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Rank-deficient and discrete ill-posed problems : numerical aspects of linear inversion / Per Christian Hansen.
- Format:
- Book
- Author/Creator:
- Hansen, Per Christian.
- Series:
- SIAM monographs on mathematical modeling and computation ; 4.
- SIAM monographs on mathematical modeling and computation
- SIAM monographs on mathematical modeling and computation ; 4
- Language:
- English
- Subjects (All):
- Equations, Simultaneous--Numerical solutions.
- Equations, Simultaneous.
- Iterative methods (Mathematics).
- Sparse matrices.
- Physical Description:
- 1 electronic text (xvi, 247 p.) : ill., digital file.
- Place of Publication:
- Philadelphia, Pa. : Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 1998.
- Language Note:
- English
- System Details:
- Mode of access: World Wide Web.
- System requirements: Adobe Acrobat Reader.
- Summary:
- 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.
- Contents:
- 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.
- Notes:
- Title from title screen.
- Includes bibliographical references (p. 215-242) and index.
- Digitized and made available by: Books24x7.com.
- Title from title screen, viewed 12/30/2010.
- ISBN:
- 0-89871-969-0
- Publisher Number:
- MM04 SIAM
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