actually performed through the related parameter A = l/2cr. The initial values for the parameters are found by perturbing the "true" values by various relative

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Figure 4.21: Data for Lorentz inverse problem. Time is in scaled seconds, electric field in volts/meter. Table 4.10: Convergence results over U>Q and up. % Distance True values: 20% 30%

Coo

OJ 0

OJ

r

1.0000 4.0000 xlO 1 6 4.4721 xlO 1 6 3.57 x 10 -16 1 0001 3.9958 x 1016 4.4675 x 1016 3.57 x 10 -16 1.0001 4.0000 xlO 1 6 4.4720 xlO 1 6 3.55 x 10 -16

Residual 0.0 2 4 311 x 10~ 3 1 564 x ID"

values, i.e., UQ = (1 + x)^- A value °f X — 0-05 yields a 5% perturbation of WQ, for example. The first attempts on the Lorentz model with the l^ trust region code used with the Debye model encountered difficulties arising from the scale and sensitivity to the parameters. The algorithm quickly became saddled with a trust region which was too small to allow sufficient progress to the minimum, or which caused successive steps short enough to trigger the convergence criterion prematurely. This was remedied by switching to a line-search method and scaling the parameters to give them similar magnitudes. The scaling factors used with each parameter are 1 for €00 > 1 x 108 for UJQ and cc^ and 1 x 106 for A. This produced the results given in Table 4.10. Table 4.10 suggests that initial parameter values which are within 30% of the true values are suitable for initial iterates, at least in the absence of noise in the data. Note that the parameter values we have chosen here are suitable for the representation of real materials.

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Table 4.11: Results of Lorentz inverse problem in the presence of noise. % Noise True values: 1.0% 2.0% 3.0% 4.0% 5.0% 6.0%

^oo

1.0 1 00016 1 00000 1 00038 1 00012 1.79856 1 87854

UJQ

4 000 x 4 006 x 4 019 x 4 028 x 4 034 x 2 540 x 2 356 x

16

10 1016 1016 1016 1016 1016 1016

up 4 472 x 4 480 x 4 496 x 4 472 x 4 516 x 1 715 x 1 445 x

T 16

10 1016 1016 1016 1016 1016 1016

3.570 x 3.239 x 3.238 x 3.236 x 3.233 x 3.100 x 2.976 x

10- 16 10- 16 10~ 16 10- 16 10- 16 10~ 16 10- 16

Figgure 4.22

The inverse problem is then attempted in the presence of noise of uniform relative amplitude applied to the observed data in the manner described in Section 4.2 for the Debye problem. The initial values for the parameters are perturbed by 10% from the correct values (with one exception noted below) and the results of these optimizations are given in Table 4.11. We notice that the quality of the results deteriorates at 5.0% noise and that r is the most difficult of the parameters to recover accurately. When considered as a function of r only, the objective function exhibits considerable sensitivity with respect to noise, as shown in Figure 4.22. Here, we can see that the objective function is much less sensitive to the value of r even in the presence of a small amount of noise. The minimizing values of r in the presence of various amounts of noise are given in Table 4.12.

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Banks, Buksas, and Lin Table 4.12: Results of identifying % noise 1.0% 2.0% 3.0% 4.0% 5.0% 7.0%

Estimated r 1.1291 x 10~16 6.1779 x ID"17 4.5353 x 10-17 3.6834 x ID"17 3.1432 x 10-17 2.4687 x 10~17

r.

Chapter 5

Computational Methods for General Polarization Models 5.1

The Forward Problem

We begin with the differential equation (2.11) and develop a variational form of the problem much as described in Section 4.1. The time and polarization variables are scaled in the same fashion, but we do not apply the method of mappings to scale the space variable. In order to model materials of various thicknesses, we allow the computational domain to be of arbitrary size, i.e., £1 = [O,/] for I > 0. (Conceptually, the right boundary of the material domain need not be z^ = / but must coincide with one of the nodes in the spatial discretization. Hence, z2 = ZR, & = {^I^R\I and ^o = ^ — ^- For our examples we do always take z^ — I.) We allow a more general problem by considering both the supraconductive boundary condition at z = I or allowing an absorbing boundary condition similar to the one at z = 0. Although it is not a generally used part of these models, we also retain the possibility of instantaneous polarization with er > I . As always, we will assume a zero initial condition for the electric field. Our more general weak form of the equation is

The new term is a contribution of the absorbing boundary condition at z = I and the inner product has been redefined as 89

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The Galerkin finite element approximation proceeds as in Section 4.1. Our two cases for the boundary condition at z = I are implemented by either retaining or omitting the function (J>N(Z) from the finite dimensional space of test functions. Let ./V denote the index of the highest test function included and VN = span {00^,0^, • • • 5^} De the space spanned by these functio supraconductive boundary is implemented with N = N — I while the absorbing boundary condition is implemented with N = N. For polarization, we use the weak form of the alternative (but equivalent) equation (2.14); that is, we use the equation

with coefficients given by

Applying the same scaling as before (f = ct, P = P/CQ) and using the approximation of E in (4.2) we obtain the following variational equations:

Note that the contributions of the term (/3£"(0, •),} in (5.1) vanish because of our chosen initial condition of £(0, z) = 0. We find the variational form of the equation by allowing 0 = (f>j for 4>j in VN, obtaining a system of size N + 1. For a function a(t, z) which varies in both variables, the integral term becomes

We make the following definitions:

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91

Note that A(t) = [j4(£)]ij is a time- varying matrix-valued function, while Mr , M, and X are constant matrices of size N. We can now write our system as

and /(£) = ~f]Qjs(t)e\^ where e\ is the unit vector with 1 in the first component.

5.1.1

Galerkin Methods for the History Approximation

We choose a method for approximating the history of e(t) which maintains the same order of accuracy as the discretization of the rest of the problem. This is prudent since the order of the accuracy of the solution is generally limited by the lowest order in accuracy of each of its several approximations (time, spatial, history). To do this, we use a Galerkin discretization of the history variable on a finite domain with a degree matching the degree of the discretization in the space variable z. For all of the results presented here, we use degree one elements for each approximation. We can limit the history variable to a finite domain because the hysteresis models of interest to us are all fading memory models, i.e., the kernel function a decays to zero as s —> — oo. The integral over the history of the electric field with this kernel function can be well approximated by truncating it to a sufficiently long interval. We consider an abstract problem with the appropriate conditions to approximate the history. Let -u(t, s) denote the solution to this problem and represent the truncated history of e with finite duration r via u(t, s) = e(t + s) for (£, s) £ [0,oo) x [— r, 0]. By its relationship to e, we see that u(t,s) satisfies,

The appropriate initial condition is w(0, s) = 0 since e(s) = 0 for all s < 0. For boundary conditions, we suppose that the state variable e(i] is known at all times t, and we can thus impose the boundary condition w(t,0) = e(t). Since it is a first order hyperbolic problem, there is a unique solution specified by u(t, s) = 0 when t < — s and u(t, s) = e(t 4- s) when t > —s. We create a Galerkin finite element approximation in the history variable s, much as was done for the space variable z in Section 4.1. Let i/jf* be continuous basis functions defined on the interval [— r, 0] with inner product (7/>i,i/>j} = J_ri^i(s)il)j(s}ds. There are a number of choices one might make for these elements, e.g., piecewise linear, quadratic, or higher order polynomial elements. Since our space variable approximations were made with piecewise linear

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elements, we chose the same type of elements here for compatibility of order. These functions are defined on the uniform grid Si = — r + j^i for i = 0 , . . . , M, where ^i(sj) = 6ij, The past history of e is approximated as an element of the space spanned by these basis functions:

Recall that e is a vector of size TV, and so each of the Ui are vectors of the same size. Let the matrix U denote these unknowns taken together as rows:

Storing the coefficients Wj as rows in this matrix means that the operators we will derive for the history-evolution are applied on the left side of U. This was deliberately chosen to simplify the numerical implementation, since one of these operators needs to be inverted, and the available linear-algebra routines are formulated with matrices applied on the left. Matrices representing operators in the space variable will be applied to the right side of U or to the left side of UT.

