Ill-posed problems : theory and applications
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Bibliographic Information
Ill-posed problems : theory and applications
（Mathematics and its applications, 301）
Kluwer Academic Publishers, c1994
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Includes bibliographical references and index
"This monograph is a new and original work based on two books by the same authors previously published in Russian: Iterative Methods for Solving Ill-Posed Problems, Moscow, Nauka c1989, and Ill-Posed Problems, Numerical Methods and Its Applications, Moscow, Moscow State University Press c1989" -- T.p. verso
Description and Table of Contents
Description
Recent years have been characterized by the increasing amountofpublications in the field ofso-called ill-posed problems. This is easilyunderstandable because we observe the rapid progress of a relatively young branch ofmathematics, ofwhich the first results date back to about 30 years ago. By now, impressive results have been achieved both in the theory ofsolving ill-posed problems and in the applicationsofalgorithms 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 ofconstructing 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.
Table of Contents
Preface. Introduction. 1. General problems of regularizability. 2. Regularizing algorithms on compacta. 3. Tikhonov's scheme for constructing regularizing algorithms. 4. General technique for constructing linear RA for linear problems in Hilbert space. 5. Iterative algorithms for solving nonlinear ill-posed problems with monotonic operators. Principle of iterative regularization. 6. Applications of the principle of iterative regularization. 7. Iterative methods for solving nonlinear ill-posed operator equations with non-monotonic operators. 8. Application of regularizing algorithms to solving practical problems. Bibliography. Index.
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