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Discrete inverse problems. Insight and algorithms. (English) Zbl 1197.65054
Fundamentals of Algorithms 7. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-96-2/pbk; 978-0-89871-883-6/ebook). xii, 213 p. (2010).
This textbook provides an introduction to linear inverse problems, with a focus on basic mathematical and computational aspects. The presentation starts with a summary of the most important properties of linear Fredholm integral equations of the first kind, with the singular value expansion of the kernel function as a basic tool. Then discretization methods are discussed, specifically quadrature methods and Galerkin methods, followed by an introduction of the singular value decomposition of a matrix and its relations to the singular value expansion. The next chapter is devoted to the regularization of discrete (i.e., finite-dimensional) linear inverse problems. This includes the truncated singular value decomposition of a matrix and Tikhonov regularization, and different forms of stochastic noise are considered. It is followed by a chapter on methods for choosing the regularization parameter, e.g., the discrepancy principle, generalized cross validation and the L-curve criterion. The next chapter deals with iterative methods for discrete inverse problems, including Landweber iteration, Kaczmarz’s method and Krylov subspace methods. A discussion of some real-world problems follows, including image deblurring, 2D tomography, depth profiling and 2D gravity surveying. The text concludes with a chapter on generalized smoothing terms for Tikhonov regularization.
The book assumes only a basic knowledge of calculus, linear algebra and functional analysis. It includes a number of tutorial exercises involving numerical experiments with the MATLAB package ‘Regularization Tools’, and numerous graphical illustrations are presented. Each chapter contains a comparison of the considered methods.
This carefully written textbook provides a very readable survey for graduate students, researchers and professionals in engineering and other areas that depend on solving inverse problems. It certainly will be appreciated by the reader.

MSC:
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45B05 Fredholm integral equations
47A52 Linear operators and ill-posed problems, regularization
65F22 Ill-posedness and regularization problems in numerical linear algebra
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
65J22 Numerical solution to inverse problems in abstract spaces
65R30 Numerical methods for ill-posed problems for integral equations
65R32 Numerical methods for inverse problems for integral equations
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