5.1.2 Approximating the History of e(t) We consider two different methods for rinding approximate solutions to (5.5), which differ in their variational expression and in how the boundary condition is imposed. The first method is to use a variational form which implies no boundary conditions, consider its resulting finite dimensional approximation, and then use the approximate boundary condition to eliminate an equation and unknown from the system. We will see that this approach has a particularly simple implementation. The second method is to write a variational form of the equation which imposes the boundary condition in a weak sense. Care must be taken in this approach not to introduce a spurious boundary condition at the other boundary s = —r.

Computational Methods for General Polarization Models

93

Imposing u(t, 0) = e(t) exactly: Consider the trivial variational form

No boundary conditions are implied in a variational sense by this equation, since no integration by parts is performed in order to relate it to the original PDE (5.5). We create a finite dimensional version by restricting ifj to members of the basis set T/>J in (5.8) and using the approximation (5.6). The following system of equations results: where Mij = (V'^V'j) and W»j = (ipi^j). This system is determined in the sense that the derivative is well denned, but it is not clear which PDE it approximates, since it has nothing to say about the boundary condition necessary in a well-posed continuous problem. We consider the system of (5.9) too large by one equation and one unknown, which we eliminate with the auxiliary condition arising from the boundary data. Using the same approximation (5.6) for u(t, s) we find

We assume that V ; Af(0) 7^ 0 (which is the case for the piecewise linear spline approximations used here) and solve for the coefficient UM(£)- We obtain

For the elements we use in the numerical simulations, this reduces to UM(^) = e(t), because V'i(O) = <*>0iWe seek a simple way of implementing the removal of the coefficient UM (t) from the system of (5.9). We can use Gaussian row- reduction to eliminate UM from the left-hand side, and then substitute the expression (5.11) wherever •UM(£) appears on the right. This produces a smaller system of the form MU = WU + ve(t), where v is a vector of coefficients. We avoid this process by noting that the same process of Gaussian operations can be applied just as well to both sides of (5.9) after performing the matrixvector multiplication, as they can to the matrix system itself. This allows us to substitute the value of WM(£) in (5-11) before the elimination operations. This yields the following procedure for evaluating the derivative of the reduced system (which is missing (1) Compute the correct boundary value of WM(*) from (5.11). (2) Compute U from (5.9) using the entire state it;, i ~ 1, . . . , M.

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This also produces a value for UM (t) which we ignore as the evolution of this term is determined by the boundary condition. Another possible advantage of this method is that the boundary condition is satisfied exactly, instead of in a weak sense. The effects of this on the quality of the rest of the approximation are not known. Since the above method departs from the standard finite-element approach, we can no longer say that our approximation of u is the least-squares projection into the finite dimensional subspace. For this reason we consider another approximation scheme which imposes the boundary condition as part of the weak form. Imposing the Condition u(t, 0) = e(t) weakly: As the variational formulation of the problem we choose

This implies that Ut = us and w(0) = e(0) under the standard variational arguments. Notice that the term —i/j(—r)u(—r) cancels a term which would otherwise yield the condition u(—r] = 0 after integration by parts. In the usual manner, we find a system of differential equations from the variational form (5.12),

We define the matrices M., W by Mij = (V'i, V'j then express the system of differential equations (5.13) in matrix form as

where # = (t/>0 (0) , fa (0) , . . . , V>M(O)) T - We note that the M + 1 rows of U are vector- valued functions in R lxAr+1 . This differs from the convention of treating e and e as column vectors, hence the appearance of eT in (5.14). The matrix equation (5.14) is appropriate for implementation; however, we seek to simplify the implementation further. Through integration by parts we find

Computational Methods for General Polarization Models where Wij = (V>i,V'j)

95

an

d (5-14) becomes

where W^ = Wij — ^(0)^(0). This equation can also be written as MU = WU + ty(e(t}T — tyTU). Here we can more clearly see the presence of the boundary condition in the weak form. The boundary condition can itself be expressed as e(i]T = Y^i=i ^i(fyui = ^TU. Note that the time varying terms in (5.14) are very few provided that for only a few z, 1/^(0) ^ 0. For the standard first order splines we use, Vi(0) = &iM so only one element is nonzero.

5.1.3 Implementing the Hysteresis Term We express the integral term in (5.4) in terms of the Galerkin approximation. We have

where A-3 = f _ r A ( s ) i ( ) j ( s ) d s . Recall that A(s) is a matrix function, and so the Ak are matrices of size N + I x N + I, where the ij element of Ak is

This gives us the complete discretization of the history terms in the second order system of equations (5.4), which we now can write as

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where f ( t ) = -r]0Js(i)ei and ^ = (V'o(O), V'i(O), • • • >M(O)) T . We convert this system to first order by letting x = e, y = e.

The state variables of the system we need to solve are the length N + 1 vectors x,y and the M + 1 x N + 1 matrix U. The matrices Ak, K, M, and M are as defined in (5.3). The matrices M. and W and the vector ^ depend only on the basis functions chosen for the history approximation, while M r , M, and the Ak's depend on physical parameters we will want to estimate.

5.1.4

Specific Implementation: Constant Material Parameters

We consider the special case of materials which are homogeneous throughout the domain 0. Functions which describe dielectric properties of these materials are constants modulated by the indicator function of the domain fi. These are given by

We have introduced new parameters a and go for the conductivity and zero value of g, and the kernel function a(s). As in Section 4.1, we further assume that the boundaries of the material coincide with the domains which define the test functions. This way, the material parameters are constant over each domain over which an integral is evaluated. The matrices in (5.3) become

where

and

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97

representing the contributions of the boundary conditions. For the case of a supraconducting boundary at z = 1 we have Bij = 0j_i(0) 0jv-i}We use (5.16) to find the specific form of the matrices Ak:

Thus where a^ = J_ a(s)i{jk(s}ds. Letting a = (c*o,... ,C*M) T , we can express the approximation of the hysteresis approximation in (5.15) as

The full system of (5.17) is now as follows:

where the last equation may still be replaced with

5.2 5.2.1

Results of Simulations with the General Model Dielectric Response Function Representation of the Debye Model

We measure the accuracy of solutions to the general polarization model by comparing them with those obtained from the differential Debye model of Sec-

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tion 4.1. The correct function to be used in the general polarization model to represent a Debye material is found from the variation of constants solution to the differential equation. In the scaled version of the equations we have

Use of an integrating factor yields

where g(s} = 6dXe~Xs. From this we calculate

The function a which appears in the definitions (5.19) is ot(s) = —€^A 2 e As and the remaining parameters are er = COQ, go — e^A. We compute the solution to a test problem using both the differential and hysteresis Debye polarization models. The results are summarized in Table 5.1. The physical parameters are a — 0.0, e^ = 5.5, es =80.1, and r — 8.1 x 10~12 s, and the simulation is run to t — 1.67 x 10~9 s. The boundaries of the material medium are z\ — 0.005 m and z2 — O.I m. The discretization in space is a first order finite element approximation with N — 100 for both the differential and hysteresis models. The hysteresis simulations also use a first order finite element approximation in the history variable. That is, the functions fa and ifri are both piecewise linear functions defined on discretizations of the intervals z € [0,1] and s 6 [—r, 0], respectively. The maximum difference is computed as \ed — £h\oo and the percent difference is \ed — eh\oo/\ed\oo, where e^ and e^ are the solutions obtained from the differential and hysteresis models, respectively. The last column is an average value of the actual computer time in seconds taken to solve the test problem. These figures should be contrasted with the 6.65 seconds time (on an IBM RISC/6000) for the differential Debye simulation using the same discretization in the space variable. We see that agreement between the two methods is very good when sufficiently accurate discretizations of the history variable are used. This requires that a sufficiently long history be retained in the simulation (as measured by r] and a sufficiently fine discretization of this interval (measured by As = r/M). The results show that a history duration of at least r = 0.03 (about 10 ns, in the unsealed time variable) is necessary for results with small relative error. Comparing results from the table, we see that similar results are obtained by M = 10, r = 0.03 and M = 13, r = 0.04 while the computation with M 13 takes about 14% longer to execute. Although more accurate results are obtained for higher values of M with r = 0.04 than with r = 0.03, these are

