Rank Deficient and Discrete Ill-Posed Problems
Numerical Aspects of Linear Inversion
Will be published approx. on 31. December 1997
Book
Paperback/Softback
263 pages
978-0-89871-403-6 (ISBN)
Description
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.
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.
More details
Series
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Product notice
Paperback (trade)
Unsewn / adhesive bound
Dimensions
Height: 255 mm
Width: 179 mm
Thickness: 13 mm
Weight
514 gr
ISBN-13
978-0-89871-403-6 (9780898714036)
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Schweitzer Classification
Content
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.
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.