Ill-Posed Problems: Theory and Applications

Mathematics and Its Applications
Volume 301
by
A. Bakushinsky Institute for System Studies, Russian Academy of Sciences, Moscow, Russia
and
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
III-Posed Problems: Theory and Applications
by
A. Bakushinsky Institute jor System Studies, Russian Academy oj Sciences, Moscow, Russia
and
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C .I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-4447-9 ISBN 978-94-011-1026-6 (eBook)
This monograph is a new and original work based 01)
two books by the same authors previously published in Russian: Iterative Melhods for Solving /ll-Posed Problems, Moscow, Nauka © 1989, and /ll-Posed Problems, Numerical Melhods and IIS Applicalions, Moscow, Moscow State University Press © 1989.
Printed on acid-free paper
All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint ofthe hardcover 1st edition 1994
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DOI 10.1007/978-94-011-1026-6
1 General problems of regularizability 4 1.1 Definition of regularizing algorithm (RA) 4 1.2 General theorems on regularizability and principles of con-
structing the regularizing algorithms . . . . . . . . . . . .. 7 1.3 Estimates of approximation error in solving the ill-posed prob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Comparison of RA. The concept of optimal algorithm 19
2 Regularizing algorithms on compacta 23 2.1 The normal solvability of operator equations. 24 2.2 Theorems on stability of the inverse mappings. 26 2.3 Quasisolutions of the ill-posed problems . . . . 28 2.4 Properties of 6-quasisolutions on the sets with special structure 36 2.5 Numerical algorithms for approximate solving the ill-posed
problem on the sets with special structure . . . . . . . . . .. 40
3 Tikhonov's scheme for constructing regularizing algorithms 43 3.1 RA in Tikhonov's scheme with a priori choice of the regular
ization parameter . . . . . . . . . . . . . . . . . . . . . . . .. 43 3.2 A choice of regularization parameter with the use of the gen-
eralized discrepancy 47 3.3 Application of Tikhonov's scheme to Fredholm integral equa-
tions of the first kind . . . . . . . . . . . . . . . . . . . . . .. 57 3.4 Tikonov's scheme for nonlinear operator equations . . . . .. 61 3.5 Numerical implementation of Tikhonov's scheme for solving
operator equation . . . . . . . . . . . . . . . . . . . . . . . .. 68
v
vi
4 General technique for constructing linear RA for linear problems in Hilbert space 73 4.1 General scheme for constructing RA for linear problems with
completely continuous operator . . . . . . . . . . . . . . . .. 74 4.2 General case of constructing the approximating families and
RA 77 4.3 Error estimates for solutions of the ill-posed problems. The
optimal algorithms . . . . . . . . . . . . . . . . . . . . . . .. 90 4.4 Regularization in case of perturbed operator. . . . . . . . . . 100 4.5 Construction of linear approximating families and RA in Ba-
nach space 114 4.6 Stochastic errors. Approximation and regularization of the
solution of linear problems in case of stochastic errors .. . . 122
5 Iterative algorithms for solving non-linear ill-posed problems with monotonic operators. Principle of iterative regularization 127 5.1 Variational inequalities as a way of formulating non-linear
problems 128 5.2 Equivalent transforms of variational inequalities. . . . . . . . 131 5.3 Browder-Tikhonov approximation for the solutions of varia-
tional inequalities. . . . . . . . . . . . . . . . . . . . . . . 136 5.4 Principle of iterative regularization . . . . . . . . . . . . . . . 141 5.5 Iterative regularization based on the zero-order techniques . . 142 5.6 Iterative regularization based on the first-order technique (re-
gularized Newton technique) 150 5.7 RA for solving variational inequalities 155 5.8 Estimates of convergence rate of the iterative regularizing
algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Applications of the principle of iterative regularization 164 6.1 Algorithms for minimizing convex functionals. Solving the
non-linear equations with monotonic operators .. . . . . . . 164 6.2 Algorithms for minimizing quadratic functionals. Non-linear
procedures for solving linear problems .. . . . . . . . . . . . 168 6.3 Iterative algorithms for solving general problems of mathe
matical programming. . . . . . . . . . . . . . . . . . . . . . . 172 6.4 Algorithms to find the saddle points and equilibrium
points in games . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7 Iterative methods for solving non-linear ill-posed operator equations with non-monotonic operators 185 7.1 Iteratively regularized Gauss - Newton technique for operator
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.2 The other ways of constructing iterative algorithms for general
ill-posed operator equations . . . . . . . . . . . . . . . . . . . 195
8 Application of regularizing algorithms to solving practical problems 199 8.1 Inverse problems of image processing . . . . . 200 8.2 Reconstructive computerized tomography . . 208 8.3 Computerized tomography of layered objects 213 8.4 Tomographic examination of objects with focused radiation 218 8.5 Seismic tomography in engineering geophysics 222 8.6 Inverse problems of acoustic sounding in wave approximation 227 8.7 Inverse problems of gravimetry . 236 8.8 Problems of linear programming 238
Bibliography 242
Index 254
Preface
Recent years have been characterized by the increasing amount of publications in the field of so-called ill-posed problems. This is easily understandable because we observe the rapid progress of a relatively young branch of mathematics, of which the first results date back to about 30 years ago.
By now, impressive results have been achieved both in the theory of solving ill-posed problems and in the applications of algorithms using modem computers. To mention just one field, one can name the computer tomography which could not possibly have been developed without modem tools for solving ill-posed problems.
When writing this book, the authors tried to define the place and role of ill posed problems in modem mathematics. In a few words, we define the theory of ill-posed problems as the theory of approximating functions with approximately given arguments in functional spaces. The difference between well-posed and ill posed problems is concerned with the fact that the latter are associated with discontinuous functions. This approach is followed by the authors throughout the whole book. We
hope that the theoretical results will be of interest to researchers working in approximation theory and functional analysis.
As for particular algorithms for solving ill-posed problems, the authors paid general attention to the principles of constructing such algorithms as the methods for approximating discontinuous functions with approximately specified arguments. In this way it proved possible to define the limits of applicability of regularization techniques. The possibility of constructing iterative procedures for approximating
solutions of the ill-posed problems is thoroughly investigated for a wide range of linear and non-linear problems.
The authors have acquired extensive experience in applying the methods of solving ill-posed problems. This allowed them to demonstrate the efficiency of algorithms using model and applied practical problems of mathematical programming, linear algebra, functional optimization, solving the operator equations.
Some new approaches concerned with computer tomography in the frameworks of both geometrical optics and wave approximation are presented in a monograph for the first time ever. We hope that these sections of the book will attract the interest of researchers specializing in computer tomography of seismic geophysics or those applying tomographical approaches in industry. The presentation of the material in this book was strongly affected by many
years of scientific contacts with colleagues from the USA, Japan, France and Italy. It was also influenced by numerous discussions with Academician A. Tikhonov and scientists of his school, with which the authors associate themselves.
ix
x
This book is based on two monographs of the same authors which have been published in Russian earlier (Bakushinsky et aI., 1989a, 1989b). Compared to the Russian edition, the text of this book is substantially changed and extended to include some of the most recent results. Chapters 7 and 8 have been specially prepared for the present edition. The list of references was substantially extended. The authors are grateful to Dr. LV. Kochikov for translating this book into
English, and to E.O Drevyatnikova for wordprocessing the book.
A. Bakushinsky A. Goncharsky
Introduction
The impressive and steadily increasing opportunities provided by modern computers stimulate a continuous growth of the fields of human activities where mathematical models are applied.
In some of these models the algorithms for approximate solution of the correspondent mathematical problems can be formulated within the framework of traditional computational mathematics. In this case it is often possible to prove the formal convergence of algorithms and to evaluate the approximations errors. The technical problems related to numerical implementation of such algorithms are associated with rounding errors, data representation, etc. and do not usually lead to significant difficulties.
However. it is often the case when data available to a researcher can only be interpreted with a formal model which does not allow the application of traditional computational algorithms.
Such models usually lead to formulation of the ill-posed problems for which there exist no theorems on solvability in some natural functional spaces. Moreover, these problems lack stability (in the classical sense) of the solution with respect to errors of input data. This is significant since we almost never deal with the absolutely exact values of input parameters. The theory of solving ill-posed problems is a relatively new branch of
computer science which began to separate into an independent science shortly after the publications by A. Tikhonov (Tikhonov. 1963a, 1963b). In these publications the fundamental concept of regularizing algorithm was introduced.
Here we formulate this concept for a mathematical model represented by operator equation Az = U where operator A acts from a metric space Z into a metric space U. The peculiarity of the ill-posed problems is that operator A is not assumed to be continuously invertible (in local or in global sense). Let us assume that for the "exact" values of uand A there exists the "exact" solution zwhich we are interested in. Let. for the sake of clarity, operator A be known exactly, and instead ofuwe
are given its approximation uf, E U such that p(Ul)' u) ::; O. Here 0 is a numeric parameter characterizing the errors of input data ul). It is natural to require that a numerical algorithm for solving the operator equation should have the following main property: the less is the error 8, the more close approximation to zcan be obtained. This approach underlies the formal definition of regularizing algorithm.
The regularizing algorithm(RA) is the operator R which puts into correspondence to any pair (Uf,,<i) the element Zl) E Z such that zf, ~ z (in the metrics of Z) as 8 ~ O. For a given set of input data, R(uf,,8) can be treated as the approximate solution of the problem.
2 Introduction
The definition of the RA given above and its generalized versions constitute the base for modem theory and practice of solving ill-posed problems.
