an:05685520
Zbl 1197.65054
Hansen, Per Christian
Discrete inverse problems. Insight and algorithms.
EN
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).
2010
b
65J10 65-02 45B05 47A52 65F22 65J20 65J22 65R30 65R32
linear ill-posed problem; Fredholm integral equation of the first kind; regularization method; Tikhonov regularization; singular value expansion; singular value decomposition; truncated singular value decomposition; stochastic noise; iterative method; Landweber method; Krylov subspace method; conjugate gradient method; a posteriori parameter choice; discrepancy principle; L-curve criterion; generalized cross validation; image deblurring; tomography; depth profiling; gravity surveying; linear inverse problems; textbook; quadrature methods; Galerkin methods; Kaczmarz's method; numerical experiments
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 \texttt{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.
Robert Plato (Siegen)