Linear and nonlinear ill-posed problems

LINEAR AND NONLINEAR ILL-POSED PROBLEMS V. A. Morozov

UDC 518.517.9

The present survey has as its central motif the concepts and methods of approximate solution of ill-posed ("unstable," "improperly posed," "incorrect") problems. The uniqueness aspect of inverse problems is glossed over lightly; the full discussion of this topic is better left to a separate treatise. In writing the survey the author has drawn from a number of bibliographic sources; worthy of special mention are the book by M. M. Lavrent'ev [i01], as well as the survey articles by V. Ya. Arsenin and O. A. Liskovets [17] and P. Medgyessy ~224]. w

Well-Posed Mathematical Problems

An important consideration in the formulation and development of an approximative method for the solution of a mathematical problem is whether it is "well posed." Hadamard [217, 218] framed the following definition. The problem of solving the equation

Au=f, y~F,

(1)

where A: DA~U+F is an operator having a nonempty domain of definition D A and acting from a metric space U into an analogous space F, is said to be well posed ("properly posed," "correctS') if the following conditions are satisfied: i) QA = A[DA] = F (solvability condition); 2) Aul = Au2, ul, g2@DA, implies ul = u2 (uniqueness condition); 3) the inverse operator A -I is continuous on F (stability condition). The fulfillment of Hadamard's conditions 1)-3) seemed a natural adjunct to any sensible mathematical problem, so much so that Hadamard propounded the notion that every ill-posed problem is without physical meaning. He conceived the classic example of an ill-posed problem, namely the Cauchy problem for the Laplace equation. It was subsequently learned that an enormous multivariety of topics in mathematics and the natural sciences takes in the large, including the continuation of analytic and harmonic functions, geophysical problems, problems in supersonic flow over bodies, etc., demands the solution of precisely that problem. The latter in fact emerged as the model problem in numerous studies of ill-posed problems, particularly in the Nineteen-Fifties. An important class of ill-posed problems comprises the so-called inverse problems [169] involving the reconstruction of quantitative characteristics of a medium on the basis of measurable physical fields determined by it. Many theoretical and practical problems in the processing of physical experiments, the reconstruction of unknown parameters in equations from knowledge of a certain system of functionals of their solutions [119], etc., are reduced to inverse problems. Tikhonov [169] formulates the following generalization of the classical (Hadamard) concept of a well-posed problem. A problem is said to be conditionally well-posed (well-posed in the sense of Tikhonov) if: i') it is known a priori that a solution ~ of problem (i) exists for some class of data in F and belongs to a given set M: u-~ M ~ Q'; 2') the solution is unique in class M; Translated from Itogi Nauki i Tekhniki

(Matematicheskii Analiz), Vol. II, pp. 129-178,

1973. 0 1976 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, ~ t h o u t written permission of the publisher. A copy of this article is available from the publisher for $15. 00.

706

3 ~) infinitesimal variations of the right-hand side of (i) such that the solution does not exceed the bounds of M correspond to infinitesimal variations of the solution. The set M in Tikhonov's definition is often called the well-posedness set (class). Tikhonov [169] first brought attention to the following well-known topological theorem articulating sufficient conditions for problem (i) to be well-posed in his sense. THEOREM (Stability of the Inverse Operator). If a nonempty compact set A 4 ~ D A satisfies conditions l') and 2'), then for continuous A the operator A -I taken on N = A[M] is continuous 9

Lavrent'ev [i01] notes that, when the conditions of the theorem are satisfied, a continuous nondecreasing function ~(t), T > 0, ~(0) = 0, exists such that for any u, v ~ M : p = (Au, Av) ~ T, the following bound is obtained: 0U(U, v) ~ ~(T). The function ~(T), which is in fact the modulus of continuity of the operator A -I on N, is sometimes called the well-posedhess function. Extensions of this theorem to metric and topological spaces are obtained by Ivanov [74, 78] for the case of a closed invertible operator A and by Liskovets [114] for a noninvertible operator. Tikhonov [169] also uses a local version of the inverse-operator stability theorem to prove the stability of the inverse problem of potential theory for a class of bodies; the uniqueness of solution of this problem is proved in an earlier paper by Novikov [142]. The first results pertaining to the Tikhonov well-posed character of certain problems in the theory of analytic functions are attributable to Carleman [207]. Rapoport [156, 157] deduces estimates characterizing the well-posedness of the planar problem of potential theory in a very restricted class of solutions. V. K. Ivanov [69, 70] refines certain conditions of Novikov's theorem and determines new classes in which the inverse logarithmic potential problem is well posed. Landis [105] obtains uniqueness and stability theorems for the solution of elliptic equations. Mergelyan [125] investivates the harmonic approximation problem in the class of harmonic polynomials and in the class of bounded harmonic functions, and gives error bounds in terms of known initial data for the problem. John [220] verifies that a particular problem relating to the continuation of a harmonic function is Tikhonov well-posed, and determines the well-posedness function for it; the same author also [221] determines the classes in which the following problems are well posed: analytic continuation of a function onto the exterior of a circle of radius 0 > 0; the Cauchy problem for linear and general elliptic equations with analytic coefficients; continuation of the solutions of the wave equation in a three-dimensional direction; the Cauchy problem for general linear equations with constant coefficients. It is shown that the first two problems have a well-posedness function of the HSlder type ("proper" problems), as is sometimes true of the last problem, while the third and fourth problems have logarithmic continuity ("improper" problems), and their numerical solution is almost unreliable. This work is continued by John in another paper [222]. Stability estimates are obtained by Pucci [228, 229] for the Cauchy problem for the Laplace equation. He also formulates a stable method for the solution of this problem. Lavrent'ev [95] investigates the problem of determining a harmonic function in a twoor three-dimensional domain on the basis of Cauchy data on a piece of the boundary. Stability is proved in the class of bounded functions. Three methods are given for the stable solution of a problem with approximate ("rough") data. The same author [96] derives the well-posedness function for the problem of analytic continuation of a function defined on a certain infinite point set z:Iz I < i; the analogous problem is considered for harmonic functions. The same body of ideas is brought to bear on an investigation [97] of the inverse problem of three-dimensional potential theory. The author also [98] obtains stability estimates for the solutions of general elliptic equations. The general content of these papers is summarized in a book by Lavrent'ev [i01]. The Tikhonov stability of certain problems for partial differential equations is analyzed by Miller [225]. Krein [91] proves that evolutionary problems of the backward heat-conducting Tikhonov well-posed.

type are 707

Krein and Prozorovskaya [92] prove that evolutionary problems in Banach spaces with an operator generating an analytic semigroup in a certain sector of the complex plane are Tikhonov well-posed in the class of bounded solutions. In another paper [93] they prove the convergence of the method of finite differences for problems of this type. Antokhin [7, 8] investigates a number of problems associated with the uniqueness and stability aspects of ill-posed problems in the theory of elliptic systems. The Tikhonov well-posedness of problem (i) has tion. A great many papers have been devoted to the references cited above, the problem is discussed in [103] and by Romanov [158]. The uniqueness problem tions to the collective work [143].

important bearing on the uniqueness quessolution of the latter. Besides the books by Lavrent'ev, Romanov, and Vasi!'ev is also treated in many of the contribu-

Anikonov [6] establishes uniqueness theorems for general nonlinear operator equations with operators satisfying the quasi-monotonicity condition. Several uniqueness theorems are proved by Prilepko [150-152] for the problem of determining the shape and density of a body on the basis of the internal and external potentials without the usual condition of a "starshaped" body, Prilepko also ['153] obtains stability estimates for various classes of bodies, including convex bodies. Ostromogil'skii also treats the uniqueness of solutions of inverse problems in potential theory [144] and proves [145] a uniqueness theorem for the problem of determining the forces responsible for vibrations of a certain elastic medium, recorded at the boundary of a domain. Glasko [43] investigates the uniqueness problem for certain inverse problems encountered in the study of the deep interior structure of the earth. w

The Methods of Lavrent'ev for the Approximate Solution of Tikhonov Well-Posed Problems

The first method, proposed by Lavrent'ev [94] initially for the effective (i.e., with predeterminied error) approximate solution of the Cauchy problem for the Laplace equation, involves the replacement of the operator for the original problem by a similar operator that renders the transformed problem Hadamard well-posed. The method is formulated in a later work [i00] for arbitrary operator equations of the first kind. It is assumed that the operator A, [IAH< I, is full-continuou_s and U = F is a Hilbert space. Problem (i) is assumed to be uniquely solvable for f = f; the compact set chosen in the problem is Yd----{~EU:u----Bv,[Iv][~<1}, where B is a completely continuous linear operator. The goodness function ~(r) is considered to be known a priori, so that for any ~CA4: ! ] A u I I ~ , we have the bound Hu]]~<~(~). The simplest method is formulated for the case in which A is positive and the operators A and B commute. Approximate solutions u T are determined from the equation of the second kind

(Au+~E)u~=f~, in which T > 0 is a parameter side of the inequality

determined

f6CU: [[7--f~lI~<~,.

from the minimization

(2) condition

for

the

right-hand

§ or as the root of the equation m(m) = 61T. In either case, clearly, u z § u as 6 + 0, i.e., Eq. (2) provides for the effective construction of approximate solutions. If the operator A does not satisfy the stated condition, the formulation of the method is more complicated. Thus, if the operator B commutes with A*A (A can be nonpositive), we consider the equation of the second kind

(r + A'A) a~~- A'f6, a n d m:m(C~T) r

= 6.

