Computational methods for inverse problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
2002
|
| Series: | Frontiers in applied mathematics
23 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XVI, 183 S. graph. Darst. |
| ISBN: | 9780898715507 |
Staff View
MARC
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| 245 | 1 | 0 | |a Computational methods for inverse problems |c Curtis R. Vogel |
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| 490 | 1 | |a Frontiers in applied mathematics |v 23 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Inverse problems (Differential equations) |x Numerical solutions | |
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Record in the Search Index
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| adam_text | Contents
Foreword xiii
Preface xv
1 Introduction 1
1.1 An Illustrative Example 1
1.2 Regularization by Filtering 2
1.2.1 A Deterministic Error Analysis 6
1.2.2 Rates of Convergence 7
1.2.3 A Posteriori Regularization Parameter Selection 8
1.3 Variational Regularization Methods 9
1.4 Iterative Regularization Methods 10
Exercises 11
2 Analytical Tools 13
2.1 Ill Posedness and Regularization 16
2.1.1 Compact Operators, Singular Systems, and the SVD ... 17
2.1.2 Least Squares Solutions and the Pseudo Inverse 18
2.2 Regularization Theory 19
2.3 Optimization Theory 20
2.4 Generalized Tikhonov Regularization 24
2.4.1 Penalty Functionals 24
2.4.2 Data Discrepancy Functionals 25
2.4.3 Some Analysis 26
Exercises 27
3 Numerical Optimization Tools 29
3.1 The Steepest Descent Method 30
3.2 The Conjugate Gradient Method 31
3.2.1 Preconditioning 33
3.2.2 Nonlinear CG Method 34
3.3 Newton s Method 34
3.3.1 Trust Region Globalization of Newton s Method 35
3.3.2 The BFGS Method 36
3.4 Inexact Line Search 36
Exercises 39
ix
x Contents
4 Statistical Estimation Theory 41
4.1 Preliminary Definitions and Notation 41
4.2 Maximum Likelihood Estimation 46
4.3 Bayesian Estimation 46
4.4 Linear Least Squares Estimation 50
4.4.1 Best Linear Unbiased Estimation 50
4.4.2 Minimum Variance Linear Estimation 52
4.5 The EM Algorithm 53
4.5.1 An Illustrative Example 54
Exercises 57
5 Image Deblurring 59
5.1 A Mathematical Model for Image Blurring 59
5.1.1 A Two Dimensional Test Problem 61
5.2 Computational Methods for Toeplitz Systems 63
5.2.1 Discrete Fourier Transform and Convolution 64
5.2.2 The FFT Algorithm 66
5.2.3 Toeplitz and Circulant Matrices 68
5.2.4 Best Circulant Approximation 70
5.2.5 Block Toeplitz and Block Circulant Matrices 71
5.3 Fourier Based Deblurring Methods 74
5.3.1 Direct Fourier Inversion 75
5.3.2 CG for Block Toeplitz Systems 76
5.3.3 Block Circulant Preconditioners 78
5.3.4 A Comparison of Block Circulant Preconditioners .... 81
5.4 Multilevel Techniques 82
Exercises 83
6 Parameter Identification 85
6.1 An Abstract Framework 86
6.1.1 Gradient Computations 87
6.1.2 Adjoint, or Costate, Methods 88
6.1.3 Hessian Computations 89
6.1.4 Gauss Newton Hessian Approximation 89
6.2 A One Dimensional Example 89
6.3 A Convergence Result 93
Exercises 95
7 Regularization Parameter Selection Methods 91
7.1 The Unbiased Predictive Risk Estimator Method 98
7.1.1 Implementation of the UPRE Method 100
7.1.2 Randomized Trace Estimation 101
7.1.3 A Numerical Illustration of Trace Estimation 101
7.1.4 Nonlinear Variants of UPRE 103
7.2 Generalized Cross Validation 103
7.2.1 A Numerical Comparison of UPRE and GCV 103
7.3 The Discrepancy Principle 104
7.3.1 Implementation of the Discrepancy Principle 105
7.4 The L Curve Method 106
Contents xi
7.4.1 A Numerical Illustration of the L Curve Method 107
7.5 Other Regularization Parameter Selection Methods 107
7.6 Analysis of Regularization Parameter Selection Methods 109
7.6.1 Model Assumptions and Preliminary Results 109
7.6.2 Estimation and Predictive Errors for TSVD 114