It proved possible to classify ill-posed problems into regularizable (Le. those for which the RA can in principle be constructed) and non-regularizable. Besides, the general principles for constructing the RA for the wide class of mathematical models have been created. It was discovered that many classical methods (e.g. iterative techniques for solving linear operator equations) can be used to solve ill posed problems as well; or, more exactly to construct the RA for such problems The only necessary improvement is to formulate the appropriate stopping rule dependent on the errors of input data. A significant progress has been achieved in constructing the RA for non-linear problems. The principle of iterative regularization has allowed the generation of special iterative sequences which are characterized by only slight formal differences from the traditional procedures.
By now, many monographs dealing with the ill-posed problems have been published (see, for example, (Groetch, 1984, Louis, 1989, Tikhonov et al., 1977». However, they are all concerned with certain fragments of theory of the RA. As a rule, only Tikhonov's classical variational scheme is described. Many general problems and new schemes of constructing RA are mentioned briefly, and readers are referred to journal publications. The rapidly growing interest in both theory and practice of solving ill-posed
problems has stimulated the authors to write a monograph which would include all the main currently available results related to the existence and constructing of the RA.
The general structure of this book is as follows. Chapter 1 contains fonnulations and discussion of the main definitions used
throughout the book. The general theorems on regularizability and on the error estimates are proved. In particular, the theorem on necessary and sufficient conditions of regularizability (Vinokurov, 1971) is proved, which previously could be found only in journal publications.
Chapter 2 and 3 are devoted to the more widely known problems of regularizability on compacta and the general scheme of constructing RA proposed by A. Tikhonov.
In Chapter 4 we present a general scheme, first given in (Bakushinsky, 1967) of constructing linear RA for linear problems (e.g. solving linear equations and calculating the values of linear unbounded operator) in Hilbert and Banach spaces. The properties of approximations and the RAs generated within such a scheme are studied. The relations between traditional iterative techniques for solving linear equations and some of the newly created RAs are revealed.
Introduction 3
A separate section deals with regularizing linear problems under the conditions of stochastic errors. The latter problem is only briefly discussed, attention being paid to the questions that do not require significant extension of the general "deterministic" approach of the book. A complete state-of-the-art reference in linear ill-posed problems with stochastic errors can be found in (Fedotov, 1982).
Chapters 5, 6 and 7 are devoted to various aspects of implementing the principle of iterative regularization, allowing to construct iterative RAs for non linear problems. Such problems, in particular, embrace all the problems of convex optimization. With the use of the mentioned principle, it proved possible (for the first time) to formulate strongly convergent iterative procedures and RAs for solving the general infinite-dimensional problem of convex optimization. The general problem of linear programming is considered as an example for which many stable (in the sense of regularization theory) iterative algorithms can be constructed on this way. The last chapter (Chapter 8) contains examples of practical application and
numerical implementation of some of the methods developed in Chapters 2-7. In selecting the examples, the authors tried to demonstrate the opportunities that RA provide for solving the wide range of problems (from linear integral equations to linear programming). A relatively significant space is devoted to applications of RAs to image reconstruction and reconstructive tomography. This is justified by the importance of these problems for modem engineering and reflects the particular scientific interests of the authors.
The statements formulated in the monograph are, as a rule, provided with detailed proofs. Some more special results as well as theorems of functional analysis are given without proofs. However, references are always provided for the readers interested in more detailed infonnation on the subject. The reading of this book requires a certain knowledge of the basic concepts
of functional analysis (acquaintance with, for example, (Dunford et al., 1958, 1963) is sufficient). As for the reference list, it does not aim to be complete. The authors have
included in the reference list only publications directly related to problems discussed in the text.
Chapter 1
1.1 Definition of regularizing algorithm (RA)
Any mathematical model sets some correspondence between two kinds of objects. One class of objects includes characteristics of the model, and the other consists of experimentally observed attributes of the studied phenom ena. The problems of processing experimental data are always concerned with inevitable experimental errors. For the purposes of present analysis, we shall consider the objects of the second kind belonging to some metric space X, and of the first kind - to space Y. The mathematical model es tablishes the certain correspondence y =G( x) between input data x E X and characteristics of the model y E Y. Modeling is aimed at obtaining model characteristics y using approximately specified input data x. The above problem is often formulated in other terms. Let z represent
the characteristics of the model, and u - the attributes of the observed phenomenon. The values of z and u are related by operator equation Az = u, where operator A is known. It is necessary to obtain an approximation to z using the knowledge of approximately specified u. The difficulties on this way arise from the following facts: 1) operator G = A-I, in general, is not specified explicitly; 2) domain of G = A-I does not include the whole space; 3) operator G = A-I defined on its domain, is not continuous. Only items 2), 3) are concerned with the principal problems of theory.
Therefore we shall assume that mapping y = G(x) is given, though not
4
1.1 Definition of regularizing algorithm (RA) 5
everywhere in X, but rather on some subset DG eX. Mapping G is not continuous on D G •
Function G may as well represent a one-to-many mapping. In particular, the operator equation Az = u may have different solutions for the same right-hand side u. If follows that, in general, G = A-I is a one-to-many mapping. In principle, it is possible to formulate the regularizability theory for
one-to-many mappings. In most cases, however, there exists a natural way of selecting a one-to-one cross-section Go of mapping G defined by the math ematical model. All the consequent analysis is usually performed for such a cross-section. For this reason, when formulating the general problems of regularizability later in this chapter, we shall restrict our considerations with the case of one-to-one mapping G. This is sufficient for the consequent applications of the results. D e fin i t ion 1. The mathematical model is absolutely well-posed
(correct) on DG if it defines the mapping G which may be continued to all X continuously on D G . If G is relatively continuous on a subset D ~ D G ,
the model is called conditionally well-posed. The above definition of correctness is somewhat more broad than the
classical one given by Hadamard. Example 1. Consider a linear algebraic equation
where Em , En are Euclidean spaces of corresponding dimensions. Let us consider the vector u as input data distorted by experimental
errors. Let us introduce mapping G(u) which puts into correspondence to any u the solution of the.above equation with the minimal norm in the sense of the least-squares technique. Function G(u) is defined and continuous everywhere in En, i. e. it represents the absolutely well-posed model.
If elements of matrix A are also considered as input data and are known approximately, the mapping z =G(A, u) is no more continuous with respect to A, if distance between matrices is estimated, for example, with the use of convenient operator norm. This model is therefore ill-posed with respect to perturbations of matrix. These and related problems are discussed in more detail in Chapter 3. Example 2. Consider the operator equation
Az = u, z E Z, u E U,
6 Chapter 1. General problems of regularizability
where Z, U are linear normed spaces, and A is a completely continuous opera tor from Z to U. The inverse mapping G = A-1 is only defined on D G = AZ which does not include the whole U in the case of infinite-dimensional spaces. It is well known that mapping G = A-1 is unbounded on D G = AZ, i. e. this problem is ill-posed on AZ. However, if we could a priori specify a compact subset M C Z, then
mapping G '= A-1 defined on D = AM C DG is relatively continuous, i. e. the problem is conditionally well-posed on D = AM C DG .
In computational mathematics, it is essential to develop methods of cal culating G(x) using approximately specified argument x. In this case G( x) may also be found approximately. It is natural to refer to y as an approxi mation of G(x) if py (G(x), y) is "small" (py (I, g) is the distance between f and 9 in the space Y).
It is necessary to clarify the concept of "approximately specified data x". We shall represent the approximate data with a pair (x~, 6) such that Px(x, x~) ~ 6. The element x~ does not necessarily belong to DG • The theory of regularization is based on the assumption that such a pair is available. Obviously, it is very attractive to yield the approximation of G(x) as a
pair (YClE) where py(G(x),yc) ~ E. It proves, however, that such approxima tions of the solution of ill-posed problem cannot be constructed. For these problems it is only possible to find y(x~) such that y(x~ )-.G(x) when 6-.0. Constructing y( x~) is concerned with creating the so-called regularizing al gorithms. For the ill-posed models (i. e. discontinuous mappings G) the approxi
mation in the sense of Tikhonov regularizability has the special significance. Let some mapping G(X-.Y) be defined on a subset D(G)<;;'X of a metric space X, and G(D(G))<;;'Y (Y is also a metric space). Let us introduce the approximation error of G(x) for a certain fixed 6 > 0 and a priori given mapping R~(X-.y) as
D.(R~,6,x) = sup py(R~(x~),G(x)). p(r,r6)~~
De fin i t ion 2 (Vinokurov, 1971, 1987). A function Gis regularizable on DG C X (or on non-empty subset D <;;, DG ) if there exists a mapping R~(·)
(or R(·, 6), which is the same) from the direct product of spaces X ®Rt into Y such that
lim D.(R~,6,x) =0 \Ix E DG (\Ix ED). ~-o
(1.1 )
1.2 General theorems on regularizability 7
The operator R 6(x) is called a regularizing algorithm (RA) or a regu larizing operator for calculating G(x). Note that R6( x) is not necessarily continuous.
If R6 is RA for G(·), it is natural to regard R6(X6) as an approximation to G(x) using input data (X6' <5). In the regularization theory, it is essential that the pair (X6,<5) is used for approximating G(x). We cannot use the triple (X6' <5, x) because the exact value of argument is not known. Then, maybe, it is possible to create RA with the use of X6 only? Let X6 be an arbitrary element from the <5-vicinity of x, i. e. P(X,X6) ~ 8.
Does there exist any function R(X6) which does not explicitly depend on o such that Eq. (1.1) holds? It proves to be that only well-posed on D problems may be regularized with such a RA.
Theorem 1.1 Mapping G is regularizable on D(DG ) by the family R6 = R(·,o) = R(·), 0 < <5 ~ <50 , if and only if G(x) may be continued to all X, the continuation being continuous on D( DG ) in X.
