Rank-Deficient and Discrete Ill-Posed Problems

Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, seismology, astronomy, and other areas. Rank-deficient problems involve matrices that are 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 given measurements. Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about 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 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 the efficiency and reliability of the computations.

Spis treści:
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.