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.
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.
1. Preface 2. Symbols and Acronyms 3. Chapter 1: Setting the Stage. Problems With Ill-Conditioned Matrices 4. Ill-Posed and Inverse Problems 5. Prelude to Regularization 6. Four Test Problems 7. Chapter 2: Decompositions and Other Tools. The SVD and its Generalizations 8. Rank-Revealing Decompositions 9. Transformation to Standard Form 10. Computation of the SVE 11. Chapter 3: Methods for Rank-Deficient Problems. Numerical Rank 12. Truncated SVD and GSVD 13. Truncated Rank-Revealing Decompositions 14. Truncated Decompositions in Action 15. Chapter 4. Problems with Ill-Determined Rank. Characteristics of Discrete Ill-Posed Problems 16. Filter Factors 17. Working with Seminorms 18. The Resolution Matrix, Bias, and Variance 19. The Discrete Picard Condition 20. L-Curve Analysis 21. Random Test Matrices for Regularization Methods 22. The Analysis Tools in Action 23. Chapter 5: Direct Regularization Methods. Tikhonov Regularization 24. The Regularized General Gauss–Markov Linear Model 25. Truncated SVD and GSVD Again 26. Algorithms Based on Total Least Squares 27. Mollifier Methods 28. Other Direct Methods 29. Characterization of Regularization Methods 30. Direct Regularization Methods in Action 31. Chapter 6: Iterative Regularization Methods. Some Practicalities 32. Classical Stationary Iterative Methods 33. Regularizing CG Iterations 34. Convergence Properties of Regularizing CG Iterations 35. The LSQR Algorithm in Finite Precision 36. Hybrid Methods 37. Iterative Regularization Methods in Action 38. Chapter 7: Parameter-Choice Methods. Pragmatic Parameter Choice 39. The Discrepancy Principle 40. Methods Based on Error Estimation 41. Generalized Cross-Validation 42. The L-Curve Criterion 43. Parameter-Choice Methods in Action 44. Experimental Comparisons of the Methods 45. Chapter 8. Regularization Tools 46. Bibliography 47. Index.
1. Preface 2. Symbols and Acronyms 3. Chapter 1: Setting the Stage. Problems With Ill-Conditioned Matrices 4. Ill-Posed and Inverse Problems 5. Prelude to Regularization 6. Four Test Problems 7. Chapter 2: Decompositions and Other Tools. The SVD and its Generalizations 8. Rank-Revealing Decompositions 9. Transformation to Standard Form 10. Computation of the SVE 11. Chapter 3: Methods for Rank-Deficient Problems. Numerical Rank 12. Truncated SVD and GSVD 13. Truncated Rank-Revealing Decompositions 14. Truncated Decompositions in Action 15. Chapter 4. Problems with Ill-Determined Rank. Characteristics of Discrete Ill-Posed Problems 16. Filter Factors 17. Working with Seminorms 18. The Resolution Matrix, Bias, and Variance 19. The Discrete Picard Condition 20. L-Curve Analysis 21. Random Test Matrices for Regularization Methods 22. The Analysis Tools in Action 23. Chapter 5: Direct Regularization Methods. Tikhonov Regularization 24. The Regularized General Gauss–Markov Linear Model 25. Truncated SVD and GSVD Again 26. Algorithms Based on Total Least Squares 27. Mollifier Methods 28. Other Direct Methods 29. Characterization of Regularization Methods 30. Direct Regularization Methods in Action 31. Chapter 6: Iterative Regularization Methods. Some Practicalities 32. Classical Stationary Iterative Methods 33. Regularizing CG Iterations 34. Convergence Properties of Regularizing CG Iterations 35. The LSQR Algorithm in Finite Precision 36. Hybrid Methods 37. Iterative Regularization Methods in Action 38. Chapter 7: Parameter-Choice Methods. Pragmatic Parameter Choice 39. The Discrepancy Principle 40. Methods Based on Error Estimation 41. Generalized Cross-Validation 42. The L-Curve Criterion 43. Parameter-Choice Methods in Action 44. Experimental Comparisons of the Methods 45. Chapter 8. Regularization Tools 46. Bibliography 47. Index.
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