Proof. To prove the sufficiency, one may use R(·) = G where G is continuation of G, continuous on D(DG ). When necessity is proved, R(·) is the continuation we are searching for. According to the above theorem, the pair (X6' 8) is a minimal set of data
which may secure approximation of G(x )in the sense of DeL 2. Unluckily, it is generally impossible to estimate deviation of R6(X6) from
G(x) without any additional information concerning x. This is specific fea ture of the ill-posed problems (see Sec. 1.3). In general, RA guarantees only asymptotic convergence of approximations to the exact solution as 0 tends to zero.
1.2 General theorems on regularizability and principles of constructing the regularizing al gorithms
Let us try to find out how wide is the class of regularizable mappings G. Obviously, it is not empty. It can be seen immediately from the definition that all the mappings G corresponding to the well-posed models are reg ularizable (i. e. all functions G( x )continuous on Dare regularizable). In this case it may be assumed that R6 =G for VO. Since G is continuous, Eq. (1.1) is valid. All the classical computer mathematics is based on the
fact that G(X6) approximates G(x)when P(X6'X) ~ D, if G corresponds to a well-posed mathematical model. Let us deduce an immediate consequence of Def. 2 which guarantees that
the class of nonregularizable mappings is not empty.
Theorem 1.2 (Vinokurov, 1971). Suppose that G maps DG ~ X onto Y , i. e. G(DG ) == Y , where X, Y are metric spaces. If G is regularizable on
00
DGI and N is an everywhere-dense set in X, then U R1/n(N) is everywhere n=!
dense in Y.
Proof. Let y E Y and x E G-!(y). Since N is dense in X, there exists Xn E N such that p( Xn , x) ~ 1/n. In accordance with Def. 2,
In other words, we have found a sequence of points Rl/n(xn)l~ ~ 00
~ UR1/ n ( N) that converges to y. Since y is chosen arbitrarily, the theorem n=!
is proved. Con seq u e n c e 1. Let G map DG onto Y, mapping Gis regularizable
on DG and X is a separable space. Then Y is also separable. This directly follows from Theorem 1.2 if we select a numerable every
where dense set N in X. This fact allows to easily construct the examples of nonregularizable
mappings. Example 1. Let X = 100 be a space of bounded sequences, and
Y = J2 be a space of square-summable sequences; the norm of element x = (Xl, •.. ,xn, ... ) in 100 is given by equation "xlIl~ = sup Ixnl·
n
We define a linear operator A from 100 to J2 as
It is easily seen that KerA =0, and the inverse mapping G =A-I from J2 to 100 exists. In accordance with Cons. 1, it is not regularizable on D G C J2 because [00 is an inseparable Banach space (Dunford et al., 1958), while J2 is separable.
Example 2. Let A be a linear one-to-one operator from the space of functions with bounded variation V: to L 2 • The mapping A-I may not be regularizable on AV; C L 2 because V; is not separable, while L 2 is. We have already mentioned that continuous mappings are regularizable.
It is a remarkable fact that some discontinues mappings can also be regu larized. In particular, this is true for all the mappings which represent the pointwise limits of continuous mappings. It follows that the approximate solution of the ill-posed problems is possible (in the above-specified sense).
Theorem 1.3 Let function G : X ~ Y be a pointwise .limit (on a set D ~ X) of the functions Gn which are continuous everywhere on X. Then G is regularizable on D.
Proof. Suppose that Gn is a given approximating sequence. Let us define the family of coverings lIn of the space X by open sets as follows: III consists of all the open sets U ~ X with diameters :s; 1 such that
d{GI(u)} :s; 1; 112 consists of all the open sets U ~ X with diameters $ 1/2 such that
d{GI(u)}:s; 1/2, d{G2(u)} :s; 1/2; lIn consists of all the open sets U ~ Xwith diameters :s; l/n such that
d{GI(u)}:S; lin, ... , d{Gn(u)}:S; lin. The continuity of every Gn implies that any point x E X belongs to each
of lIn- Let us construct the regularizing family R6( x). Suppose we have fixed
the point x and some 15 > O. For the small 15 $ Co there exists n = n( 15, x) such that we may find the set U C II n(6,x), S( x, 8) C U;l Let n( 15, x) be the maximal number for which the latter condition is valid. We define
R 6 (x) = {Gn(6'X)(X), n < 00
GI(x), n 1= 1,2'00'
It is clear that necessary n can always be found for sufficiently small 8 :s; Co. Now let us show that the above-defined family R6(x )regularizes G on the
set D C D G eX. We select an arbitrary point x from D. From the triangle unequality we get
sup p(R6(x'),G(x)) $ P(Gn(6,f)(X),G(x)) + p(x' ,f)
+ sup p(Gn(x'), Gn(X)) $ p(Gn(X),G(X)) + lin. p(x',r)
(1.2)
lHere S(x, 6) used to denote the open globe with the center in x and with radius 6.
Note that n = n(6,x) -+ 00 when 6 -+ O. In fact, since any globe S(x',6) contains the point X, it is included into an arbitrary vicinity of x when 6 tends to zero. Hence, for the arbitrarily large N we can choose the sufficiently small 6 so that S(x, 6) is included into some of the open sets constituting covering fiN' It follows that n(6, x) ~ N. This taken into account, proof of the theorem follows from the inequality
(1.2). In the following important case the condition of continuous approxima
tion is also a necessary condition for regularizability of G.
Theorem 1.4 (Vinokurov, 1971). Suppose a separable space Y is a convex subset of a linear normed space, and G is regularizable on D G • Then G is a pointwise limit of the sequence of mappings Gn which are continuous on X.
Proof. The proof would appear trivial if the RA mentioned in the formulation of theorem were continuous. It happens that such a RA can be constructed.
Lemma 1.1 Given the conditions of theorem 1.4, it follows that there exists RA such that each R 6 acquires only a limited number of values in Y.
Proof of Lemma 1.1. Any separable metric space Y is homeomorphic to some completely bounded metric space Y' (i. e. the space where the finite 6-net exists for every 6)(Dunford et al.,1958). Let 1 be the mentioned homeomorphism. The function lG is obviously regularizable, and the fact that R6 regularizes the mapping G implies that family lR6 is a regularizing algorithm for lG. Suppose that L 6 (6 > 0) is a mapping Y' -+ Y' which maps each element y' E Y' to one of the nearest points of 6-net in Y'. Using the triangle inequality, it is easy to show that L 6lR6 is also a regularizing algorithm for lG. The family of functions l-1 L 6 lR6 is the RA the existence of which is declared by Lemma 1.1. Rem ark. The number of values that R6 can acquire in Lemma 1.1,
depends on 6.
Lemma 1.2 Given the conditions of theorem 1.4, there exists the RA such that every R 6 is a continuous function on X.
Proof of Lemma 1.2. Let R6 be a RA for G having only a finite number of values Yl, ... , Yk(6) for any 6 > O. It is proved in Lemma 1.1 that such RA exists. Let us fix some 6* > O. For the sake of simplicity we shall
assume that k(c*) = 2. A complete prototype of the point YI will be denoted as R"iol(yd =AI' Consider the open set O(Ad of all the points with distances from Al not
exceeding 6* /8. If Riol(Y2) cO(Ad, we stop with this set. Otherwise, define A2 = Ri}(Y2)\O(AI ) and construct O(A2) in the full analogy with O(Ad. It is obvious that
(1.3)
{
0, p(x, Ai) ~ C' /4, Ii (x) = 1 _ p(x, A;) ( ) c /
C' /4 ' P x, Ai < • 4.
Note that for any x E X
1 p
(1.4)
where p is the number of constructed sets Ai (in this particular case p ~ 2). The first inequality in (1.4) follows from the fact that p(x, A;) < C' /8 for at
. c* /8 1 least one t because of (1.3). Hence the value of function Ii ~ 1 - c* /4 = 2' Now we can introduce obviously continuous functions R6• /8(x) with values
if the set A2 = (j), and
2
Ly;fi(X)
LIi(X) i=1
(1.5)
otherwise. The procedure of constructing R60 /8(x) for the case when R6 has more than two values is absolutely analogous. Because of the assumptions on Y (linearity and convexity), R60 /8(x) is
defined correctly, and R60/8(X) E Y.
Now we are to show that Eq.(1.5) defines a RA for mapping G. Let us fix some x E Da . Since R6 is a RA for G, for any c there exists h(c, x) such that
(1.6)
if p(x',x). Select some c > O. Let for the sake of simplicity R6(t,F)(X) has two values (YI and Y2) and two sets A}, A 2 are constructed. If, for instance,
IIYI - G(x)11 ~ c,
then fI(x') = 0 for every x' from the vicinity of x. To prove this, note that in this case
(1.7)
--1 R6(t,f)(Ydn{p(x',x) ~ h(c,x)} = (J).
It follows that Al n{p(x', x) ~ h(c, xn = (J), and, consequently, that O(A1)
n{p(x',x) ~ h(c,x)/8} = (J), which yields fl(x') = O. Inequality (1.7) cannot be true simultaneously for all the Yi participating in construction
--I procedure (1.5) because of Eq. (1.3) and Ai ~ R6 (Yi)' If all Yi satisfied (1.7) simultaneously, it would follow that x rt X. Hence, if (1.7) is satisfied for YI, then
R6(t,f)/S(X') = Y2 if p(x', x) ~ 15/8.
If both inequalities
IIR6«,f)/S(X') - G(x)1I ~ c, p(x', x) ~ 6(c, x)/8.
This means that R6/ S(x) is the RA for G that we are searching for. The proof of Theorem 1.4 becomes simple now. Let hn -.0 be a sequence
of positive numbers. The family of continuous mappings Gn == R6n / 8 is a pointwise approximation of G on Da .
This enables us to formulate a criterion of regularizability (Vinokurov, 1971):
Theorem 1.5 For the function G : X -. Y (Y is a separable linear normed space) to be regularizable on D ~ Da C X I it is necessary and sufficient that G is a pointwise limit (on D) of a sequence of functions G n that are continuous on D.