For such

m = T(~)

Ilu - ll

(V-m)-+o,

If the operator AB is positive, we put u~ = B (AB + ~E) -1 f~,

and in the case of arbitrary A and B 708

(3)

u,----B [(AB)* AB § z2E]-~(AB)* f~. It is proved that parameters T = T(8) exist such that liu,-~II-+0, a-+0. A second method [!01] pertains to the determination of approximate solutions on the basis of successive approximations. Earlier, Carleman [207], Goluzin and Krylov [45], and Malkin [117] propose approximation methods yielding exact solutions of (i) in the limit if the initial data, i.e., the operator A and right-hand side f are exactly specified. Andreev [5] proposes a method of solution of the Cauchy problem for the Laplace equation with exact data. The method is formulated in general form by Malovichko [118]. These papers, however, omit the necessary a n a l y sis of the influence of error in the data, which is exceedingly important with regard to illposed problems. If the operators A and B are such that A = A*, A > 0, and AB = BA, we can then put uo = f~ and u~+I=u~--(Au~--f~),

n = 1, 2 . . . .

.

Now

if

n + ~.

For a special

choise

of n = n(~)

Lavrent'ev

obtains

a convergence

un § u as

~ + O.

Analogous results are given for other cases, including arbitrary A and B. Lavrent~ev [i01] also justifies the convergence of the method of successive approximations for certain classes of nonlinear operator equations. He uses the operation of projection of the resulting approximations onto the set M. The method of successive approximations is investigated by Antokhin [8, 9] under different assumptions. Lavrent'ev, Anikonov, and Fazylov [104] discuss the solution of a nonlinear equation (i) on a chain of compacta, estimating the error of the approximate solution and using their results to solve a particular inverse problem. The approach of Lavrent'ev has been further developed in the regularization method of A. N. Tikhonov (w and the quasi-solution method of Ivanov (w w

Ivanov's Method of Quasi Solutions

One of the difficulties met in the application of the Lavrent'ev method is the need of direct knowledge of the well-posedness function in the formulation of approximate solutions. This feature strongly limits the class of solvable ill-posed problems. Modifying the concept of the solution of (i), Ivanov [71, 72] succeeds in circumventing specification of the wellposedness function. The familiar concept of the least-squares method is used, but the residual is minimized, not on the entire domain DA, but on a portion (as a rule, compact) thereof: M~DA. Thus, the quasi solutions uM are given by the condition uM~M : 9e (AuM, f ) = inf @ (A u, f ) . uCM

(4)

The quasi solution generalizes the usual concept of a solution. The basic notion of the formulation of quasi solutions is encountered in the earlier work of Mergelyan [125] and Pucci [228, 229], as well as in papels by Douglas [210, 211], who proceeds from certain general notions of mathematical programming. In [210] Douglas analyzes the problem of finding a harmonic function u defined and nonnegative (well-posedness class) in the half-plane y ~ 0 on the basis of its values of the line y = c (c > 0). Given the condition that the function x(x, y) has bounded first derivatives for y ~ 0 and 0
2+(y+k2)1-1,

k>0,

the sequence of functions Fn(x): ] F n ( X ) -- u ( x , c ) [ < ( l / n ) , ]x[ < ~ , i s u s e d t o c o n s t r u c t a sequence of rational h a r m o n i c f u n c t i o n s Wn(X, y ) : wn g u i n a n y c l o s e d h a l f - p l a n e y ~ so > 0 .

The functions are chosen so that

709

n

i=--n

and t h e c o e f f i c i e n t s

mi a r e d e t e r m i n e d by t h e c o n d i t i o n ]lw,~-- FN (x)[ I ~ IIw-- F x (x)[i,

ii/]i2~ Efe (x~) ax i,

on the class of functions ~ of the indicated type. The proof of convergence rests on general properties of families of functions compact in C and the uniqueness of continuation (according to, in particular, the Stieltjes--Vitali theorem). An analogous idea is used in [211] for the numerical solution of the analytic continuation problem for a function defined on a circle of radius p > 0, onto the domain z: Izl ~ p in the class of bounded functions (well-posedness class). Douglas later [212] elaborates his method in application to the numerical solution of integral equations of the Volterra type, of the first kind: x

J k (x,

= f (x)

0

Under the condition that the approximate function f(x) is given and the compactum 7~9~(x)is known, where u is an exact solution, it is proposed that an approximate solution ~ ( x ) @ M b e sought as a solution of the extremal problem

HAu - - / H = inf IlAu --.~1. uCm Mathematical programming methods are used f o r the n u m e r i c a l s o l u t i o n

of this

problem.

An a n a l o g o u s i d e a i n v o l v i n g l i n e a r p r o g r a m m i n g m e t h o d s i s a l s o u s e d i n [212] f o r t h e a p p r o x i m a t e c o n s t r u c t i o n o f a h a r m o n i c f u n c t i o n on t h e b a s i s o f a d i s c r e t e s e t o f v a l u e s of that function. V. K. Ivanov has formulated the general concept of the method of quasi solutions and investigated fundamental problems associated with them, including the existence, uniqueness, stability under perturbations of the right-hand side, and construction of quasi solutions. Fundamental arguments are advanced for the case in which M is a compactum and A is a fullcontinuous invertible operator. In this case problem (4) is always solvable, i.e., the set of quasi solutions U M # ~. In the same situation, quasi solutions are stable in the sense of B-convergence of sets according to E. A. Barbashin If U and F are linear, if the equation Au = 0 has only a trivial solution, and if M is a convex compactum, then the strict convexity of F guarantees uniqueness of the quasi solution. For the numerical determination of quasi solutions, V. K. Ivanov [71, 73] proposes the use of a chain of finite-dimensional expanding compacta Mn, n=l, 2, .., M=uM~. In this case the sequence of solutions of finitedimensional problems n

UnEMn:U6MnPFin[ (Au, f)=PF(AUn, f),

(5)

c o n v e r g e s t o an " e x a c t " q u a s i s o l u t i o n ( p o s s i b l y i n t h e s e n s e o f B - c o n v e r g e n c e ) . The s o l u t i o n (5) i s t y p i c a l o f t h e w e l l - s t u d i e d m a t h e m a t i c a l p r o g r a m m i n g p r o b l e m and i s e f f i c i e n t l y implemented on a computer. In [71] V. K. Ivanov applies the method of quasi solutions to the numerical solution of the inverse logarithmic potential problem. Its application is particularly effective when U and F are Hilbertian on weakly compact sets 7~={uCU:IIu[I
(XE + A'A) u ~ = A * f

X > O,

and the numerical determination of the Lagrange parameter % from th% condition X:llu~l[~. The weak convergence of ux to an exact solution ~ is guaranteed if u67~m Liskovets

[ii0] uses a similar approach to solve certain ill-posed problems.

Dombrovskaya and V. K. Ivanov [61] investigate quasi solutions for continuaus linear operators A acting in Hilbert spaces U and F, on closed convex sets M. They prove the 710

stability of the quasi solutions with respect to the right-hand side in the weak topology. The method is used for the numerical solution of integral equations in convolutions. V. K. Ivanov [74] elaborates the theory of the quasi-solution method for the case of closed linear operators A having an unbounded inverse and acting from a linear topological space U into a Banach space F of the type E, M E U is a convex compactum. For quasi solu!ions the representation u M = A-IPf is given, where P is the metric projection operator onto N = A[M]. It is proved that the determination of the quasi solutions is a Hadamard wellposed problem, and the inverse-operator stability theorem [169] is generalized to the case of unbounded A. The properties of the projection P are analyzed as a function of the geometry of F. It is proved, in particular, that P is continuous if F is an E space [214]. Analogous problems relating to noninvertible operators are studied by Liskovets in terms of B-convergence [109]. The B-convergence of quasi solutions with respect to the complete data set, namely the right-hand side, the operator A, and the set M, is proved in another paper by the same author [113]. Here, essentially, the inverse-operator stability theorem deduced by V. K. Ivanov [78] is further generalized to the case in which U and F are topological spaces (F is a Hausdorff space). A somewhat more general method than the quasisolution principle may be found in [78]. Denisov [50] uses the method of quasi solutions to analyze integral equations with a kernel having the ~-property. Ramm [155] applies the same method to the solution of certain inverse problems in the synthesis of antennas. w

Tikhonov's Regularization Metho d

The method first proposed by Tikhonov [170, 171], while it basically, like the method of Lavrent'ev, necessitates the solution of a Fredholm equation of the second kind, is based on a totally different notion, namely stabilization of the minimum of the mean-square deviation of Au from a given right-hand side ~ by means of an auxiliary parametric functional. Tikhonov starts with the following problem of computing the values of an operator:

u=Rf, /e~,

(6)

where R is an operator defined in F with values of U. The solution of Eq. (I) necessitates the solution of problem (6), in particular, under the condition that the operator A is invertible. The problem R of computing an element u ~ U from a specified element f6F is said to be well posed [171] if the following conditions are satisfied: i =. Every element ~6F is associated with a certain element u E U 2*. The element u = R~ is uniquely determined by the specification of f. 3*. The element u depends continuously on the perturbations of the element ~. It is readily seen that a Hadamard well-posed problem (i) is also well posed in the stated sense if we put R = A -I. The converse is not true in general, because problem (6) is not equivalent to (I) in the general case. Tikhonov is the author of the important concept of the regularizing operator (or algorithm) for problems ill-posed in the sense defined here. Specifically [171], an operator satisfying the following conditions is called a regularizing operator and is denoted by R6f ,

~ >0. i. The operator R~ is defined for all f~F and ~ > 0. 2. If u = R~, then u s = R ~

§ u in U as ~ ~ 0.

