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

作者：Per Christian Hansen · 发表于：Medical Entomology and Zoology · 年份：1987 · 被引用次数：2145 · 研究领域：Statistical and numerical algorithms、Numerical methods in inverse problems

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 P...