Book,
Solutions of ill-posed problems
V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, (1977)Translated from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.
Abstract
MR: Let F and U be metric spaces. The problem of determining a "solution'' z∈F from the "initial data'' u∈U is said to be wellposed with respect to the pair of spaces (F,U) if for each u∈U there exists a solution z∈F; the solution is unique; and the solution is stable relative to small changes in the initial data. Problems that do not satisfy these requirements are said to be ill-posed.
The notion of a well-posed mathematical problem made its debut with discussions in Chapter I of J. Hadamard's book Lectures on Cauchy's problem in linear partial differential equations, Yale Univ. Press, New Haven, Conn., 1923; Jbuch 49, 725. Hadamard observed: "But it is remarkable, on the other hand, that a sure guide is found in physical interpretation: an analytic problem being well-posed, in our use of the phrase, when it is the translation of some mechanical or physical question.'' For a long time, it was an accepted point of view in the mathematical literature that ill-posed problems cannot describe real phenomena and objects. It is now recognized that this attitude about ill-posed problems is erroneous, and that "the majority of applied problems are, and always have been, ill-posed, particularly when they require numerical answers''.
A. N. Tihonov Dokl. Akad. Nauk SSSR 151 (1963), 501--504; MR0162377 (28 #5576); ibid. 153 (1963), 49--52; MR0162378 (28 #5577) was one of the earliest workers in the field of ill-posed problems who succeeded in giving a precise mathematical definition of äpproximations'' for general classes of such problems, and in constructing öptimal'' solutions. This monograph is a survey of the main thrust of contributions by Soviet mathematicians to ill-posed problems, with emphasis on extensions and elaborations of ideas which were originated by Tihonov and his school.
The authors give many examples from analysis, physics, and engineering that lead to ill-posed problems, for example, the Cauchy problem for the Laplace equation, integral equations of the first kind, the problem of differentiation of a function that is known only approximately, summation of Fourier series with approximate coefficients, analytic continuation of functions, design of optimal control systems, automatic processing of observational data, inverse problems in geophysics, synthesis problems, etc. Many of these problems can be formulated in the form (∗) Az=u, where A is a completely continuous operator from F to U. If the initial data in such problems are known only approximately and contain a random error, then the instability of their solutions leads to nonuniqueness of the classically-derived approximate solutions and to serious difficulties in their physical interpretation. Also, in many cases there simply is no classical solution of problems with approximate initial data. In Chapter 1, the authors define a concept of an approximate solution that is stable to small changes in the initial data, and use a selection method for deriving this solution. This is based on additional information that restricts the set of all possible solutions to a compact set M. The authors also discuss quasi-solutions of (∗), i.e., zˆ∈M minimizing the functional d(Az,u) over M, where d is a metric.
One feature (indeed the central theme) of this book is the development (in Chapter II) of the regularization method in the construction of approximate solutions of ill-posed problems that was first expounded by Tihonov op. cit., when the set of all possible solutions is not necessarily compact. An operator R(u,α) depending on a parameter α is called a regularizing operator for the equation (∗) if (i) R(u,α) is defined for every α>0, u∈U and is continuous in u; (ii) if Az˜=u˜, then there is α(δ) such that, for any ε>0, there is δ(ε) for which the condition d(uδ,u˜)≤δ(ε) implies that d(z˜,zδ)≤ε, where zδ=R(uδ,α), and α=α(δ). The usefulness of this notion resides in the fact that if d(u˜,uδ)≤δ, then one can take for an approximate solution of (∗) with approximately known data uδ the element zα=R(uδ,α) obtained with the aid of the regularization operator. Several methods are developed for the choice of the regularization parameter α and for the construction of regularization operators by variational methods.
Chapter III is a brief exposition on the solution of singular and ill-conditioned systems of linear algebraic equations by the regularization method. Approximate regularized solutions of integral equations of the first kind of convolution type are discussed in Chapter IV. In Chapter V the authors develop certain optimal regularizing operators for integral operators for integral equations of the convolution type, and discuss for this case the connection between the regularization method and optimal Wiener filtering. Regularization methods are used in Chapter VI to provide stable methods of summing Fourier series with coefficients that are approximate in the l2 metric. Stable methods of minimizing functionals and solving optimal control problems, and of solving ill-posed optimal planning (linear programming) problems, are treated in Chapters VII and VIII, respectively.
The initial data underlying ill-posed problems (generally measurements) contain random errors. Depending on the nature of this initial information, one can take either a deterministic or a probabilistic approach to the approximate solutions. The authors have generally (except for parts of Chapters IV and V) confined themselves to the deterministic approach. The authors "have not attempted a survey of the literature on ill-posed problems. Therefore, the bibliography does not pretend to be complete''. The bibliography contains 221 references, of which nearly 190 are by Soviet authors; the remaining references for the most part are not cited explicitly in the exposition.
In the Translation Editor's Preface, he concludes: "The present translation should be of special interest to mathematicians and scientists concerned with the numerical solution of applied problems.... But not just to those. Past experience suggests that the concepts and methods used in the discussion of ill-posed problems will in turn stimulate advances in `pure' mathematical analysis.'' For the latter purpose this book provides stimulating concepts and methods that have no nontrivial counterpart in mathematical analysis of well-posed problems. As far as methods of constructing solutions that are easily processed on a computer, this monograph represents the history, and not the future of numerical analysis of ill-posed problems (it does not survey recent advances in other methods, e.g., iterative, projectional and generalized inverse methods, etc.). But then it is the history articulated and coauthored by a pioneering contributor to regularization methods. It is also the first book in English which is fully devoted to ill-posed problems in the framework of operator and integral equations.
