Ill-Posed Problems: Theory and Applications

✍ Scribed by A. Bakushinsky, A. Goncharsky (auth.)

• Publisher: Springer Netherlands
• Year: 1994
• Tongue: English
• Leaves: 267
• Series: Mathematics and Its Applications 301
• Edition: 1
• Category: Library

✦ Synopsis

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

Front Matter....Pages i-x
General problems of regularizability....Pages 4-22
Regularizing algorithms on compacta....Pages 23-42
Tikhonov’s scheme for constructing regularizing algorithms....Pages 43-72
General technique for constructing linear RA for linear problems in Hilbert space....Pages 73-126
Iterative algorithms for solving non-linear ill-posed problems with monotonic operators. Principle of iterative regularization....Pages 127-163
Applications of the principle of iterative regularization....Pages 164-184
Iterative methods for solving non-linear ill-posed operator equations with non-monotonic operators....Pages 185-198
Application of regularizing algorithms to solving practical problems....Pages 199-241
Back Matter....Pages 242-258

✦ Subjects

Computational Mathematics and Numerical Analysis; Math. Applications in Chemistry; Earth Sciences, general; Optimization; Functional Analysis