For the majority of computational applications it is sufficient to con sider linear normed separable spaces, or even Hilbert separable spaces X, Y. These cases are completely covered by the above theorem. Still, it is possible to formulate a more general criterion of regularizabili ty
for metric spaces. D e fin i t ion 3. Function G : X --+ Y (X,Yare metric spaces) is
called a first-class B-measurable function if the prototype of each open set in Y is a set of the type F6 , i. e. it can be represented as a countable sum of closed sets.
Theorem 1.6 For the function G from Do ~ X into a separable space Y to be regularizable on D ~ Do ~ X, it is necessary and sufficient that the contraction of G on D is a first-class B-measurable function.
Hence, the regularizability offunction D on X (or Do) depends only on the properties of this function on the mentioned subset. We omit the proof of this theorem. It is based on the results of Theo
rem 1.5 and existence of certain standard homeomorphisms. Example. Dirichlet function from [0,1] to [0, 1] is not regularizable ac
cording to Theorem 1.4 because it is not a pointwise limit of a sequence of continuous functions (this is proved in theory of functions ofreal arguments). On the contrary, it is obviously regularizable on the subset of rational points of [0,1), the RA being R6 == o. Theorem 1.3 shows the close correspondence between the problem of
constructing the RA and the problem of constructing the approximating sequence of continuous mappings for G. Later in this book we shall consider not only families Gn ( x) where n is a positive integer, and n --+ 00, but as well the more general family {Ga(x)} where Q belongs to some ordered infinite numerical set, and
lim p(Ga(x),G(x)) =0 for "Ix E Do 0-00
(1.8)
Constructing the family ROt possessing the property (1.8) is a classical problem of computational mathematics. In the classical computational mathematics different classes of continu
ous mappings G are considered which originate from mathematical models (solving the differential, algebraic, integral equations, etc.). Constructing approximating sequences (1.8) for them is generally equivalent to creating an approximate technique for calculating G(x).
To clarify this point, suppose {a} to be the set of positive numbers and 00 = O. Since Go are all continuous, it is true that
lim lim p(Go(x'),G(x)) = 0, p(x',x) ~ b 0--+06--+0
(1.9)
irregardless of G being continuous or not. However, for the continuous map pings G the limits in (1.9) can be interchanged:
lim lim p(Go(x'),G(x)) =O. 6-00-0
(1.10)
Eq. (1.10) to some extent justifies the technique frequently used in a classical computational mathematics, when for the approximately given x', such that p(x', x) ~ b, the approximation Go(x') with the smallest possible a is taken. For example, this is the case when iterations in the iterative process are continued as long as the time allows. If we neglect the accumu lation of rounding errors in such a process, this technique can be justified by consideration that GO~.n(x') is "close" to G(x'), while G(x') is close to G(x) because of the continuity of G. Eq. (1.10) in many cases may be made even more strong for continuous
mappings:
(1.11)
Sometimes the error p(Go(x'), G( x)) may be estimated by some function <p(0, b) such that <p(0, b) --+ 0 when 0, b --+ 0:
p(Go(x'), G(x)) ~ <p(a, b).
If the condition (1.11) is satisfied, it is possible to choose any dependence a(b) compatible with (1.11), e. g. a = b. If the estimate <p(a, b) is known, a(b) may be defined from the condition
a(b) =argmin<p(a,b) o
The way from constructing the family (1.8) to obtaining some kind of approximation to continuous mapping G is, in general, clear and is widely studied in the classical computational mathematics (e. g. for constructing finite differential methods of solving differential equations, both in complete and partial derivatives).
It turns out to be that efficient approximating sequences of continu ous mappings may as well be created for the wide range of discontinuous mappings, i. e. it is possible to approximate the solutions of the ill-posed problems. When the approximating family is constructed, Theorem 1.3 allows to
speak of G being regularizable. The natural way of transfer from G(a) to R 6 (or from Gn to R6 ) is to specify the function a(x',6) (or n(x',6)) such that the family Ga(.r/,6)(X') == R6(x') (or Gn(.r',6)(X') == R6(x')) satisfied the definition of RA (1.1). This method has already been implemented while proving Theorem 1.3. Unfortunately, the same approach as used in that proof may not be applied to a real problem. Therefore the other methods of specifying the parameter a( x', 6) (or n( x', 6) for iterative techniques) have been developed. When these methods are considered, it is very important to ensure that
a (or n) depends only on x', <5 and no additional information about x is used. If such an information (be it implicit or explicit) is used, it is only possible to speak of RA on a certain subset D ~ Do defined by the used information.
It must be specially mentioned that there exists a wide range of map pings and approximations allowing to specify function a(x',6) (or n(x',6)) independent of x'.
Theorem 1.7 Let mapping G(x) may be approximated on Do by mappings Ga( x) satisfying Lipschitz conditions with the constant La in X, i.e. for any X,X1 E X p(Ga(x),Ga(xd) ~ LaP(x,xd. Then it is possible to specify a = a(b') independently of x' so that Ga(6)(X) is a RA for G(x) on Do.
Proof. Let x be an arbitrary fixed element from Do such that p(x', x) ~ 6. Then
p(Ga(x'),G(x)) ~ p(Ga(x'),Ga(x)) + p(Ga(x),G(x)) ~ Lab' +p(Ga(x),G(x)).
Suppose we define a(6) -> 0 when <5 -> 0 so that
(1.12)
(1.13)
Combining (1.12) and (1.13), we find that R6 == Ga (6)(X) is the RA for G on Do.
Absolutely analogous treatment is valid if every G" satisfies Lipschitz condition L,,(S) in some open globe S(O,p) including 6o-neighborhood of D(G) (or of its subset D). In this case we again specify a(6) so that (1.13) is true which enables us to construct a RA Go (6) on D (or on the whole of Do). Running a little ahead, we would like to note here that conditions of
Theorem 1.7 are satisfied for mappings generated by linear operator equa tions in the pair of arbitrary Hilbert spaces: G(X) =A-lX, if KerA = 0, or, in general case of one-to-many mappin~, G(X) = A(j"IX where A(j"l is an appropriate cross-section of one-to-many mapping. Theorem 1.7 is also applicable to non-linear mappings associated with the solution of monotonic variational inequalities, e. g. the problems of convex optimization, etc.
1.3 Estimates of approximation error in solving the ill-posed problems
In the theory of regularization, a new concept approximation error of the solution has been introduced. By definition, if the values ofG are calculated with the use of some regularizing algorithm, it is the function 6. in the DeL 1 that is called the approximation error of RA at the point x.
If mapping G is regularizable on D( Do), the error tends to zero when 6 ~ °at any point x E D(Do ). Like the traditional theory of estimating errors, the theory of RA studies
the behavior of 6(6, x) as a function of 6. Various estimates are constructed to determine the rate of convergence of the error to zero when 6 ~ 0. A simple but very characteristic result concerning the approximation error is that, in general, there exists no uniform (for all x E Do) estimate 6. if the problem of calculating G(x) is ill-posed.
Theorem 1.8 If function G( x) is regularizable on D by mappings R6 , and there exists a uniform on D estimate of approximation error 6.(R6 , 6, x) ~
~ <p(6) such that lim cp(b) = 0, then contraction GI D is uniformly continuous 6-0
on D.
Proof. Let XllX2 E D, and P(X\>X2) ~ b. Due to the triangle inequal ity, P(G(XI),G(X2» ~ p(R6(xd,G(xd) + p(R6(xd,G(X2» ~ 2cp(b). This means that GID is uniformly continuous. Con seq u e n c e 1. Let mapping G = A-I be the inversion of the
injective linear completely continuous operator acting in Hilbert space H.
1.3 Estimates of approximation error in solving the ill-posed problems 17
It is shown in Chapter 3 that this mapping is regularizable everywhere in the domain of A-I, and the regularizing mappings R6( x) are continuous. However, A-I is not uniformly continuous in its domain, hence there exists no uniform estimate of error of such a RA on the whole DA-I.
Let us now clarify the point on what linear functionals depending on the solutions of the ill-posed problem may be approximately calculated with the uniform (relative to the input data) estimate of errors, depending on error t of the input data. Con seq u e nee 2. Let A be a linear continuous operator in a
Banach space B such that A -I exists and is densely defined. Let 1 be a linear continuous functional in B. Consider the mapping G(u) = l(A-1u). A regularizing algorithm for calculating G(u) with the uniform estimate
of approximation error (everywhere in DA-I) exists if and only if 1E DAo-., i. e. the equation A· q = 1 is compatible. Indeed, if 1 E DAo-., then G(u) = l(A-1u) = q(u) where q is a linear
continuous functional. Such a mapping is no doubt regularizable uniformly on D A -. (and even on the whole of B), the RA being R 6(u) == q(u). Vice versa, if R 6(u) with the formulated properties exists, then l(A-1u)ID
A _
I is
uniformly continuous on DA-I due to the results of Theorem 1.8. It is easy to show that in this case the linear functional l( A-IU ) may be continued to yield a linear continuous functional everywhere on B. Using the defini tion of a conjugate operator, we come to 1E DAo-1. This result completely defines the class of linear functionals calculating
the values of which on the solutions of the ill-posed problems is an (abso lutely) well-posed problem.
It is of great significance from the point of view both applications and theory to classify the manifolds which allow to create a uniform estimate of error. The results obtained in this field so far are not absolutely complete. It seems that the manifold allowing the uniform estimate is very "slim". This not very accurate statement is illustrated by
Theorem 1.9 Let G be a linear mapping inverse to a linear condensation in Banach space, and G is unbounded in its domain. Then the set D allowing to estimate the approximation error is first-category set in X I i. e. D is the countable sum of nowhere dense sets.
We omit the proof of this theorem. It is of interest how far can the conditions of this theorem be extended
(e. g. for non-linear mappings and more general spaces).
