An Introduction to Inverse Problems with Applications:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer
2013
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 243 S. Ill., graph. Darst. |
| ISBN: | 9783642325564 |
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MARC
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| 245 | 1 | 0 | |a An Introduction to Inverse Problems with Applications |c Francisco Duarte Moura Neto ; Antônio José da Silva Neto |
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Record in the Search Index
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| adam_text | Titel: An introduction to inverse problems with applications
Autor: Moura Neto, Francisco Duarte
Jahr: 2013
Contents
Foreword VII
Preface XI
Acknowledgements XV
0 Road Map 1
0.1 Inverse Problems in a Snapshot.................... 1
0.2 Applied Inverse Problems....................... 1
0.3 Crossing the Science-Technology Gap................ 3
0.4 Chapter by Chapter.......................... 4
1 Mathematical Modeling 7
1.1 Models ................................ 7
1.2 Observing Reality........................... 8
1.3 The Art of Idealization: Model Characterization........... 10
1.4 Resolution of the Idealization: Mathematics............. 13
1.5 Idealized Data Processing: Solution Operator............ 17
1.6 Mind the Gap: Least Squares..................... 17
1.7 Optimal Solution........................... 21
1.8 A Suboptimal Solution........................ 22
1.9 Application to Darcy s Law...................... 22
Exercises.................................. 23
2 Fundamental Concepts in Inverse Problems 27
2.1 Condition of Function Evaluation................... 28
2.2 Condition as a Derivative Bound................... 29
2.3 Other Derivatives and Other Notions of Condition.......... 29
2.4 Condition of Vector-Valued Functions of Several Variables..... 30
2.5 Stability of Algorithms of Function Evaluation ........... 33
2.6 Questions on Existence and Uniqueness............... 37
2.6.1 Exact Data.......................... 37
2.6.2 Real Data........................... 38
2.7 Well-Posed Problems......................... 43
2.8 Classification of Inverse Problems.................. 44
2.8.1 Classes of Mathematical Models............... 44
XX Contents
2.8.2 Classes of Inverse Problems................. 46
Exercises .................................. 48
3 Spectral Analysis of an Inverse Problem 53
3.1 An Example.............................. 53
3.2 Uncertainty Multiplication Factor .................. 55
3.3 Regularization Scheme........................ 56
3.4 Tikhonov s Regularization...................... 58
3.5 Regularization Strategy........................ 61
3.6 Reference Value in Tikhonov s Method ............... 63
3.7 Steepest Descent Method....................... 65
3.8 Landweber Method.......................... 66
3.9 Discrepancy Principle......................... 69
3.10 Conjugate Gradient.......................... 70
3.10.1 Conjugate Gradient Algorithm................ 70
3.10.2 Two-Step Optimization.................... 71
3.10.3 A Few Geometric Properties................. 73
3.11 Spectral Analysis of Tikhonov s Regularization........... 77
Exercises .................................. 79
4 Image Restoration 85
4.1 Degraded Images........................... 85
4.2 Restoring Images........................... 88
4.3 Restoration Algorithm........................ 90
4.3.1 Newton s Method....................... 91
4.3.2 Modified Newton s Method with Gain Factor........ 92
4.3.3 Stopping Criterion...................... 92
4.3.4 Regularization Strategy.................... 93
4.3.5 Gauss-Seidel Method..................... 93
4.3.6 Restoration Methodology .................. 94
4.4 Photo Restoration........................... 95
4.5 Text Restoration............................ 99
4.6 Biological Image Restoration..................... 99
Exercises .................................. 102
5 Radiative Transfer and Heat Conduction 109
5.1 Mathematical Description....................... 109
5.2 Modified Newton s Method...................... Ill
5.3 Levenberg-Marquardt s Method ................... 113
5.4 Confidence Intervals for Parameter Estimates............ 114
5.5 Phase Function, Albedo and Optical Thickness........... 115
5.6 Thermal Conductivity, Optical Thickness and Albedo........ 121
5.7 Refractive Index and Optical Thickness............... 125
Exercises.................................. 127
Contents XXI
6 Thermal Characterization 129
6.1 Experimental Device: Hot Wire Method............... 129
6.2 Traditional Experimental Approach ................. 131
6.3 Inverse Problem Approach...................... 134
6.3.1 Heat Equation......................... 134