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99

Table 5.1: Comparison of general and differential Debye simulations, r is the history duration, M the number of history elements, and As the discretization of history interval. r 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04

M 5 7 10 20 7 10 15 30 10 13 20 40

As

3

4.0 x 10~ 2.86 x 10-3 2.0 x 10~3 1.0 x 10~3 4.28 x 1Q-3 3.0 x 10-3 2.0 x 10-3 1.0 x 1Q-3 4.0 x ID"3 3.08 x 10~3 2.0 x ID"3 1.0 x 10-3

Max difference 0.4038 0.5705 0.6824 0.6662 0.4011 0.1119 0.0149 0.0120 0.3101 0.1117 0.0223 0.0016

% Difference 1.04 1.47 1.76 1.73 1.04 0.29 3.84 x 10~2 3.10 x 10-2 0.801 0.289 5.94 x 10~2 4.24 x 10~3

Execution time 36.08 40.27 46.09 68.29 40.13 45.79 56.3 91.13 45.92 51.93 67.75 113.44

only incremental improvements in the quality of the simulation. Since we are interested in the inverse problem which requires repeated simulations of the system, the time of execution is a primary concern. We show graphically in Figure 5.1 the close fit of results generated with the two formulations (i.e., differential Debye versus hysteretic Debye polarization models). This plot is for M — 4 and r = 0.03 in the hysteresis formulation. For more accurate simulations the graphs are indistinguishable in the plot. Figure 5.2 exhibits a case where an insufficiently refined discretization of the history variable leads to a very poor approximation. This result was computed with M = 3 and r = 0.03.

5.2.2

DRF Representation of the Lorentz Model

The representation of the Lorentz model of polarization is plagued by computational difficulties which make it impractical for simulations. The plot of the hysteresis kernel using the parameter values employed in the inverse problem of Section 4.4 is given in Figure 5.3. This function is the derivative of the impulse response function which appears in the variation of constants solution to the differential equation. Because of the oscillatory nature of the function, a much finer approximation of the history variable is required, making the size of the resulting system of equations prohibitively large. For example, the plot of the kernel function in Figure 5.3 suggests that a history duration of r = 2 x 10~15 is necessary before the kernel function drops to a level where it can be neglected. This includes twelve complete oscillations. To approximate this function accurately would require on the order of 36 basis

100

Figure 5.1: Results from differential tric field in volts/meter.

Banks, Buksas, and Lin

and DRF Debye models, z in meters, elec-

Figure 5.2: Breakdown of DRF Debye model, z in meters, electric field in volts/meter. elements in the Galerkin discretization of the history variable, and this allows for a meager three elements per cycle. Furthermore, stability requirements limit the discretization in time of the entire system to being no larger than the discretization of the history variable.

Computational Methods for General Polarization Models

Figure 5.3: DRF kernel for Lorentz model with r = 3.57 x 10

101

. s in seconds

This arises from the discretization of the evolution equation of u in (5.5). The resulting discretizations in (5.9) and (5.14) are subject to a stability condition which limits the size of the discretization in time. Since we do not want the propagation speed of the numerical scheme (As/At) to be faster than the actual propagation speed of the solutions (1, in the scaled variables), the time step can be no larger than the discretization used in the history variable. In the example considered here, the resulting discretization in the history variable would be approximately As = 5.6 x 10~17, suggesting a time step of similar size. (The similar differential problem in Section 4.3 required an even smaller step size of 1.0 x 10~17 for an accurate result.) A simulation of duration 6.67 10~14 (as used in the inverse problem of Section 4.4) would require about 1200 steps. Because of the large number of basis elements in the history variable, the resulting computation is prohibitively expensive.

5.3

The Inverse Problem with the General Polarization Model

Having successfully duplicated the quantitative behavior of the Debye model of polarization with a hysteretic polarization model, we turn to using this method of simulation in an inverse problem. Different inverse problems can be formulated in this way depending on the way in which the hysteresis kernel depends on the set of parameters being optimized. One option is to express the kernel as a function of the same set of parameters which appear in the differential

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Banks, Buksas, and Lin

equation, <7, e^, COQ and r and hence in the variation of constants solution. For the Debye differential equation, this function was computed in Section 5.2.1 as g(s) = £d\e~Xs, with the kernel function a(s] = —ei-i> V'j-i)- The values which appear in the forward simulation are «& = f_ra(s)^k(s)ds and the coefficients of the Galerkin approximation of o; are given by h = M~la, where the vector a is a = (ao,ai,... ,QM) and h = (h0,hi,... ,/IM)We adopt a standard test problem for comparing the relative ease of reconstructing the hysteresis function through the Debye physical parameters and through Galerkin coefficients. The true values for the physical parameters are a = 0.0, €00 = 5.5, es = 80.1 and r = 8.1 x 10~12. (Again, these are the typical values for water — see [BF95, p. 1505].) The data are gathered from a forward simulation of the problem using the differential Debye polarization model and are depicted graphically in Figure 5.4. We attempt two inverse problems over different sets of parameters using this test problem. These different sets of parameters arise from the different ways of representing the Debye kernel function, either as a function of the Debye parameters or through the coefficients of a Galerkin approximation. Both of these inverse problems use the result of a forward simulation from the differential Debye model to provide the data. Since no noise is added, we expect to be able to reproduce the parameter values with a high degree of accuracy, depending on the precision of the hysteresis approximation. We do not attempt to identify

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103

Figure 5.4: Data for Debye test problem. Time in scaled seconds, electric field in volts/meter. Table 5.2: Results of estimating hysteresis Debye model from differential Debye data.

M True values: Initial values: 4 5 6 7 8 9 10

Estimated es 80.100 83.8300 81.3481 79.7209 80.4458 79.9598 80.2243 80.0418 80.1558

Estimated T 8.1000 x 10~12 7.7142 x 10~12 8.2190 x 10~12 8.8766 x 10~12 7.9891 x 10~12 8.3541 x ID"12 8.0389 x 10~12 8.1978 x ID"12 8.0660 x ID"12

Residual

Iterations

1.14808 0.37468 0.30911 0.13821 0.11429 5.83645 x 10-2 5.19723 x 10~2

14 11 9 10 8 12 14

the parameters a and e^. In Section 4.2 the low sensitivity of the objective function to these parameters was established. Instead, each of these parameters have their values fixed at the value used in generating the data. The results of fitting the hysteresis Debye model to data generated by the differential Debye model are summarized in Table 5.2. All of these attempts used r = 0.03 for the duration of the history approximation. The initial values for the two parameters were TO = 7.7142 x 10~ 12 ,e s0 = 83.83. These results demonstrate that a sufficiently accurate hysteresis implementation of the Debye

104

Banks, Buksas, and Lin Table 5.3: Results from estimating all coefficients of kernel.

M

/lQ

True values: Initial values: 4* 5 6*

h* x 1.1 -683759 -765083 -809484

0(0) 2264.20 2490.63 2336.54 2365.97 2349.39

Residual

Iterations

2.159 x ID"1 1.645 x 10~2 1.323 x 10-1

900 188 829

Table 5.4: True and initial values of h$ for various M.