In general, the set of uniform estimate depends on X, G and R6•
Some more special problems allow to efficiently describe (using the oper ator of the problem) the set of uniform estimates of convergence in Eq. (1.1) and to obtain the uniform estimates of errors on this set. Examples of such problems are to be found in Chapters 3-7. So far we have spoken only about the upper bounds of error in Eq. (1.1).
It happens that there exist universal uniform lower bounds of this error dependent on the mapping G and independent of the particular RA (Vi nokurov, 1973).
Theorem 1.10 Let the set E ~ DG ~ X consist of more than one point, and there exist at least two points Xl>X2 E E stich that P(Xl>X2) ~ 6. Then
where
w(6,G,E) = sup p(G(Xd,G(X2)) :r,.:r.EE
(1.14)
(Function w in the right-hand side of (1.14) does not depend on the particular RA and is called the continuity module of mapping G on E). Proof. Let XI,X2 E E, and P(Xl>X2) ~ 6. Then
b.(R6 , 6, xd ~ p(R6(xd, G(xd),
b.(R6• 6, X2) ~ p( R6( xd, G(X2)),
Using the triangle inequality, we obtain
b.(R6,6,xd +b.(R6 ,6,X2) ~ p(R6(xd,G(xd)+
(1.15)
where Xl, X2 are the points from E that we have chosen. Taking into account that Xl> X2 may be chosen arbitrarily (as long as distance between them does not exceed 6), we obtain the inequality (1.14). Note that in the above theorem we nowhere used the condition that R6
is a RA for G. For any fixed 6 we could as well use any mapping R : X -+ Y defined everywhere on X instead of R6 •
1·4 Comparison of RA. The concept of optimal algorithm 19
1.4 Comparison of RA. The concept of optimal al gorithm
The function sup iJ.(R6 , 0, x) or, more generally, sup iJ.(R, 0, x) where R is an xEE xEE
arbitrary mapping X ~ Y, provides the opportunity to compare algorithms with respect to approximation error on the set E (at some fixed 0 = 00). Algorithm R~l) !: R~2) (i. e. R} is better than RD on E at 0= 00 if
sup iJ.( R~~), 00, x) ~ sup iJ.( R~~), 00, x). E E
D e fin i t ion 4. Algorithm R6 is optimal on E at a fixed °= 00 if
supiJ.(R6 ,o,x) = inf sup iJ.(R,o, x). xEE R:X-Y xEE
(1.16)
Algorithm R6 is called optimal if it is optimal in the sense of this defini tion for any 0. Analysis of behavior of the error as a function of 0 when 0 -+ 0 justifies
the following definition: D e fin i t ion 5. Algorithm R6(x,o) is of the optimal order (or
order-optimal) on E if for any 0 such that 0 < 0 ~ 00
supiJ.(R6 ,o,x) ~ k inf supiJ.(R,o,x) (1.17) xEE R:X-Y xEE
where k 2: 1 is a constant independent of 0. The general idea underlying the construction of optimal and. order-op
timal algorithms is as follows. First, we try to obtain as accurate lower boundary estimates of the right-hand side of Eq. (1.16) as possible. Then we are to construct an algorithm with the upper boundary of error equal to the above - mentioned estimate multiplied by a constant independent of 0 (this leads to the order-optimal algorithm).
It happens that the lower boundary estimate given by Theorem 1.10 is in many cases sufficient to obtain the order-optimal algorithms. Let E again be a subset of metric space X, and E C Da . Consider the
following technique for approximate calculating G(x) (Vinokurov, 1973): for any x E X
(1.18)
where
{
X, if x E E, P6(X) = element X· E EnS(x,6) , if x rt E,EnS(x,6) 1= 0,
elementxQ , if x rt E ,EnS(x,6) = 0·
Let w(6,x) be a local continuity module of GIE at the point x E E.
Lemma 1.3 The following estimate is true:
(1.19)
Proof.
xEX xEX
p(x',x):5 26
The lemma is proved. Calculating the upper boundaries of both sides of the inequality (1.19)
leads to
(1.20)
The inequalities (1.20) and (1.14) cannot be compared directly because in general the relation between w(26, G, E) and w(6, G, E) is not known. The method based on Eq. (1.18) is obviously order-optimal for the map
pings G and sets E, for which there exists k ~ 1 such that
w(26,G,E) ~ kw(6,G,E). (1.21 )
Lt Comparison of RA. The concept of optimal algorithm 21
We can say, at least, than the class of pairs G,E defined by Eq. (1.21) is not empty.
Example. Let X =Y =B be Banach spaces, and A be a linear injective operator in B. Consider the mapping G = A-I and the set E =A(S) where S = {x : Ilxll $ I}. In this case
w(26,G,E) = sup IIA-1(xl - x2)11 = x"x,eE
IIx,-x,II~26
(1.22)
Suppose that VI, V2 are arbitrary elements from S such that
then
II VI - v211 =211vd2 - v2/211 =211w I - w211
since WI, W2 E Sand IIAwl - AW211 ~ 6. Finally we obtain for any pair (VlJ V2) in Eq. (1.22) the following inequal
ity:
We have found that (1.21) with k =2 is true for the considered example. The fact that E is the image of a unit globe is not essential. It was used only to secure that element w/2 belongs to S if w does. In particular, the set S may be substituted by L(S) where L(S) is the image of the globe created by a linear mapping L. The technique based on Eq. (1.18), being the order-optimal algorithm, is
not necessarily a RA for the mapping G even on the subset E. It is therefore natural to clarify the point if there exist optimal and order-optimal methods among the regularizing algorithms. The general answer to this question is not known. It is obvious, however,
that mapping defined by Eq. (1.18) is a RA for G on E if limw(8, G, E) = O. 6-0
Certain results concerned with constructing order-optimal algorithms can be found in (Groetch, 1984, Ivanov et al., 1978, Vainikko et al., 1986). It is shown, for example, that Tikhonov RAs are order-optimal. In the rest of this section we shall briefly describe the problem of con
structing the optimal methods. The estimate of lower boundary given by (1.14) is too rough for this purpose. In some more special cases it is possible
to replace this estimate by the more accurate one. This latter estimate is sometimes sufficient for obtaining the optimal algorithm. At present, the optimal algorithms are developed only for linear problems
in Hilbert spaces. The examples of such algorithms are to be found in Chapter 4. We shall give here the estimate of error which will be of use later instead of (1.14) (Strachov, 1970).
Theorem 1.11 Let Z, U be Hilbert spaces, and G = A-I be the inversion of a linear bounded operator A defined on Z. Let the set E = A(M) be the image of centrally symmetrical set M E Z produced by mapping A. Then for any mapping P : U -+ Z
sup 6(P, 0, z) ~ WI (0, M), zEE
where
IIA z lI:S6
Proof of this theorem can be found in (Vainikko et al., 1986, Morozov, 1987).
Chapter 2
Regularizing algorithms on compacta
Henceforth we consider how to construct regularizing algorithms for solv ing the ill-posed problems. The typical ill-posed problem is the operator equation of the first kind
Az = u,z E Z,u E U. (2.1)
Here Z, U are Banach spaces, and A is a continuous injective mapping of Z onto U. In this chapter we consider the case when a priori considerations allow to specify a certain set M C Z (the correctness set) including the exact solution i of Eq. (2.1), the set M being such that the inverse operator A-1 is defined and continuous throughout AM CU. In this case the approximations to zmay be easily constructed. It should
be noted, however, that A-1 Ub may not be used as such an approximation because Ub does not necessarily belong to the set AM. The idea of approximating the solution of the ill-posed problem on a spe
cial set was first suggested by A. N. Tikhonov as early as in 1943 (Tikhonov, 1943). In the mentioned paper he has formulated the concept of conditionally well-posed (or Tikhonov well-posed) problem. The problem is conditionally well-posed if the following requirements are met: 1) it is known a priori that a solution of Eq. (2.1) exists and belongs to
a specified set M; 2) the operator A is a one-to-one mapping of M onto AM; 3) the operator A -1 is continuous on AM CU. This definition is really a reformulation of DeL 1 given in Chapter 1.
23
24 Chapter 2. Regularizing algorithms on compacta
In contrast to absolutely well-posed (Hadamard well-posed (Hadamard, 1932)) problems, for a conditionally well-posed problem: a) it is not required that Eq. (2.1) is solvable over the whole space; b) the requirement of continuity of A-lover all U is substituted by the
requirement that A-I is continuous over the image of correctness set. Establishing the links between the concept of conditionally well-posed
problem and that of regularizability (see Sec. 1.1), we could say that the main result of Chapter 2 is the proof of regularizability of the mapping G = A-Ion the set AM cU. For the conditionally well-posed probleI1ls, it is possible (due to Theo
rem 1.1) to construct regularizer R6(·) on -M dependent only on U6, i. e. R6(·) = R(·). Moreover, the isolation of correctness set allows not only to construct RA, but to obtain a uniform estimate of approximation error on M.
2.1 The normal solvability of operator equations
The aim of this section is to disclose the relations between the structure of the range of operator A: AZ = R(A) and the stability-of the solution of Eq. (2.1). More strictly, it is to be found out, when the correctness set M coincides with the whole space Z. Let us consider the simplest case of linear injective operator A and Ba
nach spaces Z, U. De fin i t ion 1. A graph of operator A defined on D(A) C Z is a set
of pairs z, Az where z E D(A). A graph of operator is a subset of the direct sum of spaces Z and U. D e fin i t ion 2. Linear operator A 'is called closed if its graph is a
closed set in the direct sum of spaces Z and U. The operator being closed means if Zn E D( A), Zn -+ Z and Az~ -+ u,
then z E D(A) and u = Az. It is clear that operator A is closed if D(A) = Z and A is linear and bounded. It is also easy to see that closedness of operator A leads to operator A-I being closed. Really, the graph ofthe operator A-I may be written as Az, z, z E D(A),
i. e. it is obtained by interchanging z and Az in graph of operator A. There fore graph of A-I is also a closed set in the direct sum of Z and U. The following statement is true:
Lemma 2.1 The range R(A) of the linear continuous closed operator A is a closed set.