The family u s = R ~ 6 is called the regularized family of approximate solutions. Tikhonov regards as fundamental the problem of solving the Fredholm integral equation of the first kind b

Au~-Ik(x, ~)u(~)d~=f(x), c ~ x < d ,

(7)

a

with continuous kernel k(x, ~), a ~ x ~ b, c ~ ~ ~ d. The operator A is considered to be the continuous map C[a, b] + L2[c, d]; it is assumed here that Eq. (7) has the unique solution ~($)CC [a, b]. The fundamental assumption is taken to be ~@W/~~)[a, b], i.e., the solution has a square-summable derivative. Now the regularization method is implemented by the 711

following algorithm.

A smoothing parametric functional is introduced: M s [u; f~] - N [u; f~] + ~ (u),

uGW~~,

where ~ > 0 is the parameter and

N [u;/61 =--llAu--/~[l~,, ~

(u)=llull~,~,

and the following minimization problem is considered: Find an element tt~CW~ I) for which

inf .,'V/~[u;f~] =M~ruS".,t a, fa]. ,,ew~ I)

(8)

I t i s ~ p r o v e d t h a t f o r any a > 0 and f o r e v e r y elementf~fiLz[o, d]:problem (8) has a u n i q u e solution u~, which is found as the solution of the Euler equation

(~E + A'A) u = A* f~,

(9)

in which

A*:(Au, g)L,=(U, A*g)~l), gEL2. The solutions u~ thus found have the following property: If the parameter ~ is constant with the error 6 of the approximate right-hand side f6 in the sense that the ratio ~V~=O(1),

then the family u~ ~ u~ ~ u as ~ § 0. ness in C of the sets U M = {u:~(u) ! M

(i0)

The proof of this fact relies heavily on the compact= const} and the membership~CU~for a certain M.

It is also proved in [170] that finite-difference methods can be used for the numerical solutions of an equation (9) of the second kind, and the applicability of projection methods is noted. Arsenin [12] justifies the convergence of the regularization method when the solution is a piecewise-smooth function. We point out that, although the construction of the compact embedding of W$ ~) in C is used in the regularization method, no use is made of information on the compactum containing the exact solution of (7). This feature accounts for the popularity of the method in practice. The construction of a compact embedding of the space W~ n+l) in C(n) is used in [171]. The regularizing functional ~(u) in this case is given as

The foregoing definition makes it possible, under_the same condition (i0), to prove the convergence of the approximate solutions u 6 H u~ to u in C (n) space (provided that the exact solution ~CW~n+~)): For the case in which the smoothing functional ~(u)----jlullL,,Tikhonov verifies only weak convergence of the regularized solutions.

Bakushinskii [19] and Morozov [126] subsequently succeed in proving the strong convergence of regularized solutions for arbitrary Hilbert spaces (assuming the operator A to be linear, self-adjoint, and positive). The regularized solutions in this case are determined as the solution of the operator equation of the second kind

(=E + A) u = f ~

(11)

with the choice of parameter a:~/a § 0 as ~ § 0. The necessity of this condition for convergence of the regularized solutions of (ll) has also been proved [19]. The strong convergence of the regularization method has been verified for a closed linear operator [68]. Tikhonov [171] confirms the possibility of regularization in the form (!i) and demonstrates it in the examples of the problem of continuation of a potential in the direction of perturbing masses as well as the backward heat-conduction problem, both the standard type and the corresponding problem of determining climate in backward time. Fadeeva [191] studies the algorithm (Ii) for systems of linear algebraic equations with a symmetric nonnegative matrix. An algorithm for refinement of the approximate solutions is based on the notion of suppressing the "high-frequency" components of the approximate solu712

tions, and numerical calculations are carried out. Korkina [86] investigates the regularization algorithm (ii) for closed linear operators in B spaces. V. K. Ivanov [77] investigates the necessary and sufficient conditions for convergence of the regularized solutions of (8) in L2 and C (~(~)=llulI~}~ L ' It is established in [19] that the necessary and sufficient condition for convergence in L2 is ~2/a + 0, ~ § 0. Ivanov shows that the condition a = 0(62) is necessary and sufficient for weak convergence (Korkina [89] obtains an analogous result for general closed linear operators). The conditions for convergence in C are complexly formulated and are related to the behavior of the C norm of the operator R a = (aE + A'A)-I A* as a § 0 (sufficient conditions for convergence in C are given in an earlier paper by Khudak [194]). Analogous considerations for Eqs. (8) with a Green kernel are studied by Korkina [87] and in a later work by Khromova [193]. Mel'nikov [124] applies Ivanov's procedure to the case in which the operator (7) acts from M[a, b] into L2[a, b]. The proposed method is verified numerically in [184] in the example of solution of the model problem for an integral equation of the form (7): I

(x,D~(~)~=f(x),

--t~<X~t,

k(X,~)=-- ~ (x--~)~+h ....

which corresponds to the solution of certain inverse potential problems, as well as spectroscopy. The influence of both computational errors and "instrument" errors modeled in a particular way is exposed. It turns out that the regularizing algorithm yields high precision; on the other hand, the standard algorithms for solution of the problem either have a very low precision or are rendered totally inapplicable by overloading of the digital-place array of the computer in the computational process. A. N. Tikhonov has also developed the regularization method in application to nonlinear integral equations. The convergence of regularized solutions in C is proved in [173] for a nonlinear integral equation. The conditions under which the regularized problem (8) has a unique solution are investigated (for the nonlinear case). The regularization method receives the most thorough treatment with regard to the solution of operator equations in a paper b~ Tikhonov [174]. There also, the general concept of an S-compact embedding of metric space U into another metric space U is introduced. The following conditions are necessary for this operation: i) U c U is the set-theoretic sense. 2)_The U metric is majorant with respect to U metric, i.e., PU(U, v) ~ p~(u, v) for any ~, vCU. 3) Every sphere S c(~0)= {~CU :ps(~,~0) k ~ 0, etc. 2 Assuming that a certain Hilbert space U is S-compactly embedded in the space U and that U is the uniqueness set for an operator A (i.e., Au: # Au2 if u~ # u2 for all ul, u2~), Tikhonov introduces the following parametric functional for solvable equations:

where ~(~)-----I[ul]~,and he discusses the minimization problem (8). It is proved in [174] that regularized solutions exist in U for all a > 0 and any f ~ @ F : p F ( ~ f s ) ~ , and that they converge in U. Sufficient conditions are given, guaranteeing uniqueness of the solution of the parametric problem (8). We note that the construction of an S-compact embedding is well suited to practical applications of the regularization method, specifically by virtue of the fact that the approximate solution of problem (i) does not rely on any quantitiative information about the unknown solution (cf. the quasi-solution approach). The regularization algorithm is numerically substantiated in [183] in the example of solution of a nonlinear integral equation of the type l

An------~ K (x, $; u @))d~ = 7 (x), --L ~<x < L --l

713

with kernel

K ( x , $ ; u ) ~ ~I in (x_~)2+(H_u) (x--S) '+H2 ~, t t > O . Equations of this kind emerge in certain inverse problems of gravimetry. The solution of problem ~8) is sought bn an a-net such that as+ 1 = ~as, s - 0, i, 2, ..., ~ < i, where ao is the initial value of the regularization parameter, by the Newton successive-approximation method. A suitable practical value of the regularization parameter (on a discrete net) is determined in this case from the condition of minimization of the function z(~)=II u~ --~IIu 9 This principle of quasi-optimal evaluation of the regularization parameter corresponds to selection of the "smoothest" element from the "cluster" of regularized solutions u~ (the same principle is treated from another angle in [131]). A numerical experiment shows that the regularization method admits stable approximate solutions even in the face of a high level of error on the right-hand side. An important contribution to the evolution of the regularization method is Tikhonov's treatise [176] on the solution of degenerate systems of linear algebraic equations (possibly overdetermined). This paper is the first investigation of a problem having a nonunique solution. It is shown that the regularization method provides solutions converging to the normal solution of an exact algebraic system (to the solution with minimum norm). The influence of preturbations in the specification of the matrix A is also analyzed. The regularization parameter is chosen as a function a(~, h) of both the error d of the right-hand side and the error h in specification of the matrix A, where a(~, h) + 0 as d, h § 0. A more detailed treatment is offered in [178]. This result is generalized in [132] to the case in which U and F are Hilbert spaces and A is a normally solvable operator (i.e., its value domain QA is a subspace of F). In the same paper is introduced the concept of a pseudosolution of (i), namely a minimum-norm element minimizing the residual llAu--~l]~, u G U. It is shown that the problem of determining the pseudosoluti0ns is Hadamard well-posed in the case of a normally solvable operator A. The application of the regularization method to such problems shows that it is sufficient for the parameter a to be consistent only with the error h of the operator A. The regularization method of a Hilbert Space U has been generalized to the case of an unbounded (closed) linear operator A [37, 67, 68, 131] and to the case of unbounded nonlinear operators [134]. In the latter case the B-stability of Tikhonov regularization is proved. Analogous questions are discussed by Liskovets [iii, 114] for topological spaces when the sets 97~={u:~(u) < C } are relatively bicompact in U. Vasin [35] proves the applicability and B-stability of projection methods for the solution of the regularized problem under conditions somewhat generalizing the conditions of [134]. Typical numerical calculations are given. Gavurin [42] gives a topological treatment of the regularization method. [38] is concerned with general aspects of Tikhonov regularizability.

Vinokurov

A special regularization technique known as self-regularization has been proposed 54] for the solution of integral equations having singularities in the kernels.

