Rank-Deficient and Discrete III-Posed Problems - Per Christian Hansen

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Résumé :

• 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–...

  Biographie:
  Presents 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....

  Sommaire:
  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.
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