2.1 The normal solvability of operator equations 25
Proof. Let Z be a limit point of D(A), and Zn -t z, where Zn E D(A), The sequence AZn is fundamental in U because
Using the completeness of U, we get AZn -t f, fEU. To prove the lemma, it is sufficient now to use the closedness of operator A.
I
Theorem 2.1 Let A be a linear continuous injective operator defined on D(A) == Z with the range R(A) ~ U, where Z and U are Banach spaces. For the inverse operator A-I acting from R(A) to Z to be bounded (IIA- 1 11 < +00), it is necessary and sufficient that R(A) =R(A).
Proof. Necessity. Taking into account that operator A is continuous, and D( A) = Z, we obtain that A is a closed operator. Therefore A-I is closed as well, and, due to the conditions of the theorem, it is continuous. Using Lemma 2.1, we arrive at R(A) =R(A).
Sufficiency. If R( A) == R( A), then the linear operator A is a one-to-one continuous mapping of Banach space Z onto Banach space R(A). It follows from Banach theorem (Dunford et aI, 1958) that operator A-I is continuous and therefore bounded, i. e. IIA- I II < +00. We have found that for the linear one-to-one continuous mappings, the
stability of the solution and solvability of the problem are closely related. In particular, for the ill-posed problems when A-I is unbounded in its domain, the set R(A) is not closed and, consequently, Eq. (2.1) cannot be solved for the whole space U. For example, the common Fredholm integral equation of the first kind with the closed square-summable kernel (the operator acts from L 2 to L 2 ) is obviously not solvable in the whole space. This follows from the fact that operator of a direct problem is completely continuous and therefore may not have a continuous inversion. The above results mean that for the ill-posed problems, when A-I is un
bounded in its domain, two correctness conditions of Hadamard are violated: that of solvability and of continuity of the inverse operator. The normally solvable equations (i. e. those for which R(A) is closed) may
yield the ill-posed problems in U only due to the condition R(A) = R(A) i i- U. In this case the approximately given element U6 E U such that IIu6 - iLll ~ 8 does not necessarily belong to R(A). However, G == A-I is regularizable on R( A) C U, and the regularizing algorithm may be con structed, for example, in the following way.
Let P6 be the operator mapping any element U6 E U from the 6-vicinity of it into the arbitrary element u~ E R(A) such that lIu~ - u611u ~ 6. Then G = A-I P6 is a RA for the problem (2.1).
2.2 Theorems on stability of the inverse mappings
In this section the structure of the correctness set is determined for the case when it does not cover the whole Z. Any compacturn in Z can be" taken as the correctness set M C Z. This
statement is based on the following well-known lemma.
Lemma 2.2 Let Z and U be Banach spaces, and operator A produce a con tinuous one-to-one mapping of compadum M onto AM cU. Then the inverse operator A-I is continuous on AM.
Let us consider some examples of compacta in Banach spaces. Example 1. Any bounded closed set in a finite-dimensional space is
a compactum. This example may seem trivial but the approach using this property was widely used for solving the ill-posed problems until the modern powerful regularization technique was created. To find the approximate solution of the ill-posed problem, the functions to be found are parametrized by a finite number of parameters in accordance with a priori information on the solution. If the range of parameters is bounded, the correctness set introduced in such a way is a compactum in Z.
Example 2. Let S be a sphere in reflexive space V, and B be a completely continuous linear operator from V to Z. Then we can take the image of a sphere produced by mapping B a.s a compactum in M, i. e. M =BS.
Example 3. When the operator equations (2.1) are solved, and the func tion to be found is z( t), t E [a, b], some features of the solution are some times known a priori. We may know, for example, that z(t) is bounded, or monotonic, convex, etc. Let us consider the set of bounded nongrowing functions Zie such that z(t) E Zie if 0 ~ z(t) ~ C, z(td ~ z(t2 ), t l ~ t 2 ,
t,t h t2 E [a,b]. The set Z Ie is a compactum in Lp(p > 1) (Goncharsky, 1987). Really, according to theorem on choice (Dunford et aI., 1958), we can always select(from any sequence Zll"" zn) a subsequence zn,(t) that converges everywhere to some function z(t) E Z Ie . The latter statement, together with the uniform boundedness of this sequence, yields the conver gence in Lp •
2.2 Theorems on stability of the inverse mappings 27
Example 4. It may be shown in a full analogy with Ex. 3 that the set Zc of convex upwards functions such that 0 ::; z(t) ::; C for any t E [a, b] is a compactum in Lp(p > 1) (Goncharsky et al., 1979). Therefore, the compacta may serve as the examples of the correctness
sets for Eq. (2.1) with a continuous (not necessarily linear) operator. For the linear continuous operators, the correctness set M may be ex
tended to algebraic sum M = I( +£ where I( is a compactum, and £ is a finite-demensional subspace of Banach space Z.
Theorem 2.2 (Ivanovetal.,1978). Let A be a linear continuous one-to-one operator from Z to U where Z, U are Banach spaces. If the set M C Z can be represented as M = I( +£, I( n £ = (J), where I( is a compactum, and £ is a finite-dimentional subspace in Z, then operator A-I is continuous on AMCU.
Proof. Since A is a one-to-one mapping, and I( n £ = (J), it follows that AI( n A£ = (J). Therefore the inverse operator All is defined on AI(, and operator A2"l is defined on A£, the operators being such that A-I'll = = Aliv +AzIw, V E AI(, w E A£. The operator Az
1 is finite-dimensional and hence continuous. The continuity of All is proved by Lemma 2.2. The
only thing to complete the proof is to notice that convergence Un ~ u implies that V n ~ V, W n ~ w. This means that operator A-I is continuous on AM. Now we shall give some examples of the correctness sets M representable
as the algebraic sum of compactum and a finite-dimensional subspace. Example 1. Let M be a set of continuously differentiable functions cp( x)
1
such that f[cp' (x )pdx ::; 1. It is easy to see that o
I<;>(x) - <p(0)1 < i<p'(s)ds ~ [!(<P'(X))'dX] 'J' < 1,
l<p(x,) - <pix,)1 ~ VI<P'(x )J'dx]'" lx, - x.1 ~ lx, - x.l· Let us introduce a subset I( C M of the functions ep( x) such that ep(O) =
= 0 . According to Arzela theorem, the set I( is a compacturn in C[O, 1]. The set M consists of the functions cp( x) representable as cp = ep( x )+c and therefore can itself be represented as M =K +£ where £ is one-dimensional subspace.
Example 2. Consider the set offunctions from Lp(p > 1) with variations not exceeding a given constant c. This set we denote as Ve • Let the set o 0
VeC Ve include functions tP( x) from Ve such that tP(O) = O. The set V e is a compactum in Lp (Goncharsky et al., 1979b). The set Ve is representable as
o Ve = K + I:- where K =Ve , and I:- is a one-dimensional space. In a certain sense, the requirement for the correctness set M = K + I:
where K is a compactum, and I:- is a finite-dimensional space, cannot be weakened. D e fin i t ion 2. The set M C Z is ooundedly compact if any bounded
subset of M is a compactum. It is clear that the set M is boundedly compact if it is representable as
M = K + I:- where K is a compactum and I:- is a finite-dimensional space. For the linear completely continuous operators the set M being boundedly compact is a necessary condition for the continuity of A-Ion AM (Ivanov et ai., 1978).
Theorem 2.3 Let M be a convex closed subset of the reflexive space Z, and A be a completely continuous linear one-to-one operator from Z to U. For A - I to be continuous on AM C U, it is necessary that M is boundedly compact.
Proof. Let the opposite be true, i. e. there exists incompact bounded sequence Zn E M. This means that its subsequence Zn' exists that does not converge strongly. Since M is reflexive, we can always assume that subsequence Zn' --+ Z· in a weak sense as n' --+ 00.
The set M is convex and closed, therefore it is weakly closed, and Z· EM. Since A is a completely continuous operator, the sequence Un' = Azn , con verges strongly to Az· = U· in U. We have got that Un' --+ U· and, because of the theorem conditions, A-Iun , = Zn' also converges strongly. This con tradicts the assumption that Zn' does not converge, hence the theorem is proved.
