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

TLDR: In this article, the authors present a survey of regularization tools for rank-deficient problems and problems with ill-conditioned and inverse problems, as well as a comparison of the methods in action.

Abstract: Preface Symbols and Acronyms 1. Setting the Stage. Problems With Ill-Conditioned Matrices Ill-Posed and Inverse Problems Prelude to Regularization Four Test Problems 2. Decompositions and Other Tools. The SVD and its Generalizations Rank-Revealing Decompositions Transformation to Standard Form Computation of the SVE 3. Methods for Rank-Deficient Problems. Numerical Rank Truncated SVD and GSVD Truncated Rank-Revealing Decompositions Truncated Decompositions in Action 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 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 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 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 8. Regularization Tools Bibliography Index.