M 4 5 6

True HQ -640889 -706575 -753628

Initial ho -704978 -777233 -828991

model yields reasonable estimates for the parameter values in an inverse problem in the absense of noise. The identification of the Galerkin coefficients proved to be considerably more difficult than identification of the Debye model parameters. The true values for the parameters are found by taking the vector of coefficients a which are computed to perform the forward simulation and computing the coefficients of the Galerkin approximation h. These true values are then perturbed by 10% to generate the initial values used in the optimization. As summarized in Table 5.3, attempts to identify all of the coefficients h and the value of g(0) generally failed to converge in a reasonable number of iterations. The results for M = 4 and M = 6 (denoted by 4* and 6*, respectively) were taken before the program terminated and indicated convergence. (At the time that the values were taken, both programs had been running continuously for at least nine days.) We report the values of the two parameters g(0) and the first coefficient of the Galerkin approximation of the kernel /IQ. Since the value /IQ depends on the discretization, its true and initial values both vary with M. These values are summarized in Table 5.4. This result strongly suggests that the problem is overparameterized. To alleviate this problem, we note that ho is considerably larger than the other coefficients, owing to the decaying exponential form of the kernel function. This parameter, along with the value ofg(0), is likely to have the most influence over the objective function, with the remaining coefficients playing minor roles. We attempt to identify just these two most significant parameters while holding the others fixed at their true values. This smaller optimization problem also proved to be very difficult, although acceptable results were eventually obtained, as indicated by the results in Ta-

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105

Table 5.5: Results of estimating g(0) and

M True values: Initial values: 4 5 6

h0 —

v/lQ -640888 -706574 -753625

2(0) 2264.20 1.1 2490.63 2264.20 2264.20 2264.20

Residual

Iterations

1.15539 x 10~5 1.63708 x 10~5 1.98944 x 10-5

83 80 95

Figure 5.555555555555

ble 5.5. We see that this problem required a much larger number of iterations to attain convergence than the two parameter Debye estimation problem summarized in Table 5.2. One possible cause of the very slow convergence is suggested by the objective function. Figure 5.5 is a plot of the logarithm of the objective function over a narrow (5%) range of reach of these parameters. We see that the minimizer lies at the bottom of a narrow trench. This is likely caused by the two coefficients g and ho being nearly interdependent in their influence on the solution. On one side of the trench we see that the values of the objective function rise slowly, while on the other they increase exponentially fast.

5.3.1

Practical Conclusions

Some conclusions can be drawn from the inverse problem calculations outlined in the previous section. First, if one has a parameterized differential representa-

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Banks, Buksas, and Lin

tion for a concrete polarization mechanism, one can either treat it as a separate differential constraint as done in Chapter 4 or one can use a parameterized kernel in an impulse response convolution (variation-of-parameters) representation for polarization. In either case, the Debye model dielectric parameters can be identified. On the other hand, a secondary approximation for a general kernel representation does not produce reasonable identification results. For oscillatory kernels such as in the Lorentz model, one should retain the differential equation form of the polarization law in inverse problems (and simulations). The convolution kernel representation should then be used only for theoretical considerations.

Chapter 6

Computational Methods for Acoustically Backed Dielectrics 6.1

The Forward Problem with an Acoustically Backed Layer

In previous chapters, we saw that one can use a metal or supraconductive backing on a slab to determine both dielectric and geometric parameters using electromagnetic pulses. If one removes the supraconductive back boundary, reflected signals from the back boundary may be much less pronounced. If there is a significant difference in the material properties at this interface, one might expect a reasonably strong (depending highly on properties of the slab and interrogating signal) reflected signal. In this chapter, we explore the possibility of using an imposed acoustic field behind the second interface to enhance any reflected electromagnetic signal from the interface. We shall see that even in a worst case scenario (where there is no instantaneous polarization in the second material layer, nor any other polarization effects except those produced by the acoustic field), one can substantially enhance reflected signals from the back interface of a material slab to permit accurate estimation of geometry. The implications and potential applications, which we discuss in the concluding chapter below, are enormous. In this chapter we treat the forward and inverse problems for the acoustic reflector geometries introduced in Section 2.3.1. In strong form, the model is given by (2.21)-(2.25), where we note that a Debye polarization law is assumed for the dielectric medium. 107

108

6.1.1

Banks, Buksas, and Lin

Numerical Methods

We employ a spatial finite element scheme similar to the one described in Section 4.1 to compute approximate solutions to the forward problem with the acoustically backed layer. Here, we look for approximate solutions in VN C J71(0, 1), the standard linear finite element space defined on a partition of the interval (0, 1) and with a set of basis functions |<^}^Lo- We partition the time variable with a finite difference time-stepping scheme corresponding to particular points with At = tl—tl~1, i = 1, . . . , M. We leave the system in second order (in time) form, as opposed to the first order formulation of the model used in previous chapters. This provides some evidence that our approach does not depend in an essential way on the particular discretization scheme used. As in Chapter 4, we begin with the weak form of the model partial differential equation (2.21) expressed in the following equivalent form:

for any (f> £ Hl(Q, 1), where {•,•) denotes the standard L2(0, 1) inner product. From this, we can formulate the following finite element discretization of (2.21):

where

and the following finite difference notation is used for the average

along with the difference quotients

We note that as in Chapter 4, we are suppressing the notation that should indicate dependence on the spatial discretization index ]V as well as dependence

Computational Methods for Acoustically Backed Dielectrics

109

on the temporal discretization index n, i.e., we should write EN'n(z),PN'n(z) as the approximates for EN(tn, z), PN(tn, z). Since N is fixed throughout, no confusion will result in suppression of the index N. With these definitions, the Galerkin formulation (6.1) results in a linear system of equations which can be solved to give En+l in terms of En,En~l, and Pn. As a scheme to compute En+l, assuming that Pn is given exactly, the above scheme has the accuracy

provided that the solution is smooth enough. To update Pn at each time step, it is preferable to choose a scheme that has good stability and whose accuracy matches that used to compute En. Hence we use the following A- stable second order Adams-Moulton scheme to discretize (2.22):

According to the given initial conditions, we should set

for the finite element approximations at the Oth time level. By the Taylor expansion and applying (2.21), (2.22), (2.24), and (2.25), we have

Therefore we can let

Putting all of these discretizations together, we have the following algorithm to generate approximations to both E(t,z) and P(t,z): Begin with E° and P° given by (6.3). Compute El by (6.4). Then for each n = 1, 2 , . . . , M — 1, we use (6.2) to compute Pn w P(t n ,z) and use (6.1) to compute En+1 w E(tn+1,z).

6.1.2

Simulation Results

In this section, we use numerical simulations to demonstrate the extent to which the electric field is affected by the acoustic pressure. In particular, we would like to know how and to what extent the acoustic pressure can change the reflected electric wave from the interface at z = z 2 . Moreover, the relaxation parameter r is a characteristic of the material in \z\, z2] which affects the transmission of

110

Banks, Buksas, and Lin

electric waves through this region. Consequently, we will examine in more detail how r can effect the electric field. As described in Section 2.3, we assume that a time windowed electromagnetic point source input is given at the left boundary point z = 0 such that

The frequency in the source is assumed to be in the microwave range, i.e.,

The pressure is assumed to have the form

where (3p is the phase shift in the acoustic pressure. The frequency of the acoustic pressure is assumed to be in the ultrasonic range, i.e.,

and gp(z] = /[ Z2j i](z) in all the computations. We first consider the effects of the acoustic pressure on the electric wave. Since the acoustic pressure appears in the coefficient a(t, z) of E(t, z) in wave equation for the electric field, any changes in the acoustic pressure will affect the propagation of the electric wave. There are three parameters in the acoustic pressure model which may potentially affect the electric wave: the amplitude K, the frequency u>p, and the phase shift /3p. A necessary physical constraint on the choice of these parameters is that a(t, z) > 1; otherwise, the wave will propagate at a velocity faster than that of light in the region [z2,1]. We will observe this requirement in our computations here (by restricting the time intervals in the case that a = 1 + Kg To investigate the effects of acoustic pressure modulation, we carried out a group of simulations with er — I so that no instantaneous polarization is present in the acoustic layer and with parameters (see Table 6.1) similar to those for water in the material slab, except for the relaxation time r whose value here is somewhat larger than those given in the literature (the reason for this will be discussed below). Snapshots of a typical electric field at several moments are presented in Figures 6.1-6.4, where the phase shift in the acoustic pressure is /3P = £, and the vertical lines indicate the interfaces between the three layers of material at z — z\ and z — z2. The electric field recorded at the left boundary in this simulation is shown in Figure 6.5. The first oscillations on the left are the original signal emitted from the point source at z = 0, the group in the middle is the first reflection from the interface at z = z\, and the last group on the right is the reflection from the interface at z = z2. This information can be used as data to interrogate the material in the region of \z\, z2]. We will apply it to the inverse problem of determining the material depth z2 — z\.

Computational Methods for Acoustically Backed Dielectrics

111

Table 6.1: A set of typical parameters used in numerical simulations.

tf

4.0xicr10

a

l.OxKT 5

K

0.9

OJp

7T X 106

Us

7T X 1010

At

t//1600

h

1/900

T

Coo 5.5

CS

Fogir e 6.1: The elect rio c wave bevore it re in meter rs ele ctric fie ld in volts /meter .