2.3 Quasisolutions of the ill-posed problems
In the previous section we have formulated certain theorems which ensure the continuity of the inverse operator A-Ion the image of correctness set M. These results allow to construct RA for the ill-posed problem (2.1) on the set M. However, the inverse operator A-I cannot be used as a RA because
2.3 Quasisolutions of the ill-posed problems 29
it is not defined throughout the whole U for the conditionally well-posed problems. There exists a relatively simple but sufficiently general technique for the
conditionally well-posed problems; it is known as a quasisolution method. In a sense, it is based on continuing the mapping A-I from the set AM throughout U in such a way that the resulting continuation is continuous everywhere in AM . Let the operator A in Eq. (2.1) be continuous operator from a Banach
space Z to Banach space U, and M be the correctness set. De fin i t ion 3 (Ivanov, 1966, lvanov et al., 1978). A quasisolution of
Eq. (2.1) given on M is the element z; E M which minimizes the discrepancy:
IIAz; - u~lIu =min IIAz - u~lIu. 'EM
Since A is continuous, the functional g(z) = IIAz - u~1I is continuous either. Assume at first that the correctness set M is a compactum in Z. Therefore g(z) attains its exact lowest bound on M for any fixed element u~ E U. It means, in particular, that a quasisolution exists for u~ E U such that Ilu~ - ull ~ 0. Let A : M ~ U be a one-to-one mapping which maps z E Manto u E AM. Let us denote an arbitrary element from the set of quasisolutions as z;. We have formulated a rule z; = R~(u), the operator R~(·) being defined throughout the whole space. It is clear that
Since z~ and z belong to M, and operator A -I is continuous on AM C U, we finally get that liz; - zllz ~ °when 0 ~ 0, i. e. the mapping Rb( u) is a RA for A-Ion M. Operator Rb( u) really depends only on argument u, R~(u) == R(u).This
is in accordance with the results of Theorem 1.1. The conditions of this theorem imply that for RA of the kind R(·) to exist, it is necessary that A-I (defined on AM) could be continued throughout the whole space, the con tinuation being continuous on AM. The RA that we have just constructed plays the role of such a continuation. The question if a quasisolution is unique is of a certain interest. To an
swer it, one can use the well-known theorem on the uniqueness of extremum point of a rigorously convex functional in Banach space. De fin i t ion 4. Functional g(z) defined on a convex set Q of Banach
space Z is rigorously convex if for any A E (0,1) and any Zh Z2 E Q, Zl :I Z2,
the following inequality is true:
(2.2)
(If we substitute sign < by ~, and use A E [0,1], we yield the definition of the convex functional on Q C Z). D e fin i t ion 5. Banach space is rigorously convex (or rigorously
normed) if the equation IIx + yll = IIxll + lIyll is only possible for y= Ax where A is real number. Note that in a rigorously convex Banach space the functional Ilxll is not
rigorously convex. In fact, if X2 = aXl where a is a real parameter, then for any AE (0,1)
However, .the functional IIxll2 is rigorously convex.
Lemma 2.3 In a rigorously convex space functionalllxW is rigorously con vex.
Proof. Suppose the opposite is true, and there exist a· (a· E (0,1», and x, y(x :j; y) such that
Using the triangle inequality and Eq. (2.3), we come to
Let us assume, for instance, that x :j; 0 and x, yare not proportional. Since the space is rigorously convex, the inequality (2.4) is then strict:
Hence (2.5)
However, a· E (0,1) and, consequently, inequality (2.5) is contradictory. Therefore the assumption that x, yare not proportional is false,and x = Ay, A :j; O. In this case Eq. (2.3) leads to
[a· A+ (1- a·W = a·A2 + (1- a·)
it follows that A = 1, or x = y.This contradicts the initial assumption on x, y. The case x = 0 is also impossible because of the conditions a* :f. 0, a* :f. 1.
It is well known that spaces lp, Lp (for p > 1) are rigorously normed, while l}, L 1 , era, b] are not. Any Hilbert space is rigorously normed. For rigorously convex functionals in Banach space the following theorem about attaining extremal points is true (Rocafellar, 1970):
Theorem 2.4 A rigorously convex continuons functional g( z) defined on a convex compact closed set M of Banach space Z attains its exact lowest bound at a unique point z· EM.
The conditions on M may be weakened if the additional restrictions are imposed on the space.
Theorem 2.5 (Rocafellar, 1970). A rigorously convex functional g(z) de fined on a convex bounded doused set M of a reflexive Banach space attains its exact lowest bound at a unique point z* EM.
Now we may return to the problem of obtaining a quasisolution of Eq. (2.1). Obviously, the problem of seaching for quasisolution on the set M may be formulated as a problem of defining the element z* such that functional I\Az-u61lb attains its exact lowest bound at Z·. (The only difference between the latter statement and definition of quasisolution is that the functional I\Az - u611u is replaced by IIAz - u61Ib).
If the space is rigorously normed, and A produces a one-to-one mapping of M on to AM C U, then the functional IJAz - u611b is rigorously convex for any U6 E U. Hence, the following theorem is true:
Theorem 2.6 Let A be a linear continuous one-to-one operator, M be a convex compactum in Z, and U be a rigorously convex Banach space. Then a unique quasisolution of Eq. (2.1) exists for any u E U, and it is a continuous function of u.
Proof. According to theorem conditions, the functional g(z) = IIAz -u6l1b is rigorously convex for any u, and is defined on the closed convex compact set M C Z. The continuity offunctional g(z) immediately follows from continuity of operator A. Therefore Theorem 2.4 is applicable, and g( z) attains its exact lowest bound on M at a unique point. Hence, the unique quasisolution exists for any u E U.
The continuity of quasisolution as a function of u is a trivial consequence of the set M being compact and of the continuity of operator A -1 on the set AMCU.
Rem ark. If the correctness set M is extended to M = K + £- where K is a compactum in Z, and £- is a finite-dimensional space, the analogous result may be proved with the additional requirements imposed on the space U. All the results of Theorem 2.6 are still valid, if U is rigorously convex and reveals E-property (Efimov-Stechkin property), i. e. U is reflexive, and strong convergence Un --. U (n --. (0) follows from the weak convergence (un w~ly u) and convergence of the norms (lIun ll --. Ilull). To find the approximations to quasisolutions on the correctness set M C Z, the well known methods for optimizing functionals may be applied. These methods are best developed for the case of rigorously convex functional g( z) = IIAz -u6l1lr (this is the case, for instanse, when operator A is linear, and U is Hilbert space), and M being a convex compactum in Z. In this case the problem of finding a quasisolution is a problem of convex programming. With the use of various iterative optimization techniques (e. q. conjugate gradients projection, sleepest descent technique, second-order methods, ets.) one can obtain the sequence Zn minimizing functional g( z) on M. Since M is a compactum, the sequence Zn converges to a unique point of minimum of g(z) on M (provided the operator A is injective), i. e. to the quasisolution of Eq. (2.1). All these aspects are concerned with the classical well-developed extremal methods. They are widely reflected in publications (Goncharsky et al., 1979a). We shall not discuss such algorithms in detail here.
By now, the use of additional information about the solution in cases when this information is sufficient to convert the problem to Tikhonoov (conditionally) correct one, has made it possible to create efficient packages of numerical algorithms for solving the wide range of inverse problems in mathematical physics.
Since iterative algorithms applied to problems with domain restrictions usually do not solve the problem of minimizing a functional in a finite (known) number of iterations, the quasisolution may only be found approxi mately. Therefore in the practical numeric algorithms based on the concept of quasisolution, the quasisolution technique is used in a somewhat modified way. The essense of such a modification can be explained as follows. Con sider the ill-posed problem given by Eq. (2.1). Let Z, U be arbitrary Banach spaces, and A produces a continuous one-to-one mapping of a compactum M C Z onto AM cU. Instead of exact value of u = Ai we are given its
approximation U6 E U such that lIu6 - ullu ~ b. We may use the following RA instead of a quazisolution technique: for the given pair (U6' b) the RA puts into correspondence any element Z6 E M such that IIAz - u611u ~ b. Such an element obviously exists because IIAi - w511u ~ 6, i E M. It is clear that IIAz6 - Azllu ~ 26. Since i, Z6 EM, and M is a compactum in Z , then Z6 ~ z when 6 --+ 0, i. e. the described algorithm is a regularizing one. It will be referred to as b-quasisolution technique. The use of such a RA allows to overcome problems concerned with the
uniqueness of a quasisolution and with the impossibility of absolutely accu rate solving the problem of finding a quasisolution. To find the approx imate solution, the above-mentioned techniques of optimizing functional g(z) = IIAz- -u611~ are used, yielding the sequence Zn which minimizes g(z) on M. It is sufficient to continue the process of minimizing g( z) on M until the er ror level of 62 is achieved. It must be mentioned, however, that, contrary to a quasisolution technique, such a RA uses more information about the approximation to a right-hand side of Eq. (2.1): the value of b as well as U6 E U. Solving the problem on a compactum is attractive in one more aspect:
it is possible not only to obtain the approximate solution Z6 EM, but to estimate the deviation of Z6 from i in the metrics of space Z as well.This latter estimate is based on the characteristics of A and M. The approximation error may be estimated with the use of a quasiso
lution technique or its modification descibed above. In fact, if Ai = u, lIu - u611u ~ 6, and Z6 is selected from the set M n {z : IIAz - u611u ~ 6}, then
Therefore, using notation introduced in Chap. 1,
(2.6)
where w(26, A- 1 , AM) is continuity module of operator A- 1 on the set AM. Moreover,
Comparing (2.6) to inequality (1.14) we can see that the described tech niques are quasioptimal on AM if condition (1.21) is satisfied for continuity module of A-1 on AM.
The problem of evaluating the continuity module of the inverse operators on various compacta is well investigated. The most detailed results are available for the case when compactum M is an image of a sphere S = {v E
E V : II v II ~ c} of a reflexive space V, the mapping of a sphere S c V onto the space Z being produced by linear completely continuons operator B, i. e. M = BS. In this case condition (1.21) is obviously satisfied, and methods like quasisolution technique are quasioptimal on AM. So far we have considered the case of exactly known operator A. Now let
in Eq. ( 2.1) both right-hand side u and operator A be given approximately, i. e. instead of A we deal with a family of continuous operators A h such that
Here 'lj;(h, y) is a function continuous with respect to both arguments at h ~ 0, y ~ 0 , monotonically non-decreasing with increasing y, non-negative and satisfying the condition 'lj;(h, y) -> 0 as h -> 0 uniformly for y from a certain segment [0, co]. If operators A, Ah are linear, then IIAhz - Azllu ~
~ hllzll, i. e. 'lj;(h, Ilzl!) = hllzll. Suppose it is known a priori that solution z of Eq. (2.1) belongs to a
compactum M c Z, and operator A produces a one-to-one mapping of M onto AM CU. Denote"., = (6, h) and define M'1 as a set of vectors z E M such that
The set M'1 is not empty for any"., > 0 because it contains the element z EM. Since M is a compactum in a normed space, there exists a constant Co such that
sup IIzl1 ~ Co zEM
Let z'1 be an arbitrary element from M'1' Then
Taking into account properties of the function 'lj;(h, y) and continuity of operator A-l on AM, we yield z'1 -> z when"., -> 0 . Hence, any element z'1 E M'1 may be taken as the approximate solution. It means that on compacta the problem (2.1) with approximately specified operator is solved in a full analogy with the case when operator is known exactly.