[53,

The self-regularization method is based on the consistency of the characteristic "width" of a kernel singularity and the step of the quadrature formula. This method yields stable numerical solutions in the case of kernels having delta-function properties. Denchev [52] elaborates the regularization method for the determination of solutions of (i) in the weak sense (Hilbert spaces U and F) and uses his version to solve boundaryvalue problems for equations of hyperbolic type. The weak convergence of the regularized solutions is proved, but it is easily shown that strong convergence to a normal solution is valid. Algorithmic schemes for the selection of the regularization parameter in such a way as to ensure convergence of the approximate solutions are important in the practical application of the regularization method. These problems are investigated in a series of papers by V. A. Morozov. In [126], for example, under the fundamental postulates of A. N. Tikhonov a technique is established for selection of the parameter according to the values of the functional Ma[u; f~] on regularized solutions from the condition

714

:i~

[u s , f 4 - o 2 .

In [127~ 128] a t e c h n i q u e i s p r o p o s e d and j u s t i f i e d for linear operator equations in Hilbert s p a c e s U = U and F, w h e r e b y t h e r e g u l a ~ i z a t i o n p a r a m e t e r i s s e l e c t e d i n a c c o r d a n c e w i t h t h e residual principle (the practicality o f t h i s mode o f s e l e c t i o n o f t h e r e g u l a r i z a t i o n parame t e r i s a l s o n o t e d i n [62, 7 5 ] ) , w h i c h r e p r e s e n t s an a d a p t a t i o n o f t h e w i d e l y u s e d p r a c t i c a l criterion for the precision of approximate solutions to ill-posed problems. Thus, the regularization parameter can be selected as the solution of the equation

p (~) = ~2, p (~) --II Au~ - - f8 tl~The same parameter-selection scheme has also been justified for the case in which the operator A is approximately specified, but now certain information on the exact solution is required. Specifically, it is assumed that there is a known sphere Sc={u:llUllu
2

[]Au--fs[[~+~[lu--u~llu,

uEU, AC[U-+F],

O)

where u~ minimizes (i) the same functional with the substitution therein of

(0

u~ = O.

It is shown

(2) 0 ) du= that u~u~--a~5-and on t h e n e t a j = ~ a j _ l , V = 1 t h e s e l e c t i o n o f q u a s i - o p t i m a l v a l u e s o f the regularization p a r a m e t e r i s e q u i v a l e n t t o m i n i m i z a t i o n o f t h e norm o f t h e s e c o n d t e r m o f the indicated decomposition. B a k u s h i n s k i i [25] j u s t i f i e s the principle of selection of the regularization principle in terms of the residual in the solution of operator equations of the first kind by still other modifications

of the regularization method.

Bakushinskii's generation of the regularization method is based on variational principles and t:he spectral properties of the operator P~ = (A*A + aE)-:A *. In [20], for example, the following construction is introduced for Eq. tive self-adjoint operator A in Hilbert space:

(i) with a posi-

Let ~(~,~) be a bounded real function defined for hE Tr A and all a > 0 and satisfying the conditions sup]~(k,a)I ! ~ : K ~ < o c , ~(0,~----0 for a > 0 and lim~(~,~]=l for I > O.The opera~SpA

a~O

tor function ~(A, a) corresponds to the numerical function

~(~'~=[L,

[LI.=0

is investigated. The family u ~ = ~(A, a).f6 then has regularizing propeities , to wit: For a definite choice of a = a(6) th~ elements u~ tend to the exact solution u of the problem in the original space. Also stated in the same paper is the problem of optimal selection of functions @(I, ~) guaranteeing minimization ofllu~--~l[ subject to the condition K e ~ K. Bakushinskii's approach makes it possible to treat manyregularizing algorithms from a unified point of view, including successive-approxlmation, relaxation, and other schemes. The give n approach is fully developed in the monograph [21]. Khudak [194] considers the uniform convergence of the regularized solutions of (8) obtained by means of the ~'zero-order" regularizer, i.e., when the fanctional ~(u)------[!~il[. A negative answer is obtained in the general case; if, however, the solution u of (7) is representable in the "source" form u-----A*~, v@L2, then uniform approximations can be obtained by the appropriate regularizing algorithm. Morozov [129] obtains an analogous result for the case in which A*:U + B, where U is a Hilbert space~ B is a Banach space, and the "source" condition holds for the solution. In a latter work [195] Khudak succeeds in extending the foregoing results by indicating necessary and sufficient conditions for convergence of the zero-order regularizing algorithms in a particular Banach space. The algorithm regularizing (I) in the form (9) is clearly untenable if the operator A is unbounded. Maslov [122] recommends the following approach: First, the element f5 is

715

"smoothed" by solving the equation

(~AA * + E) g~ =~8, ,~>0, and then the Tikhonov algorithm (9) is automatically applicable. The algorithm is justified for closed linear operators A. The same author shows in [123] that convergence of the regularization process and solvability of the original problem in the case of Hilbert U and F are equivalent. An analog of this proposition for the problem of computing the values of an unbounded operator is proved in [139]. The same problem is investigated in [196] for other algorithms regularizing (I). Chechkin [197] develops a special construction of the regularization method for the solution of multidimensional integral equations with a kernel depending on the difference of the arguments. Standard spectral ~echniques are used to arrive at the solution; the method essentially entails a specialized choice of the functional ~(u). Applications to various theoretical antenna problems are described. Possible applications of Tikhonov regularization to the solution of unstable nonlinear boundary-value problems are investigated by Gorbunov [49]. Various types of convergence of regularized solutions to generalized solutions of the original problem are studied. Regularized computational processes for methods of the Newton-Kantorovich type are investigated by Aleksandrov [i]. The algorithms are used to solve the problem of determining exponentials from approximate data. The possible stability of the solution of (i) with approximate data under the additional assumption of known basis functions for the kernel of A is proved in [29]. Marion and Varga [223] investigate a general regularization scheme in the frequency domain for the solution of integral equations in convolutions. Korolyuk uses the regularization method (after application of the Fourier transform) to obtain uniform approximation to the solution of the Cauchy problem for the Laplace equation [90]. The regularization method is analyzed in Banach spaces by Kalyakin [84]. Likht [115] discusses stability considerations in the computation of functions on the solutions of equations of the first kind. Strakhov [161] constructs the two-parameter regularizer S= n = ,

~, ~(n)Ek where E ~ ~,

is the

k~--~

operator of translation by the argument a at points x = Xk, for the solution of integral equations in convolutions. The quantities c~ n) are determined by a certain condition for approximation of the kernel of the inverse operator. Bakushinski and Strakhov [26] consider a regularizing algorithm based on an iteration technique, for the solution of the equation Au=f, [tAil=l, IIE--A]I----l,when U = F is a reflexive B space. The Tikhonov regularization method has been applied to the numerical solution of a large number of ill-posed problems, of which we cite only a select few. For example, Tikhonov and Glasko [185] reduce the temperature measured at an intermediate point of a semiinfinite rod in a certain time interval to the temperature at the initial endpoint of the rod. This problem is equivalent to the analytic continuation problem for the solutions of equations of parabolic type. In a lengthy treatise [181] Tikhonov gives the results of a numerical analysis of the following problem: spectroscopy; numerical differentiation; the inverse gravimetry problem in the linear approximation; inversion of the Laplace transform in a real domain; determination of the phonon spectrum of a crystal from the specific heat; automatic control; optimal control. Solutions of inverse problems associated with the processing of physical experimental data on the cross sections of photonuc!ear reactions are given in [188]. Zaikin [66] develops a procedure for numerical inversion of the Laplace transform; the numerical determination of a pulse transfer function from sampled data of the input and output process is carried out in [27] with the use of regularization; the analogous problem using the fast Fourier transformation algorithm is considered in [141], in which the method is noted to have a high efficiency. The accentuation (enhancement) problem for a radial distribution function (backward heat-conduction problem) by the regularization method is solved by Zhidkov and others [63].

716

The approach is substantiated in real problems, and excellent results are obtained. The applicability of the method is justified for accentuation of the Parkinson function. Marchuk and Atanbaev [120] establish a basis for the regularization method in application to difference schemes approximating ill-posed problems of evolutionary type. A stability theorem is deduced for the regularized difference scheme, and the convergence of certain iterative processes is verified~ Marchuk and Vasil'ev [121] analyze the conditional convergence (the number of iterations necessary to achieve a prescribed precision depends on the quantity h characterizing the error of! specification of the operator and right-hand side) of universal iterative procedures for regularized operator equations. Chudov [198] develops another method for the regularization of differences schemes approximating the Cauchy problem for the Laplace equation on the basis of smoothing (suppression of high-frequency components) by the convolution of net functions with a Dirichlet kernel. The method is substantiated in numerical ~xamples. The applicability of displays for the solution of certain ill-posed problems is discussed by Nedyalkov [226]. Bellman and others [204] propose dynamic programming methods for the solution of a regularized system of linear algebraic equations. We bring attention here, however, to the (in our opinion) significantly more efficient algorithm proposed by Voevodin [41]. V. V. Kvanov and Kudrinskii [67] propose, and verify the convergence of, a series of numerical methods for the solution of regularized problems of the projection type. Cabayan and Belford[206] numerically implement the Tikhonov regularization method in application to inverse problems for a planar Newtonian potential. Error bounds are estimated for tile approximations. Arsenin has published a survey of numerical methods that strongly involve the regularization of ill-posed problems of mathematical physics [14]. w