7.6.3 Estimation and Predictive Errors for Tikhonov Regular¬
ization 116
7.6.4 Analysis of the Discrepancy Principle 121
7.6.5 Analysis of GCV 122
7.6.6 Analysis of the L Curve Method 124
7.7 A Comparison of Methods 125
Exercises 126
8 Total Variation Regularization 129
8.1 Motivation 129
8.2 Numerical Methods for Total Variation 130
8.2.1 A One Dimensional Discretization 131
8.2.2 A Two Dimensional Discretization 133
8.2.3 Steepest Descent and Newton s Method for Total Variation 134
8.2.4 Lagged Diffusivity Fixed Point Iteration 135
8.2.5 A Primal Dual Newton Method 136
8.2.6 Other Methods 141
8.3 Numerical Comparisons 142
8.3.1 Results for a One Dimensional Test Problem 142
8.3.2 Two Dimensional Test Results 144
8.4 Mathematical Analysis of Total Variation 145
8.4.1 Approximations to the TV Functional 148
Exercises 149
9 Nonnegativity Constraints 151
9.1 An Illustrative Example 151
9.2 Theory of Constrained Optimization 154
9.2.1 Nonnegativity Constraints 156
9.3 Numerical Methods for Nonnegatively Constrained Minimization ... 157
9.3.1 The Gradient Projection Method 157
9.3.2 A Projected Newton Method 158
9.3.3 A Gradient Projection Reduced Newton Method 159
9.3.4 A Gradient Projection CG Method 161
9.3.5 Other Methods 162
9.4 Numerical Test Results 162
9.4.1 Results for One Dimensional Test Problems 162
9.4.2 Results for a Two Dimensional Test Problem 164
9.5 Iterative Nonnegative Regularization Methods 165
9.5.1 Richardson Lucy Iteration 165
9.5.2 A Modified Steepest Descent Algorithm 166
Exercises 170
Bibliography 173
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:27Z |
| institution | BVB |
| isbn | 9780898715507 |
| language | English |
| lccn | 2002022386 |
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| owner_facet | DE-29T DE-20 DE-91G DE-BY-TUM |
| physical | XVI, 183 S. graph. Darst. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Frontiers in applied mathematics |
| series2 | Frontiers in applied mathematics |
| spelling | Vogel, Curtis R. Verfasser (DE-588)120340896X aut Computational methods for inverse problems Curtis R. Vogel Philadelphia Society for Industrial and Applied Mathematics 2002 XVI, 183 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Frontiers in applied mathematics 23 Includes bibliographical references and index Inverse problems (Differential equations) Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Inverses Problem (DE-588)4125161-1 gnd rswk-swf Inverses Problem (DE-588)4125161-1 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Frontiers in applied mathematics 23 (DE-604)BV001873790 23 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009856412&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vogel, Curtis R. Computational methods for inverse problems Frontiers in applied mathematics Inverse problems (Differential equations) Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Inverses Problem (DE-588)4125161-1 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4125161-1 |
| title | Computational methods for inverse problems |
| title_auth | Computational methods for inverse problems |
| title_exact_search | Computational methods for inverse problems |
| title_full | Computational methods for inverse problems Curtis R. Vogel |
| title_fullStr | Computational methods for inverse problems Curtis R. Vogel |
| title_full_unstemmed | Computational methods for inverse problems Curtis R. Vogel |
| title_short | Computational methods for inverse problems |
| title_sort | computational methods for inverse problems |
| topic | Inverse problems (Differential equations) Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Inverses Problem (DE-588)4125161-1 gnd |
| topic_facet | Inverse problems (Differential equations) Numerical solutions Numerisches Verfahren Inverses Problem |
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