zbMATH — the first resource for mathematics

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.

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
Full Text: DOI