Principle of the Residual

The conceptual origins of the residual method are to be found in the works of Phillips [227] and Kantorovich [85]. The latter describes certain principles for the reduction of unstable problems to the solution of mathematical programming problems. A significant feature here is the use of information about the error of the problem input data. Examples are given of the solution of approximation problems and the inverse gravimetry problem. The paper does not give any theoretical foundation for the proposed approach. The residual method is formulated more thoroughly by Phillips [227] in application to the numerical solution of Fredholm integral equations of the first kind (7). The approach is based on the following reasoning. The approximate specification of the right-hand side of (7) determines a certain set U of (formal) solutions, which also includes the exact solution of (7). A principle is formulated for the selection of a "regular" solution ~ according to the optimality condition b

b

(12)

= ! Writing rive at

the discrete the solution

a n a l o g o f (12) and a p p l y i n g t h e method o f L a g r a n g e m u l t i p l i e r s , of a system of linear algebraic equations of the form

(kC+ ArA) ~ = ? ,

we a r (13)

in which aij = ~ i k ( x i ~ ~j), i, j = i, n (~_.jare the quadrature coefficients, and xi, ~i are nodes of a uniform nee in x and ~), the elements of the nonnegative symmetric matrix C are determined by a method of approximation of the differential term in (12), and the parameter I is evaluated from the condition n 2

~

: "~, et =~2, e, = ~ x,k,ju] - - f , , i=I

j=l

where the vector 6 k is the solution of (13). We point out that Phillips himself uses the inverse matrix A-~ in deriving Eqs. (13), thereby restricting severely the domain of applicability of the method. A proper derivation is given by Twomey [235]. The stability of the

717

proposed method is left unsubstantiated in [227], but sample calculations of methodological problems exhibit good agreement of the numerical data with the theoretical results. A conceptually similar, but inefficient method of solution is proposed by Dombrovskaya [58] for the case in which A is a linear operator, U and F are Hilbert spaces, and the inverse A -I exists but is unbounded. Thus, assuming that an exact right-hand side of (i) is given, the author introduces ~-solutions in the guise of the elements U,: u~EUs~-{u:IIA~--fl I ~e, e>0}, for which

in~ IlulL=lJu~ll, u~Ue t h e norm i s minimum. I t i s p r o v e d t h a t u e + u, t h e l a t t e r b e i n g t h e & x a c t s o l u t i o n ( i n U), and t h a t u~ i s a c o n t i n u o u s f u n c t i o n i n t h e weak t o p o l o g y s e n s e o f f and e.

i.e.,

The r e s i d u a l method f i r s t a p p e a r s i n r e a s o n a b l y g e n e r a l f o r m i n p a p e r s by V. K. I v a n o v [ 7 5 ] , Dombrovskaya [ 5 7 ] , and Morozov [ 1 2 8 ] . U s i n g a compact embedding c o n s t r u c t i o n , Ivanov postulates the existence, together with the normed space U, of a Hilbert space H that is mapped full-continuously into U by the linear operator B. It is assumed that the solution has the membership uEQ B in the domain of values of B. Putting C = AB (C is a completely continuous, but necessarily linear operator acting from H into the Hilbert space F) and assuming that T is replaced by f~, Ivanov defines the set ~={z6/-f:llCz--f~ll gates the following extremal problem:

Find an element

II z, II=

inI

~<~}

and investi-

zs6~ such that

II z 11.

(14)

z6Q6 The solvability of this problem is proved, and the element u~ = Bz~ is taken as an approximate solution. It is proved that u~ + n in U, i.e., the residual method yields stable approximations to the solution of problem (i). The method of Lagrange multipliers is used to reduce problem (14) to an extremum-search problem for a certain constraint-free functional. These results are generalized in [57] to the case in which A is a closed linear operator and B is a completely continuous linear operator. An analogous technique is applied in [56] to the numerical differentiation problem. Morozov

[128] considers the numerical solution of integral equations of the type b

Au ~ a (x) u (x) + I k (x, ~) g (~) d~ = f(x), a

in which a(x) is a certain function and, rather than ~(x), the functionf~x):[f(x)--f(x)I-~<~(x), ]l~(x)llL-+O is known. Approximate solutions u(x) are given by the condition

in which

the

ll6]]eJ0(the

~=l.CW~S:l~u--f(x)l<~ix)}.------

infimum set is compact embedding notion

It

is

proved

that

t1~--~I1~,)-~0

as

is not Used'here). Numerical implementation of the

method is discussed. The residual method is further developed in [131] in application to the case of an approximately specified unbounded linear operator A acting from a uniformly convex space into a Hilbert (or normed) space, and in [134] for unbounded nonlinear operators. The 8-stability of the residual method is corroborated for the latter case. The case in which a bounded linear operator A acts from one Hilbert space into another is also treated in [202]. ~asin and Tanana [37] generalize the residual method to the case of a closed linear operator A acting from a normed space U into a reflexive normed space F. The scheme used here is similar to the compact embedding scheme, namely it is assumed that the solution of ( i ) i s u = Bz, where B is a continuous linear operator acting from a normed space Z satisfying the Efimo%~Stechkin conditions, i.e., reflexive and endowed with the property that every weakly convergent sequence together with the sequence of norms is strongly convergent [233]. This property, of course, is possessed by every Hilbert space. Finite-dimensional models of the solutions of regularized problems are considered in [37], and their convergence is proved.

718

Vinokurov [40] investigates the existence of approximate solutions and the convergence of the residual method for nonreflexive Banach spaces. V. K. Ivanov [78] discusses the residual method in topological spaces. With reference to the solution of Fredholm equations of the first kind Goncharskii and Yagol [46] investigate the residual method in the following form: The operator A acts from C[a, b] into La[a, b] and seek a solution in the class of monotone functions in C satisfying the condition ]]A~--f~I]L,~<~. It is assumed that the exact solution is also a member of this class. Then it is proved that the approximate solutions converge uniformly as ~ + O. The method is used to calculate the brightness of eclipsed stars. The following approach is recommended in [48] for the solution of (7). Let A h be a linear integral operator acting from W~ :) [a, b] into L=[a, b] and satisfying the approximmtion condition l[A-ZAnl[a~L,~h. If the exact solution of (7) lies in W! ~), an approximate solution is obtained as the solution of the extremal problem ~E~

2

2

D={uGg~'~:llAu--foHx,,~+hllUltL~}.

in which the infimum set is This approach has the singular feature that quantitative information about the unknown solution is not used in constructing the approximate solutions, even when the operator is approximately specified. The existence of the approximations and their convergence in W(~) to the exact solution is proved. The method has the drawback of excessively stringent approximation conditions, which, however, are easily lifted. Furi and Vignoli [216] investigate the residual method for problem (i) with an unbounded nonlinear operator A acting from a uniformly convex space U into a normed space F. ~6. Stable Computation of the Values of an Unbounded Operator; Summation ~f Series If the operator R is specified directly and is not continuous, problem (6) can be solved by the methods discussed previously, provided that R can be inverted, i.e., that problem (6) can be reduced to problem (I). It is not always practical to do so. A number of papers have been devoted to the direct solution of problem (6). They all have the following construction in common. We consider the operator equation

u=~ with approximately specified right-hand side quired to find elements g6@F such that:

f~:PF(f, f~)

(15) <- ~"

The element f ~ D R. It is re-

i) g~CD~;

u~=Rgs, tz=Rf.

2) limpu(U~,u)--O, 6--+0

This type of problem was first solved by A. N. Tikhonov. u

where

= i=I

Specifically, we let F = Lz,

and fi are the Fourier coefficients of the continI=I

uous function f(x) with respect to the system of orthonormal eigenfunctions Un(X) of the oo

operator

L;~--utl--qu-~ku,

~(a)=~(b)~-O.

It is shown that the series g = ( x ) = ~

h

~n(x)

(%i are t_he eige_nvalues of L) admits stable (under the perturbation ~) and absolute summation to u(x) = f(x)' for xG(a, b) in which th_e function u(x) is absolutely continuous, for a, ~2/~ = 0(i). It also holds that g~ ~ u(x) on every closed subset of (a, b) under the indicated conditions [172]. Morozov [130] considers the case space U = H L obtained on the basis of tive definite self-adjoint operator L U, VGff. In this case the operator R =

of a normed space F in which is embedded a Hilbert the completion of a Hilbert space H, in which a posiis defined in the scalar product sense: (~, V)L~-(Lu, V), L ~/=.

719

The regularization method of Tikhonov is applied in the form ~1 2

o

where ~ is the solution of (16) for the value of the parameter a: 62/a § O, 6 § O. H, then (16) is reduced to the Euler equation

If F =

+ g=?. The a u t h o r a n a l y z e s t h e c a s e i n which L = dn/ dx n, i . e . ,

the s t a b l e d i f f e r e n t i a t i o n

p ro b le m .

The f o r e g o i n g p r o c e d u r e i s e x t e n d e d i n a l a t e r work [135] i n a p p l i c a t i o n to t h e p ro b le m of computing t h e v a l u e s of any c l o s e d l i n e a r o p e r a t o r . The s e l e c t i o n of t h e p a r a m e t e r by t h e r e s i d u a l p r i n c i p l e i s s u b s t a n t i a t e d i n t h e same p a p e r . A somewhat d i f f e r e n t a p p r o a c h ba sed on s p e c t r a l d e c o m p o s i t i o n i s d e v e l o p e d by A n to k h in [10] f o r t h e s o l u t i o n o f p r o b l e m ( 6 ) . Lower bounds a r e o b t a i n e d , and a l a r g e c o l l e c t i o n o f examples i s p r e s e n t e d . A method of s o l v i n g problem (6) on t h e b a s i s of V. K. I v a n o v ' s q u a s i s o l u t i o n s i s d i s c u ss ed by Vasin [ 36] . The method e s s e n t i a l l y c o m p r i s e s t h e f o l l o w i n g . We c o n s i d e r t h e v a r i a t i o n a l problem u-min

IIg-?lb:.llRglt~
x given.

