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.

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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][~ 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:llUllu0}, 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

~ 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 ~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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