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Regularization methods in Banach spaces. (English) Zbl 1259.65087
Radon Series on Computational and Applied Mathematics 10. Berlin: de Gruyter (ISBN 978-3-11-025524-9/hbk; 978-3-11-025572-0/ebook). xi, 283 p. (2012).
The preface of the book contains a very clear summary, so – with a very few technical modifications – it is taken over here. At the end of this review some additional comments are given.
The book consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically require a Banach space setting and giving a brief glimpse of sparsity constraints (Chapter 1). Part II summarizes all mathematical tools necessary to carry out an analysis in Banach spaces, such as some facts on convex analysis, duality mappings and Bregman distances (Chapter 2). Part II furthermore includes a chapter on ill-posed operator equations and regularization theory in Banach spaces (Chapter 3), which also introduces the reader to modern ingredients of smoothness analysis for ill-posed problems like approximate source conditions and variational inequalities. In view of solution methods for inverse problems, the authors distinguish between Tikhonov-type or variational regularization methods (Part III), iterative techniques (Part IV) and the method of approximate inverse (Part V). Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. After stating the theory, including error estimates and convergence rates for general convex penalty terms and nonlinear problems (Chapter 4), the authors specifically address linear problems and power-type penalty terms, propose parameter choice rules, and present methods for solving the resulting minimization problems (Chapter 5). Part IV, dealing with iterative regularization methods, is divided into two chapters. The first one is concerned with linear operator equations and contains the Landweber method as well as numerically accelerated sequential subspace methods and the general framework of split feasibility problems (Chapter 6), and the second one deals with the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauss-Newton method (Chapter 7). Finally, Part V outlines the method of approximate inverse, which is based on the efficient evaluation of the measured data with reconstruction kernels. After a brief introduction to the method (Chapter 8), its regularization properties are investigated, and convergence rates are presented in $$L^p$$-spaces as well as in the space of continuous functions on a compact set (Chapter 9). The application of these results to the problem of X-ray diffractometry concludes the work (Chapter 10).
This concludes the summary which has basically taken from the preface of the book, and some additional comments are given next. This monograph provides a very readable introduction to the regularization of linear and nonlinear ill-posed problems in Banach spaces. The only mathematical background required to read the text is functional analysis. Many applications are presented, including graphical illustrations and numerical results. This is an excellent book which should appeal to any experts as well as graduate students interested in ill-posed problems.

##### MSC:
 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 47A52 Linear operators and ill-posed problems, regularization 47J06 Nonlinear ill-posed problems 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities 65J22 Numerical solution to inverse problems in abstract spaces 65J10 Numerical solutions to equations with linear operators 65J15 Numerical solutions to equations with nonlinear operators 44A12 Radon transform 65R30 Numerical methods for ill-posed problems for integral equations 44A10 Laplace transform 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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