It is proved, given the condition x->!IRfIIu, ~-+0, that ~=Rff~.-+~ (it is assumed that U and F are E spaces and T is a closed linear operator). A case is cited in which it is possible to lift the condition k->][Uiluwhich is troublesome in practice. Strakhov [162] introduces the concept of the regularizing family R 1 for the computation problem, i.e., a family satisfying the conditions i) IiR~ll< oo, 2) !lRf-R~fll-+O,

),-~k 0 (limiting value) for every f 6 D n .

By analogy with the work of V. K. Ivanov [76], Strakhov introduces the concept of uniform regularization on a set A4~OR. Certain notions are borrowed from filtering theory. Thus, the element f is replaced by the element g ~ = S ~ f ~ D ~ , for whichiIR~--R~[-+0 for X = X(~) § 0 as ~ § 0. Examples of filtering and regularizing operators are formulated. Paremeter-selection methods based on various criteria are discussed. Liskovets [112] investigates the applicability of the residual method to the solution of problem (6) in the case of a closed (possibly nonlinear) operator R when U is a topological Hausdorff space and F is a metric space. Tanana [167] shows that the residual method is stable in the separable strictly convex reflexive space if and only if that space has the E-property, i.e., is an Efimov--Strechkin space. The same author treats analogous problems in [168]. In both papers [167, 168] a finite-dimensional numerical method is formulated, and its convergence in the strong sense to the solution obtained by the residual method is proved. V. K. Ivanov [82] investigates problem (6) with a multiple-valued operator R in application to the solution of (i). It is assumed that U is of type E and F is a reflexive space. The residual method is used to obtain approximate solutions. Thus, we let S~={f:[If--f61[~
720

also given for the approximate solution of the stated problem. Morozov [138] studies the relationship of the problem of computing the values of unbounded operators to approximation theory. Stable summation methods of the Tikhonov type of order S with respect to the fundamental system of functions of the Laplace operator in an arbitrary N-dimensional space ~ are investigated by Furletov [192]. It is proved that for s > ~N the regularization method permits stable reconstruction of the values of a summable function at any Lebesgue interior point from the approximately known Fourier coefficients. The problem of reconstructing the values of the derivatives and the applicability of Tikhonov summation methods for s < ~N are discussed. Khudak [196] considers the relationship of generalized methods for the summation of Fourier tion by the Tikhonov method in the space C[a, b] Analogous investigations are carried out by Ramm w

various regularizing algorithms to stable series. The possibility of regular summais also investigated by V. K. Ivanov [77]. in [154].

Interrelationship of Variational Methods for the Solution of Unstable Problems

The regularization, quasi-solution, and residual methods yield approximate solutions on the basis of certain variational principles. Several papers are concerned with ascertaining the interrelationship of these methods. The beginning of this trend in the theory of methods for the solution of ill-posed problems is found in the work of Dombrovskaya and V. K. Ivanov [62]. They analyze Eq. (i) with a continuous linear operator A, which exists but is not bounded (U and F are Banach spaces). Let ~(u) be a convex functional such that the sets ~c~---{~:~)~
Methods for the Solution of Unstable Extremal Problems

A minimization problem for a real functional ~(u), specified on a nonempty set where U is a metric space, is said to be well posed if:

M~U,

I) a unique element u-~M exists such that

rain---- ini ? (u)=~ ~); u@M

(17)

2) every sequence minimizing (17) is compact in U. Tikhonov [175] introduces the notion of a well-posed variational problem. The regularization theory is elaborated in [175] in application to unstable optimal control problems. 721

It is required to solve problem (17), in which ~ (u) E F(x(u)), F(.) is a given continuous functional, x(u) is the solution of the system of equations

dx dt -- f (t,x,u), O< t < T , x(O)=C, and u(t) is a control function in a certain classM~C[a,b]. tional is defined:

o ~ 1 - ~ ( ~ +=e ~),

The following regularizing func-

(t8)

T

(koU"'+ k , u )dt, ki>O.

where a > 0 is a parameter and e ~ ) = [ 0

It is proved that the sequences minimizing (18) are uniformly convergent and stable under perturbations of 9 for a special choice of the regularization parameter a. These results are expounded in greater detail in [179]. The applicability of the regularization method to unstable optimal planning problems is substantiated by Tikhonov in [177]; the results are presented in detail in [182]. The stabilizing functional is chosen in the form ~(u) E ~o(u -- u*), where u* is a given plan, and is interpreted as the cost of reorganization operations in transition from plan u* to the required plan u. This approach is implemented in [187]; applied to linear programming problems for which conventional methods (of the simplex type) do not yield good results, the given approach provides results evincing the possibility of stable solution of such problems for real perturbations of the input data. The regularization method is applied numerically in [186] for the determination of uniform approximations to the control function (fuel consumption per second) in the maximumaltitude liftoff problem for a rocket probe. Polyak and Levitin [149] investigate the strong convergence of regularized solutions for the case in which ~(u) is a lower semicontinuous convex functional and ~(u) is uniformly convex, in reflexive Banach spaces (stability under perturbations of ~ is not investigated). Morozov [133] proves the stability of the regularization method in the original space if the latter is Hilbert, both for convex ~(u) and for full-continuous functionals specified with error. The same author analyzes the behavior of certain functionals on regularization solutions as functions of the parameter a, namely: F(a) ~ G ~ [us], ~(a) ~Hu~--u*N 2 (u* is a given element), etc. In a later paper [208] Cruceanu pursues analogous considerations under more general conditions. He also investigates the applicability of Galerkin's method for the solution of a regularized problem [209]. Budak and Berkovich [30] investigate a general Tikhonov regularization scheme for a sequence of approximating extremal problems in both Hilbert and topological spaces. They also analyze the stability of extremal problems. A study of the convergence of difference approximations in optimal control problems is described in [31]. In [32] a construction of the quasi-solution method of V. K. Ivanov on extended sets is used to formulate strongly convergent minimizing sequences. Sholokhovich [199] extends the residual method in application to extremal problems. Thus, taking as known the value of ~min :I~min--~minl~<~, he defines the sets f)~ {u~A4: ~ (u)~< p min +~}. An approximate solution is found as an element with minimum norm in D. Justification of the method is given for the case in which U is a uniformly convex B space. In another paper [200] Sholokhovich shows that the modified residual converges stably in E spaces, and only in those spaces. In [201] the stability of the quasi-solution and "residual" methods in certain B space is substantiated for extremal problems, error bounds are obtained for the approximate solutions, and an example is given. Dmitriev and Poleshchuk [55] propose a technique analogous to the method of quasi-solutions and construct strongly convergent minimizing sequences for a continuous functional under the condition that the solution of problem (17) is a function of bounded variation. Bensonssan and Kenneth [205] apply the regularization method to noncoercive functionals ~(u). The authors establish a close relationship between that method and the method of penalties in the most important situations, along with applicability of the given method to the solution of optimal control problems. 722

A great many facts pertinent to the stable solution of optimal control problems in partial differential equations by the regularization, penalty, and other methods may be found in Lions' book [108]. Also in the latter, other forms of regularizing algorithms are proposed for the Solution of unstable extremal problems. Tikhonov and Vasil'ev have published the most detailed survey of methods for the solution of unstable extremal problems [182]. w

Theory of the Error of Approximate Solutions and Optimization of Regularization Methods

for Ill-Posed Problems The value of the we!l-posedness function ~(T) plays a vital role in the method of M. M. Lavrent'ev. The latter obtains bounds for that function for a number of problems in his book [i01]. Denchev [51] develops a general method for the computation of the well-posedness function for unbounded operators. Maiorov [116] computes the same function for full-continuous self-adjoint operators having the "source" property. V. K. Ivanov [76] focuses attention on the fundamental difference between two problems: I) point convergence of a regularizing algorithm (Problem A); 2) error bounds for approximate solutions (Problem B). Whereas in Problem A the convergence s e t , o f t e n coincides with the entire space, the solvability of Problem B requires that g2'be a compact set. Various error bounds for regularized solutions (including those depending on the error of specification of the operator A) are given by Morozov [122, 127], Bakushinskii [21, 22], V. V. Ivanov [67, 68], and others. The first development of a general error theory for the solution of ill-posed problems is found in the work of V. K. Ivanov and Korolyuk [79] for a full-continuous linear operator A acting from one Banach space into another. It is assumed that the well-posedness set M is compact. Upper and lower error bounds are established for the method of quasi solutions in terms of the goodness function: ~l (x) = suP l[ gl - - U2 ]l, ~I,U2QM' I I A u l - - A u 2 [ l ~ < x . Essentially, the optimality of the quasi-solution method with respect to order of error is verified. The following estimator is defined in the work: (x, R) = sup I[ u I[, u 6 M a = {u : u = B y , II ~ 11< R}, IIAu it < ~, w h e r e B:U + U i s

a full-continuous

linear

operator.