3.16xlO-8 78.2

che sa the material i n [z1,1] Depth is

Our first observation is that a larger amplitude of the acoustic pressure leads to a larger coefficient a(t,z), which in turn implies a stronger reflection of the electric wave from the interface at z = z2. This is confirmed by the plots of |.E(£, 0)| presented in Figures 6.6 and 6.7. Each plot in these figures shows the first and second reflected signals measured at z = 0 for various values of K. From these simulations, we can see that the electric waves reflected from the interface at z = z^ become more pronounced when the magnitude of the acoustic pressure increases. Note that this is contingent upon having sin(u;p£ + ftp) > 0) otherwise, the opposite effect may occur as the coefficient a ( t , z ) is decreased. Weak reflected electric waves from the interfaces are more sensitive to measurement error. Simulations here suggest that applying acoustic pressure behind the material to be interrogated can improve the clearness of the signals reflected from the interface at z = z2- A larger amplitude K is preferable for this purpose, but there is certainly an upper bound for its value in practice imposed by the physics of the material. We note also that a small reflection is obtained

112

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Figure 6.2: The electric wave in the material before interacting with the acoustic wave in [z2,1]. A reflected wave is generated at the interface of air and the dielectric material. Depth is in meters, electric field in volts/meter.

Figure 6.3: The electric wave in the part of the dielectric material [22,1] where there is an acoustic wave. A reflected wave generated from the material interface is seen in [21,22]- Depth is in meters, electricfieldin volts/meter. even when K = 0, i.e., there is no acoustic pressure grating present. This is due to the difference in dielectric properties between the dielectric material in [21,22] and the idealized material in [22,1].

Computational Methods for Acoustically Backed Dielectrics

113

Figure 6.4 The ref;ected wave frp, tje acpistoc nea, approaches the left bound ary. Depth is in meters, electric field in volts/meter.

Figure 6.5: A typical record of the electric field at the point 2 = 0. Time is in seconds, electric field in volts/meter. We have carried out similar simulations with er = 1.95 for K ranging from K = 0 to K = 0.9 (we note that er = 2.1 corresponds to paraffin, plywood, or teflon while a choice of er = 3 corresponds to dry soil—see [Bal89, p. 50]) so that a(i, z) = er + Ksm(upt + (3P) always satisfies a(t, z) > 1 in the slab [zi, I}.

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Banks, Buksas, and Lin

Figure 6.6: Plots of magnitudes of electric fields recorded at the source point z = 0. The two groups of waves on the left and right are reflected from the interfaces at z = Zi,i = 1,2, respectively, for various values of K. Time in seconds, electric field in volts/meter. In this case there is a larger reflected signal for K = 0 (due to the instantaneous polarization in [z2, 1]) but as we increase K, the amplification of that signal behaves almost identically to that behavior depicted in Figures 6.6 and 6.7. This provides clear evidence that it is the presence of the standing acoustic wave that is responsible for any reflected signal enhancement independent of any additional polarization differences between the material in [21,22] and that in [22,1]. The frequency UJP and the phase shift /3P determine the oscillation characteristics of the acoustic pressure and this will also affect the electric wave reflected from the interface at z = z^. In this mode, we note that the coefficient

can take values between 1 — K (as we noted earlier, we avoid the regime where o(£, 2) < 1 in simulations) and 1 + K at the moment when the electric wave hits the interface, and a larger value of a(t, z) leads to a greater magnitude of the reflected part of the electric wave from the interface. Therefore, for a desirable stronger reflection, we should choose UJP and (3p such that sin(u>pt + fip) w 1 for the time t within the range when the electric wave is passing through the interface z = 22. On the other hand, we note that the electric wave propagates across materials at high speeds. For the parameter values given in Table 6.1, the electric pulse wave arrives at the second interface z = z2 at the moment around

Computational Methods for Acoustically Backed Dielectrics

115

Figure 6.7: Plots of magnitudes of electric fields recorded at the source point z = 0. The two groups of waves on the left and right are reflected from the interfaces at z — Zi,i = 1,2, respectively, for various values of n. Time in seconds, electricfieldin volts/meter. t = 2.1878 x 10 9 , and it will have passed this interface at the moment around t = 2.6377 x 10~~9. In such a small time window, the frequency OJ P chosen in the ultrasonic range has very little effect on the value of sin(u> pt + /3p) because

for u>pt < (2.5 x 106)(2.6 x 10~9) » 6.5 x 10~3. In these simulations, we see that the phase shift is therefore more pivotal for generating a stronger reflection from the interface at z = z2, and the ideal value for the phase shift in the given format of the acoustic pressure is about TT/2. Values of the coefficient a(t, z) for z € [22,1] are typified in Figure 6.8, where the values of a(t, z) with /?p = vr/2 are about 1.9 times larger than those with Pp = 0. Because the values of the coefficients change very little during the interaction of the pulse with the interface, we conclude that the phase of the acoustic wave is more significant than its actual modulation, even for larger values of t, where upt + f3p 56 f3p. The corresponding reflected electric waves from the second interface are plotted in Figure 6.9. This illustrates the fact that a suitable choice of the phase shift parameter can yield a stronger reflection. We take a closer look at the effect of the relaxation time r and the interrogation frequency us on the ability of the electromagnetic pulse to penetrate the material. Figures 6.10, 6.11, and 6.12 contain snapshots of the electric field at three times using the parameter values in Table 6.2. These figures should be

116

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Figure 6.8: Plots of a(t,z) for z € [z2,l] in the time window [2.1878 x 10~9, 2.6377 x 10~9] for various phase shift values and K = 0.9. Time in seconds.

Figure 6.9: Plots of reflected electric waves from the second interface recorded at z — 0 for various phase shift values and K = 0.9. Time in seconds, electric field in volts/meter. compared to Figures 6.1, 6.2, and 6.3 corresponding to Table 6.1. Note that the relaxation parameter used in this simulation is smaller than that in Table 6.1,

Computational Methods for Acoustically Backed Dielectrics

117

Table 6.2: A set of parameters with a smaller relaxation time. tf 600 a

4.0xl(r10 5.5 l.OxKT 5

es K

8.1xlO~ 12 78.2 0.9

U>p

7T X 106

Us

7T X 1010

At

fy/1600

h

T

1/900

Figure 6.10: The electric wave before it reaches the dielectric layer. Depth is in meters, electric field in volts/meter. and the material in [21,2:2] is comparatively "hard" in the sense that it allows much less of the signal to penetrate the first interface z = z\ at this frequency. This small transmitted wave obviously can only generate very little, if any, reflection at the second interface. (See Figure 6.13, where one cannot even discern the reflection at the second interface in the resolution given.) This demonstrates the importance of the choice of the carrier frequency ujs for effective interrogation. Whereas the comparatively high frequency of u>s = TT x 1010 rad/s is unable to penetrate the Debye material with the relaxation parameter r = 8.1 x 10"12, it works well with the parameter value T = 3.16 x 10~8. Furthermore, we saw in the numerical simulations of Chapter 4 that lower interrogating frequencies around 1.1 x 1010 rad/s can penetrate a Debye material with r = 8.1 x 10~12 and are useful for interrogating these materials. Therefore, we see that the choice of an interrogating frequency which can penetrate the material successfully is strongly influenced by the relaxation time of the material.

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Figure 6.11: Transmitted electric wave into the dielectric is weak. Depth is in meters, electric field in volts/meter.

Figure 6.12: Most of the wave is reflected back into the domain filled with air. Depth is in meters, electric field in volts/meter.

Computational Methods for Acoustically Backed Dielectrics

119

Figure 6.13: The reflection from the first interface is strong, but the reflection from the second interface is essentially zero and not discernible. Time is in seconds, electric field in volts/meter.