6.3.2 Parameter Estimation..................... 135
6.3.3 Levenberg-Marquardt.................... 135
6.3.4 Confidence Intervals..................... 136
6.3.5 Application to the Characterization of a Phenolic Foam with
Lignin............................. 136
6.4 Experiment Design.......................... 138
Exercises.................................. 138
7 Heat Conduction 141
7.1 Mathematical Formulation...................... 141
7.2 Expansion in Terms of Known Functions............... 144
7.3 Conjugate Gradient Method ..................... 144
7.4 Thermal Source in Heat Transfer by Conduction........... 146
7.4.1 Heat Transfer by Conduction................. 146
7.4.2 Sensitivity Problem...................... 147
7.4.3 Adjoint Problem and Gradient Equation........... 147
7.4.4 Computation of the Critical Point.............. 150
7.5 Minimization with the Conjugate Gradient Method......... 150
7.5.1 Conjugate Gradient Algorithm................ 150
7.5.2 Stopping Criterion and Discrepancy Principle........ 151
7.6 Estimation Results.......................... 152
Exercises.................................. 154
8 A General Perspective 157
8.1 Inverse Problems and Their Types.................. 157
8.1.1 Classification of Problems.................. 157
8.1.2 Observation Operators.................... 161
8.1.3 Inverse Problems in Practice................. 162
8.1.4 Domain of Dependence.................... 164
8.1.5 Types of Inverse Problems.................. 168
8.2 Gradient Calculation......................... 169
8.2.1 Reformulation of the Direct Problem ............ 169
8.2.2 Gradient Calculation by Its Definition............ 169
8.2.3 Interpretation of the Gradient................. 171
8.2.4 Operational Rule for the Computation of the Gradient ... 173
Afterword: Some Thoughts on Model Complexity and Knowledge 177
A Spectral Theory and a Few Other Ideas From Mathematics 189
A.I Norms, Inner Products and Tensor Products............. 189
A.2 Spectral Theorem........................... 191
XXII Contents
A.3 Singular Value Decomposition.................... 195
A.4 Special Classes of Matrices...................... 200
A.4.1 Projection Matrices...................... 200
A.4.2 Forgetting Matrices...................... 202
A.5 Taylor s Formulae........................... 204
Exercises .................................. 207
References 233
Index 239
|
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| author_facet | Moura Neto, Francisco Duarte Silva Neto, Antônio José da |
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| discipline | Mathematik |
| format | Book |
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| isbn | 9783642325564 |
| language | English |
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| physical | XXII, 243 S. Ill., graph. Darst. |
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| spelling | Moura Neto, Francisco Duarte Verfasser aut An Introduction to Inverse Problems with Applications Francisco Duarte Moura Neto ; Antônio José da Silva Neto Berlin Springer 2013 XXII, 243 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Matrizenalgebra (DE-588)4139347-8 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 Matrizenalgebra (DE-588)4139347-8 s DE-604 Silva Neto, Antônio José da Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025385595&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Moura Neto, Francisco Duarte Silva Neto, Antônio José da An Introduction to Inverse Problems with Applications Numerisches Verfahren (DE-588)4128130-5 gnd Matrizenalgebra (DE-588)4139347-8 gnd Inverses Problem (DE-588)4125161-1 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4139347-8 (DE-588)4125161-1 |
| title | An Introduction to Inverse Problems with Applications |
| title_auth | An Introduction to Inverse Problems with Applications |
| title_exact_search | An Introduction to Inverse Problems with Applications |
| title_full | An Introduction to Inverse Problems with Applications Francisco Duarte Moura Neto ; Antônio José da Silva Neto |
| title_fullStr | An Introduction to Inverse Problems with Applications Francisco Duarte Moura Neto ; Antônio José da Silva Neto |
| title_full_unstemmed | An Introduction to Inverse Problems with Applications Francisco Duarte Moura Neto ; Antônio José da Silva Neto |
| title_short | An Introduction to Inverse Problems with Applications |
| title_sort | an introduction to inverse problems with applications |
| topic | Numerisches Verfahren (DE-588)4128130-5 gnd Matrizenalgebra (DE-588)4139347-8 gnd Inverses Problem (DE-588)4125161-1 gnd |
| topic_facet | Numerisches Verfahren Matrizenalgebra Inverses Problem |
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