I t i s p r o v e d t h a t ~ ( z , R) ~ ~ l ( z ) = ~ l ( x , R) ~ ( T , 2R) and t h e e r r o r o f t h e q u a s i solution msthod is not greater t h a n ~ ( 2 6 , 2R) i n t h e g e n e r a l c a s e o r ~ ( ~ , 2R) when F i s a Hilbert space. A general technique is given for the computation of the estimator ~(~, R) for Hilbert spaces and for C[~, b] in the case of integral operators. Estimation techniques involving m(T, R) are intimately related to the theory of scales of Banach spaces. The given method is further elaborated by V. K. Ivanov in [79] and, in application to unbounded operators and the problem of computing the values of an operator on the solution of an operator equation, by Morozov [137]. Error bounds are deduced by Korkina in [88] for the problem of computing the values of an unbounded operator by the method of V. K. Ivanov. S t r a ~ o v [163] considers two problems with reference to (6): i) the computation of the function ~(~,M,T)=

inf

sup

]]Lf--Rf[],

~--~--f,

which characterizes the best precision of reconstruction of the values of R~ in a c!assMCDr,

2) the determination of the operator Rop t for which that precision is attained. 'Fne latter problem, we note, is intimately related to the problem of Strechkin [159] on the best approximation of linear operators. S t r a k h o v solves both problems, proving that the role of the algorithmRop t can be taken by a Tikhonov-regularizing algorithm. Morozov [139] poses the problem of determining rather than the optimal, the error-order optimal algorithm for the reconstruction of the values of R~ for a minimum of a priori data (in particular, without prior knowledge of the quantitative characteristics of the set M). It is proved in [140] that this property is inherent in the residual method, and error bounds 723

are obtained for the method. This approach is further developed by Strakhov [164], who also indicates alternative methods for the construction of order-optimal algorithms. Vinokurov [39] determines the error at a point for that in the case of a uniformly convex Banach space U no than the residual method, with the possible exception of category in U. An analogous result also obtains if U is

any regularizing algorithm. He shows algorithm can give better precision the set of solutions of the first a separable reflexive B space.

Aliev [2, 3] proposes a series of regularizing algorithms for the determination of approximations to the normal solution of equations of the type

where % is the characteristic value of the operator A. He also finds error bounds for the approximations as functions of the errors of the right-hand side and operator A. In [4] Aliev applies Tikhonov regularization to the Neumann difference problem and gives error bounds in the difference analogs of norms. Voevodin [41] describes a convenient algorithm for finding the regularized solutions of an algebraized problem; his algorithm significantly enhances the effectiveness of the regularization method; error bounds are obtained as a function of roundoff error. Morozov [132] investigates the influence of errors in the specification of the operator and right-hand side on the determination of pseudosolutions and estimates the error bounds. Error bounds for the regularized solutions obtained for extremal problems by the quasisolution and residual methods are given by Sholokhovich [201]. A number of optimal regularization problems are stated in general form and solved by Bakushinskii [21]; the same is done by Antokhin [ii] for integral equations in convolutions. Strakhov [160] proposes a method for the solution of integral equations in convolutions on the basis of best approximation of the Fourier transform of the kernel in weighted norm by a trigonometric series. w

Statistical Methods for the Solution of Ill-Posed Problems

Typically in the solution of ill-posed problems one specifies th e error of the righthand side and operator, along with (in some problems) the a priori well-posedness c l a s s ~ to which the unknown solution belongs. The counterparts of these data when the right-hand side and solution of (1) are stochastic are the known characteristics of the error distribution function for the right-hand side and the a priori distribution function for the unknown solution. The use of statistical problem-solving procedures often results "automatically" in regularized schemes having a sizable margin of stability by contrast with direct methods of solution. With regard to the Cauchy problem for the Laplace equation Sudakov and Khalfin [165] investigate a class of stochastic input data guaranteeing a finite expectation value for the solution, and they determine the conditions of those data for the solution to tend to zero. The class of data that they obtain, however, is too restricted for practical application of this method. Twomey [236] posed a technique using a priori constraints on the possible solutions of a system of linear algebraic equations and resulting in stabilization of the solution. The approach here is equivalent to filtering of the solution components associated with highfrequency stochastic-error terms on the right-hand side of the system. The statistical approach to the solution of ill-posed problems is treated in the most complete and ~igorous form in the work of Lavrent'ev and Vasil'ev [102]. Their approach is equivalent to the regularization and quasi-solution methods. They pose the problem of finding an optimal regularizer and then, from this point of view, analyze the solution of the convolution-type equation

AU~ ~k(x--~)u(~)d~=f(x),

--oo<X
The optimal regularizer is also sought in the form of a convolution-type operator. chy problem for the Laplace equation is investigated as an example. 724

(19) The Cau-

Using the same approach, Arsenin and V. V. Ivanov [15, 16] analyze equations of the type (19) when u(~) and f(x) are samples of stationary stochastic processes and the noise function v(x) on the right-hand side does not correlate with the solution. It is proved that one-parameter Tikhonov regularization yields a Wiener-optimal solution (in the optimal-filtering sense). The mean-square deviation of the regularized solution from the exact solution is estimated; a method of determining the high-frequency characteristics of the solution and the noise for the regularized solutions is given for two classes of kernels in (19). Another class of kernels for (19) is considered in [18], and an asymptotic estimate is obtained for the influence of white noise v(x) on the regularized solutions as ~ + 0 (a is the Tikhonov regularization parameter of order p). A method of computing an almost-optimal value of the parameter a is presented. Optimal methods of the filtration type for the summation of Fourier series with approximate coefficients are discussed by Arsenin [13]. Using the Lavrent'ev-Vasil'ev approach [102], V. A. Morozov describes a technique for computing the optimal regularizer for full-continuous linear operators A when the righthand side f of Eq. (i) is specified with an error that is a stochastic process in Hilbert space. The error of optimal regularization is determined for various classes of solutions and errors. Techniques are described for the realization of optimal solutions yielding an error of the same order as the optimum. This property is inherent, in particular, in the oneparameter Tikhonov regularization method. An interesting approach to the optimal computation of linear functionals l(u) on the solutions of (I) when A is a full-continuous linear operator and U = F are Hilbert spaces, is proposed by A. P. Petrov [146]. The right-hand side of (i) is regarded as a distorted stochastic process ~ with known correlation characteristics. Several methods of solution are described, including in particular the minimax and Bayes approaches, which yield regularized operator equations. The Bayes approach requires that the correlation operator for the unknown solution be specified a priori, and the author proposes that the solution then be determined on the basis of its smoothness conditions. An example is given from the processing of physical experimental data, where the kernel in (19) is Gaussian. The choice of "cutoff" parameter for the high-frequency components in the solution so as to ensure optimal filtering is analyzed. Turchin [189, 190] makes significant use of the specific attributes of (19). Going over to the discrete version of the problem, he investigates the approximating system of equations

~km_num=fn, n-~---N, - - N + 1..... N.

(20)

m

Applying the discrete of (20):

Fourier

transformation

to

( 2 0 ) , we o b t a i n

~q~q-~-?q, q= - - N , N,

the following

system in place (21)

where the circumflex indicates that fi_ and fq are the Fourier transforms of u m and fn, and %q is determined by the difference kernel km_ n. The formal computation of Qq = fq/%q leads to an exact solution of (20), which possesses strong instability. Assuming that the-fq in (21) are normally distributed with variance s 2 and that fiq has an a priori known normal distribution, where the variance of Qq_ is equal to y~, and taking the conditional Bayes approach, 89 the author obtains regularized equations for Qq:

w h i c h c o r r e s p o n d t o s u p p r e s s i o n o f t h e c o m p o n e n t s f a / X q o f t h e f o r m a l s o l u t i o n o f (21) w i t h 2 2 ~ multipliers ~a/(~ o2 + s 2 /Ya) depending on the ratio of the noise s 2 to the variance Ya2 of the useful signalf I~ the solution of (19) is a smooth function, it is recommended to t~ke y$ = a/[m(q)] 2 m , where m is an integer, m(q) is the discrete frequency, and a is the regularization parameter, which the author suggests be chosen according to the condition that the residual s2(a) given by the regularized solution be made equal to the quantity s 2 (residual criterion). Turchin's paper contains many vague assumptions of a physical nature. Strand and Westwater [234], in their solution of a system of algebraic equations corresponding to an integral equation of the first kind, set down a more rigorous groundwork for the Bayes approach. They prove that the solution obtained by them (which can also be determined from the regularized system of equations) has a number of optimal properties. The application of this approach to (21) yields Turchin's refined solution.

725

Zhukovskii [64] invokes filtering methods for Markov processes to find a stable solution of systems of linear equations when the right-hand side is measured in ordered succession. The distinguishing feature of this approach is the sequential use of cumulative incoming information to improve the estimate. The limiting properties of the estimates as the number of measurements n § ~ are investigated. For the solution of a similar problem Zhukovskii and Morozov [65] adopt the sequential Bayes approach. The solution obtained for n = 1 coincides with the solution given by Turchin in [190]. Shaw [232], in effect, elaborates the foregoing approach of [64, 65], postulating that the problem-solving procedure must be iterative and executed in stages, including: the formation of an initial estimate of the solution; the refinement of successive estimates on the basis of knowledge about those procedings; and the selection of a test by which the rate of convergence of the estimations can be assessed. The author indicates the large category of problems to which the given method is applicable. Franklin's paper [215] stands favorably apart from [234] in presenting a more general method applicable to a broad class of linear operator equations in a Hilbert space, where the right-hand side and solution of those equations are stochastic processes with values in the given space. The following are offered as examples: the backward heat-conduction problem; integral equations of the first kind; singular systems of linear algebraic equations; the harmonic function continuation problem. Numerical examples are given, illustrating the substantial improvement of the results over direct methods. Bakushinskii [24] extends the regularization method in Tikhonov form to very general equations (i) in a Hilbert space with a stochastic linear operator A. Lebedev [107] synthesizes an efficient algorithm for the solution of Tikhonov wellposed problems through the use of the entropy and constructive attributes of compact sets. The approach is based on the transformation of (i) to an equation whose operator has the ~-property. A. P. Petrov and Khovanskii [147] take an information-theoretical approach to the determination of a lower error bound for the solution of operator equations of the first kind. The regularization method of Tikhonov is investigated by V. V. Petrov and Uskov [148] from the standpoint of the extraction of useful information in control theory. Goncharskii and others [47] present a simpler method than in [16] for estimation of the deviations of the regularized equations from exact equations of the convolution type (19); the method is nonetheless applicable for a class of more complex kernels. The method developed in [102] is used by Vainshtein [33] for the stable solution of ill-conditioned systems of equations whose right-hand side has a stochastic perturbation with known characteristics. The distinctive aspect of this work is an alternative approach to the realization of an optimal solution. Error bounds are not given; the paper has primarily a qualitative bearing. w

The Method of Quasi Reversibility

Underlying the method of quasi reversibility is the concept, universal in numerical analysis, of finding more practical substitutes for the operators entering into a problem. The substance of this substitution may be summarized as follows: -- The auxiliary terms must be sufficiently "small" and, in the limit, yield the original problem. -

-

The known boundary values must remain absolutely undistorted.