6.2

Identification of Material Depth

In this section, we consider the problem of using the signal E(t, 0) collected at the left boundary z = 0 to estimate the material depth. To be specific, we assume that all the physical parameters other than the depth of the material are given (these can be estimated using the first reflected waves from the interface at zi, as seen in Chapter 4), and the location of the first interface of the dielectric material is fixed at z — z\. As before, the position of the interior interface between the dielectric and the acoustic domain is z^. After an electromagnetic pulse is sent into the material, reflections will be generated at the two interfaces of the material that propagate back to the left boundary. Intuitively, the delay between the detection of the first and second reflections depends on the depth of the dielectric layer. Just as in Chapter 4, we shall attempt to estimate the depth of the material from this difference. We first need a procedure to detect the time when a wave reflected from an interface reaches the left boundary. Returning to Figure 6.5, which for ease of comparison is reproduced in Figure 6.14, we see that the function .E(t,0) is essentially flat except in three subintervals. We therefore can expect its derivative E(t, 0) to become very close to zero in these three intervals as well. After the initial pulse, E(t, 0) becomes nonzero again when the wave reflected from the first interface arrives at z = 0. After another quiet period, E(t, 0) becomes nonzero again due to the arrival of the wave reflected from the second interface.

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Figure 6.14: A typical data function. Parameters in Table 6.1 are used. Time in seconds, electric field in volts/meter.

This is visible in Figure 6.15, which presents a typical plot of \E(t, 0)|. To form our estimation procedure, we define T\ to be the first time that where the values of the constants C\ and C2 will be determined from the data E(t, 0) and the input signal. The time C2 should be larger than the duration of the source pulse to avoid detecting the original interrogation signal. The constant C\ should be chosen large enough to distinguish the reflected signal from whatever measurement error may be present. Since z\ is assumed to be fixed in the identification of depth, TI should be independent of z2. Next, let T<2 be the first time such that The constant C3 can be chosen in a manner similar to that for Ci, and we can simply take 63 = C\ . The value of 64 should be chosen so that the wave reflected from the first interface has passed the left boundary at the time t — TI + €4. Clearly T2 depends on the position of the second interface, and we denote it by T2 = T2(.z2). Then we define a function of z2 as the difference of two times, and our identification problem leads to seeking a value z2 such that

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121

Figure 6.15: A typical plot o f \ E ( t , Q ) \ . Parameters in Table 6.1 are used. Time in seconds, electric field in volts/meter.

where z^ is the true position of the second interface of the material for which the data is collected. Note that z\ is unknown, but L \ ( z ^ } can be determined by the the method described above applied to experimental or synthetic data. To see the behavior of LI (22), we calculated its values for various z^ in the neighborhood of z%. A typical plot of 1/1(22), given in Figure 6.16, suggests that L\ (22) acts almost like an affine function in the neighborhood of z%, and the secant method is a good candidate for computing 22 from (6.5). We observe that the above scheme appears to be a reasonable approach only if we can have a dependable measurement for the derivative E ( t , Q ) . This approach most likely will not work if numerical differentiation must be used to obtain an estimate of E(t,Q) from a measurement of E(t,Q) with noise. This is due to the well known catastrophic behavior encountered in using numerical differentiation on error-polluted data. Even a very small amount of error in the data for E(t,Q} will make the estimate of E(t,Q} generated by numerical differentiation meaningless. Even though noises may come from many sources with different characteristics, for the purpose of testing ideas we assume that they originate from either the background or the measurement. Hence, we assume that the data we can actually use in the interrogation is of the form

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Figure 6.16: A typical plot of L\(z) = T-2(z)—Ti in the neighborhood of z^ = 2/3. Depth is in meters.

instead of the pure data E(t, 0). Here rb(t} and ri(t) are random variables with a uniform distribution between —1 and 1 which take on different values at each time t. The coefficients nib and ra^ determine the magnitudes of the two types of random noise. Since 7715^(t) is independent of the signal, this term represents the background noise. The term m,iri(t)E(t,0) represents measurement noise, since it is proportional to the magnitude of the signal. To simplify the expression of the magnitude of the random noise, in the presentation below, x% uniformly distributed background noise means raj, = x/100 times the magnitude of the second reflection of the electric wave (we choose this standard of measurement because of its relevance to the success of the inverse problem) and x% measurement noise means ra^ = x/100. Now, if the data given in Figure 6.14 is polluted by a uniformly distributed random error from the background at the 0.5% level, and no measurement error, then numerical differentiation will generate an estimate of !£"(£, 0)| plotted in Figure 6.17. We can see that the arrival of the second reflection is somewhat obscured by the presence of noise. The situation is even worse if the magnitude of the acoustic pressure is small or zero; see Figure 6.18, where the information about the reflected wave from the second interface is almost completely eclipsed by the numerical error. On the other hand, the pulse signal £"(£,0) itself appears much more robust with respect to the random noise. Figure 6.19 is a plot of the absolute value of the data for E(t, 0). Adding uniformly distributed random error in the background at the 5% level yields a plot in Figure 6.20 from which we can still

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Figure 6.17: A plot of an estimated \E(t,Q)\ generated from numerical differentiation on a measurement of E(t, 0) with only 0.5% background random error. Time in seconds, electric field in volts/meter. Parameters in Table 6.1 are used. easily discern the times when the two reflected waves arrive at the left boundary. Hence we redefine the algorithm for detecting the return of the reflection given above in terms of |.E(£,0)|. We redefine T\ and T2 as the first times such that

where the constants C*i, C^, Cj, and C\ can be determined from the measurement of E(t, 0) in the same manner as before. We then define a new function L2 of 22 as the difference of these two characteristic times:

Obviously, the construction procedure for £2(22) is similar to 1/1(22) associated with data E(t, 0), but no differentiation is used. As before, we define the solution to the identification problem as a quantity 22 that satisfies

Since the curve of this new time lag function of 22 also appears to be a straight line in the neighborhood of the exact location of the second interface 22 (see Figure 6.21), we again believe that the secant method is a good candidate for computing 22 from (6.6). The function 1/2(22) just introduced is also rather insensitive to noise in the data. See Table 6.3 for its behavior with respect to the random error in the

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Figure 6.18: A plot of an estimated \E(t,Q)\ generated from numerical differentiation on a measurement of E(t, 0) with only 0.5% background random error. Parameters in Table 6.1 are used except for K = 0.1. Time in seconds, electric field in volts/meter. background, Table 6.4 for the random error in the measurement, Table 6.5 for the random errors in both the background and measurement. Here we used the mixed finite difference/finite element solution to generate data for E(t,0) with z% = 2/3 and other parameters listed in Table 6.1. The plot for the data E(t,Q) is in Figure 6.14, and the absolute value of the two reflected waves received at the left boundary is plotted in Figure 6.19. From these plots, we decide to choose the constants C\, C<2, Cs, C^ for the definition of £2(^2) as follows:

The data E(t, 0) perturbed by random numbers at various levels was then used to generate values of L(z%) in Tables 6.3-6.5. Note that L(z^) changes little even with the data at a rather high noise level. The estimation procedure for the depth z% based on this new time lag function £2(22) ig robust in the presence of random error; see Table 6.6 for the results with background noise. Even the data with background noise at a 25% level still yields a satisfactory approximation to the location of the second interface. Similar results can be obtained for data with measurement noise, or both. Physical parameters listed in Table 6.1 are used in the computation of z% here, and the exact location of the second interface is z% = 2/3. As we have have observed in the previous section, if the acoustic pressure is turned off,

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Figure 6.19: A plot o f \ E ( t , Q ) \ , no measurement error. Time in seconds, electric field in volts/meter.

Figure 6.20: A plot of \E(t,Q)\, corrupted with 5% background random error. Time in seconds, electric field in volts/meter. the reflection from the interface at z^ becomes weaker and is more susceptible to being obscured by measurement noise. Using the data collected from this kind of uncontrolled (no acoustic pressure) configuration may yield an inferior

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Figure 6.21: A typical plot o

in the neighborhood of z% = 2/3.

Table 6.3: Sensitivity of L^z^) to the background noise with a uniform distribution in the data. No measurement noise. by data level with noise Background noise 0.2215 x 10~8 0 5% 0.221325 x 10~8 0.221275 x 10~8 10% 0.2212 x 10-8 15% 20% 0.2213 x 10~8 25% 0.22155 x ID"8 30% 0.2209 x 10~8

estimate of the material depth z%, especially in an environment at a high noise level, and this is corroborated by the results presented in Table 6.7. Comparing the results in Tables 6.6 and 6.7, we can see that at the same noise level, the data collected with acoustic pressure leads to a better estimates of z% than that without acoustic pressure. When the noise becomes higher, it will eventually become impossible to distinguish the reflected waves from both interfaces in the data £'(i,0), and the data becomes unreliable for the estimation of z%, (see Tables 6.6 and 6.7). However, this happens at a significantly higher noise level when acoustic pressure is used to enhance the data than without it.

