Rank-deficient and discrete ill-posed problems numerical aspects of linear inversion - Hansen, Per Christian 1957- (Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1998)
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Includes bibliographical references (s. 215-242) and index
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
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
Keywords: Equations, Simultaneous / Numerical solutions; Iterative methods (Mathematics); Sparse matrices; Numerische lineare Algebra; Inkorrekt gestelltes Problem; Numerisches Verfahren