It is readily seen that the quasi-reversibility method comes very close in this sense to the method of Lavrent'ev as well as the regularization method. The main area of application of the quasi-reversibility method is in the solution of noninvertible evolutionary problems in backward time; control problems for a system described by partial differential equations (where control is effected through the influence of the boundary conditions); and boundary-value problems with redundant (overspecified) data on part of the boundary and with deficient (underspecified) data on the rest (the Cauchy problem for the Laplace equation, the continuation of harmonic and analytic function into a large domain than that in which they are defined, and a great many other problems grouped under the general heading of "inverse" or "backward"). The fundamental results pertaining to the application of the method are summarized by Latt~s and Lions in the book [106].

726

Referring to [106], we give a typical application of the method. Let S be a domain in Rn, let t be the time variable, and let u(x, t) be the solution of the equation

subject

to the initial

condition

Ou hu-~-0, xE ,Q, t > 0 , (22) 0t u(x,O)=} (x), x 6 2 , a n d b o u n d a r y c o n d i t i o n u ( x , t ) = 0 , x~O~, t>O.

Also, let u(x, T; ~) ~ u(x, t), T > 0. krnen the function X(X) and quantity e > 0 are given, it is required to find at least one function ~(x) such that

I (}) = ~ Itt (x, T; }) -- Z (x)[ 2 dx ~<~2. The notion behind the solution of this unstable problem is to find an operator Pa, close to the operator (~/3t) -- A, such that the inverse problem is rendered solvable. The operator P must be evolutionary, and for any specified function X(X) the solution of the problem

P~u~ = O, xE~2, tE (0, T), trY(x, T ) = Z ( X ) , u~,=O on dO., O < t < T , must exist and be unique. Putting ua(x, 0) = ~(x), one must find an a such that ua(x, T, ~) gives the solution of the problem. The operator Pa is interpreted as Pe = (3/3t) -- A -- ah =, with the following additional conditions imposed on ua: Aua = 0, xE02, 0 < t < T. If Ua(x, t) denotes the solution of problem (22) for u(x, 0) - Ua(X , 0), then Ua(x, T) § X(X) as e § 0 in

L=(~). The method works equally well in abstract spaces for evolutionary equations of both parabolic and nonparabolic type. Its application to other problems depends essentially on their specific attributes (and in this we find a significant drawback of the method). A great many exemplary applications of the quasi-reversibility method are given in [106], ,along with calculations of various methodological problems. Nonlinear as well as linear problems are considered. Also given are examples of direct application of the Tikhonov regularization method to evolutionary-type problems and the Cauchy problem for a general equation of elliptic type. The numerical stability of difference schemes is analyzed for problems obtained through application of the quasi-reversibility method. V. K. Ivanov [80] improves the result obtained in [106], proving that the solutions U~(x, t) regularized by the quasi-reversibility principle converge in ~ = R n, not only in the mean, but also uniformly on every compactum R n. The function X(X) is assumed to be continuous and to grow at infinity no faster than '

.!

O
/a,,4/a

3 -I~/'~(for

, an~---~

C

exp b

n

I x~ 14]3

,

b>0, and ~ satisfies the relation

3#

n = i, neither the order of growth 4/3 nor the constanta, ~- 6 g

can be increased). The machinery of the Fourier and generalized functions is used in the proof of the result. Voyarintsev and Vasil'ev [28] investigate the stability problem for an implicit difference scheme formulated for the equation

g-fOu~ --a~-~O'u--~g-~,O'u x~[0, I], 0 < t < T ,

~>0,

tt(x,O)=v(x),

0= tt(O,t)=t~(1,[)=~-~uO3, t ) = ~0' itt(I,t)=O,

which corresponds to the regularized (by the method of quasi reversibility) heat-conduction equation. Weak stability is claimed for the scheme. It is proved that, given certain constraints on the original function ~(x) (absence of high-frequency components), the norm of the step operator in L2 is less than unity, i.e., the regularized scheme is stable. A counting technique based on this theorem is proposed. Tamme [166] investigates the stability of the difference scheme

yt--a1~Ayt + o=~A*Ayt--A~+ ~ A * A ~ =

~,

727

where A is a linear operator in Hilbert space, Yt = (yi+l _ yi)/T, T is the time step of the net, o and ~I are constants, and e > 0 is the regularization parameter occurring in the application of an abstract quasi-reversibility scheme for ~ > 1/2, a > 0, o~T ~ 0.94~. A scheme with a decomposable operator is proposed for the solution of the multidimensional problem, and the stability of the scheme is proved. The results are used to solve the backward-time heat-conduction problem. w

Conclusion

The theory and methods of ill-posed (unstable) problems began to be developed at a highly accelerated pace in the late Nineteen-Fifties. Their development is closely identified with the names of celebrated Soviet mathematicians, such as A. N. Tikhonov, M. M. Lavrent'ev, and V. K. Ivanov, as well as the mathematical school founded by them, who by and large set forth the guidelines and methods of the theory of ill-posed problems, which has emerged in more recent times as one of the most productive areas of advanced-state computational mathematics. In our opinion, the development of the theory and methods of solution of unstable problems has contributed enormously to the successful penetration of computers in mathematical research and the national economy and in turn, by way of natural feedback, to mastery over the flood of extremely diversified problems demanding solution in the shortest possible time to meet the requirements of the national-economic program. It has been mandatory to develop approximation methods that could be readily applied to the solution of a tremendously broad class of problems whose mathematical formulation is not shackled by the rigid framework of well-posedness. Only under this stipulation has it been feasible to cope with certain problems (mainly unstable) propounded in theoretical physics, spectroscopy, electron microscopy, automatic control, heat physics, gravimetry, electrodynamics, engineering, the theory of experimental physical data processing, approximation theory, and other areas of science and technology. The theory and methods of solution of ill-posed problems, having been created and developed mainly by Soviet scientists to meet timely needs, have unquestionably played a decisive part in the solution of this vital national-economic problem by virtue of their u n i versal character. LITERATURE CITED i. L. Aleksandrov, "Regularized Newton--Kantorovich computational processes," Zh. Vychisl. Matem. I Matem. Fiz., ii, No. I, 36-43 (1971). 2. B. Aliev, "Regularizing algorithms for finding a stable normal solution of an equation of the second kind on a spectrum," Zh. Vychisl. Matem. i Matem. Fiz., I0, No. 3, 569-576 (1970). 3. B. Aliev, "Evaluation of the regularization method for integral equations of the second kind on a spectrum," Zh. Vychisl. Matem. i Matem. Fiz., ii, No. 2, 505-510 (1971). 4. B. Aliev, "Two difference approaches to the solution of the Neumann problem in a rectangular domain," Zh. Vychisl. Matem. i Matem. Fiz., 12, No. i, 230-236 (1972). 5. B. A. Andreev, "Calculations of a spatial distribution," Izv. Akad. Nauk SSSR, Ser. Geogr. i Geofiz., No. i (1947). 6. Yu. E. Anikonov, "Operator equations of the first kind," Dokl. Akad. Nauk SSSR, 207, No. 2, 257-258 (1972). 7. Yu. T. Antokhin, "Some ill-posed problems in potential theory," Differents. Uravnen., i, No. 4, 525-532 (1966). 8. Yu. T. Antokhin, "Some ill-posed problems in the theory of partial differential equations," Differents. Uravnen., 2, No. 2, 241-250 (1966). 9. Yu. T. Antokhin, "Some problems in the analytic theory of equations of the first kind," Differents. Uravnen., ~, No. 2, 226-240 (1966). i0. Yu. T. Antokhin, "Ill-posed problems in a Hilbert space and stable methods for their solution," Differents. Uravnen., ~, No. 7, 1135-1156 (1967). ll. Yu. T. Antokhin, "Ill-posed problems for equations of the convolution type," Differents. Uravnen., ~, No. 9, 1691-1704 (1968). 12. V. Ya. Arsenin, "Discontinuous solutions of equations of the first kind," Zh. Vychisl. Matem. i Matem. Fiz., ~, No. 5, 922-926 (1965). 13. V. Ya. Arsenin, "Optimal summation of Fourier series with approximate coefficients," Dokl. Akad. Nauk SSSR, 182, No. 2, 257-260 (1968). 14. V. Ya. Arsenin, "Algorithms for the solution of ill-posed problems," in: Digital Computer Software [in Russian], Kiev (1972), pp. 79-106. 728

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