# Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion

## Per Christian Hansen

SIAM Monographs on Mathematical Modeling and Computation 4

Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas.

Rank-deficient problems involve matrices that are either exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of the given measurements.

Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about some interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data.

This book describes, in a common framework, new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and on the efficiency and reliability of the computations. The setting is that of numerical linear algebra rather than abstract functional analysis, and the theoretical development is complemented with numerical examples and figures that illustrate the features of the various algorithms.

### Audience

Written with the nonexpert in mind, this book will appeal to applied mathematicians as well as other scientists and engineers interested in numerical methods dealing with rank-deficient and discrete ill-posed problems. It is assumed that the reader is familiar with the underlying theory of inverse problems and has some background in numerical linear algebra and matrix computations. The book can be used in an advanced course on inversion methods. It "zooms in" on section 5.3 of Numerical Methods for Least Squares Problems (Björck, SIAM, 1996) and provides the computational background needed to supplement A Primer on Integral Equations of the First Kind (Wing and Zahrt, SIAM, 1991).

### Contents

Preface; Symbols and Acronyms; Chapter 1: Setting the Stage. Problems With Ill-Conditioned Matrices; Ill-Posed and Inverse Problems; Prelude to Regularization; Four Test Problems; Chapter 2: Decompositions and Other Tools. The SVD and its Generalizations; Rank-Revealing Decompositions; Transformation to Standard Form; Computation of the SVE; Chapter 3: Methods for Rank-Deficient Problems. Numerical Rank; Truncated SVD and GSVD; Truncated Rank-Revealing Decompositions; Truncated Decompositions in Action; Chapter 4. Problems with Ill-Determined Rank. Characteristics of Discrete Ill-Posed Problems; Filter Factors; Working with Seminorms; The Resolution Matrix, Bias, and Variance; The Discrete Picard Condition; L-Curve Analysis; Random Test Matrices for Regularization Methods; The Analysis Tools in Action; Chapter 5: Direct Regularization Methods. Tikhonov Regularization; The Regularized General Gauss–Markov Linear Model; Truncated SVD and GSVD Again; Algorithms Based on Total Least Squares; Mollifier Methods; Other Direct Methods; Characterization of Regularization Methods; Direct Regularization Methods in Action; Chapter 6: Iterative Regularization Methods. Some Practicalities; Classical Stationary Iterative Methods; Regularizing CG Iterations; Convergence Properties of Regularizing CG Iterations; The LSQR Algorithm in Finite Precision; Hybrid Methods; Iterative Regularization Methods in Action; Chapter 7: Parameter-Choice Methods. Pragmatic Parameter Choice; The Discrepancy Principle; Methods Based on Error Estimation; Generalized Cross-Validation; The L-Curve Criterion; Parameter-Choice Methods in Action; Experimental Comparisons of the Methods; Chapter 8. Regularization Tools; Bibliography; Index.

### About the Author

Per Christian Hansen is Professor of Scientific Computing at the Technical University of Denmark. He has published more than 40 scientific papers and written the matlab package Regularization Tools that complements this book.

1997 / xvi + 247 pages / Softcover / ISBN 0-89871-403-6
List Price \$55.50 / SIAM Member Price \$38.85 / Order Code MM04