An interesting case is when a priori information is only sufficient to se cure the compactness of M within a weak topology. It is well known, for example, that any sphere is a weak compacturn in a reflexive space. Using the same technique, we can construct an approximate solution which con verges to the exact one in a weak topology. In general, a weak topology has no metrics in the whole space. However, using the special procedure to construct approximations, it proves possible to obtain certain estimates of approximation error. We shall now give the idea of such a procedure of constructing approx
{
, N;l < t < 1.
The approximate solution (or, more exactly, its finite-dimensional ana logue) zJ.. (t) is to be found in a form
6 " 6 eiN(t)zN(t) = LJ(U6, 1/JiN)~, ,
where . (t) = {N' t E (iIN,(i + l)IN
e,NO, t ~ (i IN, (i +1)IN.
Functions 1/JtN(t) are obtained as the solutions of the following problems:
IIA*1/JtN - eiNIl = min II Av - eiNl1 IIvll5:1/v'6
The following theorem is true:
Theorem 2.7 (Gaponenko, 1976). Let it be a priori known that Iz(t)1 $ R for any t E [0,1]. Then for the finite-dimensional analogue of the approximate solution zJ..(t) and for that of the exact solution ZN(t) the following estimate of error is true:
Therefore the knowledge of solution's belonging to a weak compactum allows not only to construct approximations to the solution in a weak topol ogy, but as well to give the quantative estimates of approximation error for the finite-dimensional analogues of the solution. It must be understood, however, that for "essentially" ill-posed problems (when A-I is unbounded in its domain), the estimate obtained for any fixed h, in general, tends to infinity as N -+ oo!
2.4 Properties of 8-quasisolutions on the sets with special structure
The quasisolution technique and its modification (h-quasisolution technique) ensure strong convergence in Z of approximations z~ to the exact solution z on the correctness set M . Sometimes more delicate results can be achieved with the use of compact imbedding. This is, as a rule, possible when some more additional information about the exact solution is available. Some examples of such results are to be found in this section. Consider the operator equaton (2.1). Suppose it is known a priori that
the exact solution z ofthe ill-posed problem (2.1) belongs to a set of mono tonic nongrowing bounded functions Z!eC Lp(p > 1). Let also Z = Lp, and operator A produces a one-to-one mapping of Z!e onto AZ!eC U. As usu ally, denote the approximation to u as u~, lIu6 - ull ~ h. Introduce Z!e (h) as the set offunctions z(s) E Z!e such that IIAz - u~llu ~ h. Since Z!e is a compactum in Lp, any choice of z~ from Z!e (h) will lead to convergence
L p z~ -+ z when h -+ O. However, if we are supplied with additional information about the exact
solution, this result can be overridden. For instance, if z(s) is continuous on [a, b], we may yield as much as uniform convergence of approximations to z( s). The approximate solutions in this case still belong to Z!c, i. e. they may be discontinuous.
2·4 Properties of o-quasisolutions 37
Theorem 2.8 Suppose z(s) E C[a,bJ n Zic and [',O'J is an arbitrary seg ment of the interval (a, b). Let also Z6
n E Zic (on) and On ~ 0 as n ~ 00.
Th Cb,u] -() hen Z6n (3) ~ Z S w en n ~ 00.
Proof. At first we shall show that, when n ~ 00, the sequence Z6JS) converges to z(s) pointwise at any point s E (a,b). Select a pointwise con verging subsequence Z6 n' (z) from Z6 n (s); it converges to a certain function z(s) E Zic at any point s E (a,b). The following estimate is valid:
IIAz - Azllu ~ IIAz - AZ6nl Ilu+
+IIAz6 , - U6 I lIu + IIu6 , - ullu ~ Cn' +20n"n n n
where Cn' ~ 0 when n' ~ 00 due to the continuity of operator A from Lp
to U. Hence z~ z. Since z(s) is monotonic, and z( s) is both monotonic and continuous, then
z(s) == z(s) at any s E (a, b). Therefore z( s) = z( s) at any internal point of the segment [a, bJ. Due to the fact that it is always possible to select a converging subsequence from Z6
n (s) , and that all the selected subsequences
converge to the same element z(s) , we obtain that the sequence Z6n (s) itself converges to z(s) at any point of the interval (a,b). Now we are going to prove that the sequence Z6JS) converges uniformly
to z( s) on any closed segment [" O'J C (a, b). Fix some c > O. It follows from the uniform continuity of z( s) on a segment [" aJ that for any c > 0 there exists o(c) > 0 such that for any t h t2 E ["O'J satisfying the condition It1 - t21 < o(c) the inequality Iz(tt} - z(t 2 )1 < c/4 is true. Divide the segment [" a J by the points I = t1 , t2 ••• < tm < a in such a way that Iti+1 - til < 6(c), i = 1,2, ... ,m. Select N(c) so that for any n > N(c) the inequality
is true at any point t i , i = 1,2, ... , m. Then, because of z(t), Z6(t) being monotonic, we yield
IZ6n(t) - z(t)1 < c
at any point t E ["O'J when n> N(c). Rem ark 1. We once again stress that the elements of the set Zic are
not necessarily continuous functions. Nevertheless, sup IZ6n (t) - z(t)1 ~ 0 tE[-r,u)
when 6n ~ O.
Rem ark 2. It is clear that the set Z!c is not a compacturn in C[a, b], i. e. the result of Theorem 2.8 cannot be obtained within a standard approach. This result was achieved due to the compact imbedding and additional a priori information about the solution. Rem ark 3. The results can easily be extended to the case when exact
solution z( s) is a piecewise continuous monotonic function. In this case the convergence'of approximating sequence to exact solution is uniform on any closed segment [J,O'] which does not include points a, b and discontinuities of z(s) (Goncharsky, 1987). Hence, if it is known a priori that exact solution z(s) is a monotonic
bounded function, any element Z6 E Z!c (6) can be taken as the approximate solution. In this case not only convergence in Lp is ensured (the set Z!c is a compactum in Lp ), but also the uniform convergence to the exact solution, as it has been shown above. A similar theorem can be proved for the case when the exact solution
z(s) E f;;, where f;; is the set of convex upwards nonnegative bounded functions: 0 :::; z(s) :::; C. Morever, in this case it is possible to achieve not only the convergence of approximations, but (in a certain sense) the convergence of their derivatives (Goncharsky et al., 1979a). We shall provide one more example that also falls out of the standard
scheme: the case when exact solution is a convex (say, convex upwards) nonnegative function. Let the exact solution of the ill-posed problem (2.1) be a convex upwards
nonnegative function. The set of such functions we denote as Z and the set of functions from Z satisfying condition IIAz - u611u :::; 6 as Z(6). Let Z6 ..
be an arbitrary element from Z(6n ). We shall show that if operator A is a linear one-to-one mapping of Lp onto U, then the sequence of functions Z6 ..
for 6n --+ 0 is uniformly bounded from above. This result means that, when searching for the solution of the ill-posed problem (2.1) on the set of convex functions, there is no need to know the constant for the upper bound of the exact solution.
It is easy to prove the following statement.
~ (a +b)Lemma 2.4 Let z(s) E Z and z(s·) =z -2- :::; 2.
any s E [a, b].
Then z(s) < 4 at
Proof. Suppose the opposite is true. Then there exists point s·· E [a, b]
such that z(s··) > 4. Let, for the sake of clarity, s·· < a ~ b. The value of
2·4 Properties of b-quasisolutions 39
() h . * a +b . I h d' d'z s at t e pomt s =-2- IS greater or equa to t e correspon mg or mate of the point of a straight line connecting points {b,z(b)} and {s**,z(s**)}.
Since z(b) 2: 0, we obtain z ( a ; b) > 2. This contradicts the conditions of
Lemma and therefore proves it. Lemma 2.4 can be formulated in other words:
Lemma 2.5 Let z( s) E Z and there exists point s·· E [a, bj such that
z( s**) > 4. Then z(s*) = z ( a ; b) > 2.
Suppose that the exact solution of Eq. (2.1) z(s) E Z and z(s) ::; 1 for any s E [a, bj.
Theorem 2.9 Let Z6 n be an arbitrary element of Z(bn ). Then there exists a constant c > °such that for any s E [a, bj the condition Z6 n (s) ::; c is met (n = 1,2, ...).
Proof. Since operator A is linear, the set Z(hn ) is convex and, conse quently, the range of functions z(s) E Z(bn ) is convex at any point s E (a, b),
including s = s* =a; b. Suppose that the opposite to the Theorem ~tate
ment is true. Then there exists the sequence of functions Z6 ,(s) E Z(on' ) and the sequence of points Sn' E [a, bj such that Z6 ,(Sn' ) > n' ~ According to Lemma 2.5, Z6
n , (s*) > 2 for n' > 4. n
Note that i( s) E Z(On') for any n', and z( s·) ::; 1. Since the range of functions z(s) E Z(on') at the point s* is convex, for any n' there exists an element z~,(s) E Z(on') such that z~,(s·) = 2. According to Lemma 2.4, all the functions z~,(s) are uniformly bounded, and, due to Helly theorem
on choice, there exist a subsequence Z~/1 and a function z(s) E Z such that at any point s E [a, bj ym Z~/1 (s) = z(s). Since operator A is continuous,
n -00
IIAz - Aillu = 0, and, consequently, z~ z. However, we have obt

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Ill-Posed Problems: Theory and Applications