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Numerical methods for the solution of ill-posed problems. Rev., updated and transl. from the Russ. by R. A. M. Hoksbergen. (English) Zbl 0831.65059

Mathematics and its Applications (Dordrecht). 328. Dordrecht: Kluwer Academic Publishers. ix, 253 p. Dfl. 185.00; $ 137.00; £85.00 (1995).
[For the Russian original (1990) see Zbl 0712.65042.]
This book can be considered as a classical monograph on methods for solving ill-posed problems. The authors consider linear ill-posed problems with or without a priori constraints (such as non-negativity, monotonicity, convexity, etc.).
In Chapter 1, basic material from the theory of regularization is collected. The authors describe the circle of problems that are Tikhonov regularizable and propose constructive methods for obtaining regularizing algorithms. They also give a detailed discussion of the a priori schemes for choosing the regularization parameter and a detailed account of methods for solving incompatible equations. Chapter 2 contains a course on numerical methods for the approximate solution of ill-posed problems on compacts. In Chapter 3 the authors study in detail the numerical aspects of constructing efficient regularizing algorithms on special sets. The description of a program library for solving ill-posed problems is given in Chapter 4. This library includes:
a) various versions for solving linear integral equations of the first kind;
b) special programs for solving convolution type one- and two-dimensional integral equations of the first kind;
c) various versions for solving one-dimensional Fredholm integral equations of the first kind on the set of monotone, convex functions and on the special sets of functions.
Each program is accompanied by test examples. The Appendix contains the listing of nine Fortran programs. In Postscript the authors give the contents of the main monographs on ill-posed problems that have appeared after 1990. The list of references contains 220 items, 106 of which are written in Russian.
This book will be useful to postgraduate students of physics, mathematics, chemistry, etc., and to engineers and scientists who are interested in data processing and the theory of ill-posed problems.
Reviewer: K.Najzar (Praha)

MSC:

65J10 Numerical solutions to equations with linear operators
65R20 Numerical methods for integral equations
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65R30 Numerical methods for ill-posed problems for integral equations
45-04 Software, source code, etc. for problems pertaining to integral equations
45B05 Fredholm integral equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47A50 Equations and inequalities involving linear operators, with vector unknowns

Citations:

Zbl 0712.65042
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