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Table 6.4: Sensitivity of L^z^) to the measurement noise with a uniform distribution in the data. No background noise. Measurement noise level 0 5% 10% 15% 20% 25% 30%

1/2(22) by data with noise 0.2215 x 10-8 0.22145 x 10~8 0.22165 x 10-8 0.221575 x 10~8 0.221475 x 10~8 0.22125 x 10-8 0.22115 x ID"8

Table 6.5: Sensitivity 0/1/2(^2) to noises with a uniform distribution in the data. Noises are added in both the background and measurement. Background/Measurement noise level 0 5% 10% 15% 20% 25% 30%

£2(22) by data with noise 0.2215 x 10~8 0.22135 x 10~8 0.22125 x 10~8 0.2213 x 10~8 0.21175 x 10~8 0.221025 x 10~8 0.217325 x ID"8

Table 6.6: Estimates of z% = 2/3 by 1/2(22) with noisy data. Uniformly distributed background random errors were used in these computations. Background noise level 0 5% 10% 15% 20% 25% 30% 35% 40% 45%

Estimated z^ by data with noise 0.66666975813478 0.66641364349190 0.66634046718531 0.66623070221293 0.66637705537412 0.66672483815192 0.66579163706712 0.66112000759318 0.60731815656448 0.59860951385259

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Table 6.7: Estimates of z% =• 2/3 by L^z^} with noisy data. Uniformly distributed background random errors were used in these computations, but the acoustic pressure is turned off (K — 0). Background noise level 0 5% 10% 15%

Estimated z% by data with noise 0.66665123456790 0.66629080932785 0.66567901234568 0.59539092910678

Chapter 7

Concluding Summary and Remarks on Potential Applications In some brief remarks here we wish to summarize the findings reported above in this monograph and to comment on our views (acknowledged speculation in some cases) of the potential for electromagnetic interrogation techniques and products in diverse but important applications.

7.1

Summary of Mathematical and Computational Results

The topic of this monograph, the electromagnetic interrogation of dielectric materials, is a subject that in our treatment lies at the interface between electromagnetic theory and material sciences. The physical models for the coupling of electromagnetic fields and materials that we have used incorporate representation of radiating antenna currents, free space propagation and electromagnetic interaction with materials possessing polarization dynamics and conductive currents as well as standing acoustic pressure waves. The direct incorporation of antenna current into our materials identification scheme is, we believe, one of the unique features facilitated by the variational formulation taken in our approach. In the far field of an antenna, it is generally acceptable to use a plane wave approximation for the incident field, as we have done in the one-dimensional problems addressed in this volume. We believe, however, that complex radiating systems such as phased array and parabolic antenna systems can also be accommodated in a modification of our variational formulation. Such a development could be of great value whenever near field interrogation of materials is desired. In our treatment the medium intervening between the radiating antenna current and the interrogated medium has the properties of the vacuum. While this 129

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is likely to be an excellent approximation in the laboratory setting, it may not be appropriate in a field setting where dust, clouds, rain, and thermally determined variation in the refractive index can occur. In principle, the variational formulation we have used can also be extended to provide material interrogation in these more challenging settings. We have used the full Maxwell's equations and relatively high order constitutive relations describing material polarization. In this context, we have noted that polarization or conduction current relations within the medium can be represented using ordinary differential equations or equivalent convolution representations. Although the variational formulation permits incorporation of either formulation, our theoretical presentation has emphasized use of the convolution representation, since this provides the basis of a rather comprehensive theoretical framework that facilitates rigorous development of computational methods for both the forward and the inverse propagation problems. In this regard existence, uniqueness, and continuous dependence of solutions along with finite element convergence results are given for the forward problem. In solving the forward problem we have observed both the Sommerfeld and Brillouin precursor transients for the case of a medium following Lorentz polarization dynamics, and we have computed the Brillouin precursor in the case of the Debye polarization dynamics. While we don't pursue this direction of research, we do establish a semigroup formulation for the forward problem for models with Debye polarization as well as certain convolution representations of polarization. As noted specifically in Chapter 3, this formulation does not permit Lorentz polarization laws. A central element in the inverse method presented is the application of finite element based variational methods coupled to least squares estimation. Outlines of proofs for a theory of existence and continuous dependence on data of minimizers and convergence of approximate minimizers in these inverse problems have been provided. Least squares estimation can and has been used in other settings involving the repeated solution of forward problems to estimate material parameters. For example, one can use finite-difference time domain or integral equation solvers coupled to least squares techniques. To our knowledge, no one has provided a rigorous assessment of the existence and continuous dependence on data of the least squares estimators for these alternative formulations. Use of a supraconductive (metallic) or acoustic backing to the layered dielectric material under interrogation proved to be useful and interesting. The physical backing provides a detectable reflection that permits a two-step algorithm. In the presence of a physical backing, the reflected field can be divided into at least two signals. One signal relates to energy reflected from the vacuummaterial interface, and the second signal relates to reflection from the back wall of the dielectric layer. This separation into two components allows estimation of material parameters as an initial step. This estimate of material parameters is then used in a separate minimization scheme to estimate the depth and improve estimation of dielectric material properties. This results in improved estimation in the setting of noisy data.

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We believe results in this monograph provide methods that the physical chemist, engineer, or biologist can actually use when presented with a material whose polarization and conduction properties are unknown. These methods require only simple pulsed time domain reflection data and can provide useful estimates, particularly when the material is mounted on a reflecting backing. Thus, we provide an alternative to frequency domain reflection or transmission methods. The methods developed here may be applied to free field data or to data obtained in structures such as a transmission line or waveguide. Some materials of interest naturally come in association with a metal backing. Paints for example are often applied to metals, and paint polarization and conduction properties may be studied using short pulses and the numerical methods described here. Radar absorbing materials applied to a backing may also be assessed using these methods. We have demonstrated computationally that acoustic modification of a material may be substituted for a metallic or other strongly reflecting backing. The resulting virtual reflector can be tuned to enhance the second signal reflections mentioned in the two-step algorithm. This appears to be of significant importance since the acoustically stimulated layer might be moved through the sample, particularly in a transmission line apparatus. Thus, one could identify the polarization and conductivity parameters of a surface layer first, then move "backward" through the sample to assess material heterogeneity with regard to polarization and conductivity responses. A schematic of a proposed apparatus is given in Figure 7.1. The models used here employ rather elementary first approximations to the electromagnetic/acoustic interactions, but they clearly demonstrate the potential for exciting new interrogation techniques based on more sophisticated models as well as other types of acoustic signal generation.

7.2

Limitations and Unanswered Questions

As summarized above there are a number of positive features arising from the variational approach we use in this monograph. It permits an integrated treatment of the forward and inverse problems including proof of existence and uniqueness of solutions of the forward problem and a rigorous mathematical assessment of properties of estimators. The variational formulation also allows natural incorporation of the radiating source, the antenna, in the mathematical and computational models. A broad class of media can be considered in the forward and inverse problems, whether the material is described by a dielectric response mechanism in differential equations or convolution representation. In limited cases (including the Debye medium) a semigroup treatment is possible and is of great interest. The use of a reflecting backing is seen as an opportunity to sharpen estimates, and, in the case of an acoustic backing, can be seen as an approach to estimation of material heterogeneity.

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Figure 7.1: A proposed coaxial transmission line apparatus for the electromagnetic-acoustic probing of a liquid or shaped solid sample. The electromagnetic wave is set up between the central post and the grounded outer line structure and launched from the left. The acoustic wave is launched from the right. A stream of electromagnetic pulses is considered as "tracking" the leftward moving acoustic front. However, several questions remain for even the one-dimensional case focused on here. While it appears that the variational approach provides an accurate approximate computation of the electromagnetic transients, for example the Sommerfeld and Brillouin precursors, and finite element convergence theorems are given, the questions of a posteriori estimates of numerical error in the forward problem are not addressed. Similarly, in the inverse problems, further work is needed on the statistical properties of the parameter estimates. For example, is it reasonable to use a Cramer-Rao bound as a variance estimate? While it is clear from (4.11) that all parameters in the Debye medium are in principle amenable to estimation, methods toward a deeper understanding of the significant differences in the variability of the estimates of the model parameters would be most useful. Sensitivity of estimates to measurement noise (which is only briefly addressed here) requires further investigation. Quantitative statistical procedures are needed to determine from data whether the noise model employed in this monograph is in fact encountered in electromagnetic applications.

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Other nontrivial questions remain. For example, can preprocessing of the time domain data improve parameter estimates? It is known that windowing can improve spectral estimates. Are there analogous niters that can be used in the time domain to improve least squares estimates of dielectric parameters? Major questions in many inverse problems concern the choice of input excitation signal to the system under interrogation and this is certainly the case for electromagnetic interrogation. No realistic materials properties estimation technique would ever use a single reflection event to determine material parameters as is done in the examples presented above. However, it is not clear how information from multiple pulse events is best combined for our problems. On one hand, one could temporally align pulses to achieve a degree of noise reduction, at the expense, perhaps, of introducing phase error. On the other hand, and perhaps more attractive while being more numerically expensive, one could estimate parameters using each pulse singly and then average in some manner in the parameter space. These concerns bring up the further issue of how to best select the interrogating pulse to optimize the statistical quality of the material parameter estimates. More complex models have been employed in the study of materials than have been evaluated in this monograph. Multiple Lorentz or multiple Debye processes acting together can describe polarization in a material. Drude models can be used in describing conductivity. In principle, all of these mechanisms can be treated in the context of the framework developed here. How well our variational formulation with least squares estimation performs in these problems remains untested. Questions of a more mathematical nature are raised by our incomplete semigroup formulation. As we have already noted, Lorentz media do not satisfy the monotonicity properties for the dielectric response function used in developing the semigroup results of Chapter 3. This monotonicity is clearly sufficient for such a formulation; we do not believe it is necessary and conjecture that different arguments could lead to establishment of a semigroup formulation for electromagnetic models that include a Lorentz polarization law. More importantly, extensions of the variational theory and the associated computational methods for both forward and inverse problems in full three dimensional configurations are most desirable. Such a theory with the ability to incorporate antenna sources would lead to a most useful near field electromagnetic pulse in interrogation framework. The potential benefits of near field interrogation will be outlined in the next section.

7.3

Future Technological Possibilities

We believe the ideas presented in this monograph are a first step in the development of an electromagnetic-based remote interrogation methodology for materials and structures in civilian and military aeronautical, space, marine, and subterranean applications, as well as biomedical diagnostics. The potential

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applications are enormous and we offer some admittedly highly speculative ideas here on some of those. First of all, near field probing of objects as mentioned in the previous section would have extensive utility. For example, the evaluation of concrete structures for changes in polarization or conductivity that could be warnings of material change or fatigue could constitute an important contribution. Walls of buildings, roadways, and runways will frequently have imbedded metal bars (rebar concrete) that can be used as internal scattering objects. Alternatively, inspection of roadways or runways could be accomplished using surface acoustic wave generation in concert with near field probing. A fascinating question arises as to whether natural earth vibratory activity can be used to assist in electromagnetic subsurface probing. Extremely narrow diameter catheters have already been developed for medical use. Assessment of changes in tissue polarization dynamics and conductivity can be accomplished via scattering from catheters containing small surfaced areas of inert metals. Near-field electromagnetic probing of the human body can also be performed using electromagnetic scattering from a collimated beam of ultrasound. Far-field electromagnetic probing could also be valuable and it is probable that our existing one-dimensional algorithms can already contribute in this regime. For example, the concept envisaged here is to assess the chemical structure of bodies of water and soils using electromagnetic pulse responses. In this case there is no possibility of a metal backing to assist an airborne assessment. However, coordinated ground level acoustic events (e.g., near surface explosions as employed in seismic prospecting) could assist in the spatial localization of material property changes. It may be possible to extend the procedures described in this monograph fully into the optical regimes. In this case, probing of clouds and exhausts from urban, rural, or industrial processes would be feasible. Once again localization of material properties could be enhanced by the simultaneous application of an acoustic event into the gaseous structure that was being assessed. We anticipate that the mathematical methods described in this report may find application to aerospace and military radar applications. For example, extremely fast pulses could interrogate painted aircraft metallic surfaces. While bistatic or multistatic radar systems might be needed for the analysis, assessment of paint composition could lead to aircraft identification. Similar considerations would apply to aerospace use of radar absorbing materials. Unpainted metallic objects are likely to have oxidized surfaces. It may be possible to interrogate these surfaces using very short pulses. We envisage a rich interaction of these electromagnetic probing methods with the smart materials, acoustic and vibration control techniques already developed by our research group (e.g., see [BSW96]). For example, in near-field or far-field probing of the earth, preplaced and preprogrammed acoustic sources could be used to create surfaces of overpressure that could in essence be tracked using electromagnetic pulse sequences, exploiting the vast difference between

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sound and electromagnetic speeds. The same technique could be used to assess building structures and could assist in finding victims within collapsed buildings and rubble. Carefully designed weak external acoustic events could subtly deform aeroelastic vehicles and the elastic response can be probed using fast electromagnetic pulses whether or not there is a surface paint, other absorbing material, or oxide present. Similar applications in noninvasive biomedical interrogation can be readily envisioned.

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Index absorbing boundary condition, 19, 57 acoustic, 112 acoustic gratings, 14, 112 acoustic waves, xi acoustooptic effect, 14 acoustooptics, 8, 18

Galerkin finite element approximation, 57 general polarization, 57 Gronwall inequality, 35

conduction current density, 9 conductivity, 3, 7, 9, 10, 48, 68, 133 Drude, 133 Ohmic, 1, 10 conductivity susceptibility kernels, 3 continuous dependence, 27, 32 convolution, 10, 28 convolution kernel, 106 Crank-Nicholson scheme, 60, 61

invariant imbedding, 1, 2 inverse problem, 18, 42, 49, 57, 63

hysteresis Debye polarization models, 98

least squares functional, 49 magnetic fields, 9 magnetic flux densities, 9 Matlab, 80 Maxwell's equations, 4, 8, 9 method of mappings, 16, 22 microwaves, 7 minimization problem, 49

dielectric parameters, 4 dielectric permittivity, 9 differential Debye model, 97 dispersive materials, 1 displacement susceptibility kernel, 3,4

Ohm's law, 9 parameter estimation problems, 48 permittivity, 58 relative, 11, 12, 58 planar electromagnetic waves, 1 polarization displacement susceptibility kernel, 3, 35 polarization, 3, 7, 10, 15, 16, 28, 48 constitutive laws, 9, 16 Debye, 4, 12, 26, 35, 79, 133 dielectric response function (DRF), 4, 10 dipolar, 11 dipole, 10

electric flux density, 9 electric polarization, 3 electromagnetic waves, 7 existence, 27, 28 finite difference, 108 finite element approximations, 42, 47, 108 frequency domain, 3 Galerkin, 28, 64 145

146 displacement susceptibility kernel, 10, 35 electronic, 10, 12 frequency dependence, 9, 10 general nth order mechanisms, 12 instantaneous, 11, 12, 15 ionic, 10 Lorentz, 4, 12, 35, 79, 133 orientational, 10, 11 polarized planar wave, 18 precursors, 4 Brillouin, 4, 80, 84, 130 Sommerfeld, 4, 80, 84, 130 random noise, 71 regularity, 27, 42 Runga-Kutta, 80 scattering techniques, 2 semigroup, 35 sesquilinear forms, 28 source current, 21 source current density, 9 supraconductive, 20, 23 supraconductive material, 4, 8 susceptibility kernel, 14 uniqueness, 27, 33 wave splitting technique, 1, 2 well-posedness, 27 windowed microwave pulses, xi windowed pulse input signal, 22, 110 windowed signal, 21

Index