Michel Kern,
M Kern
Numerical Methods for Inverse Problems
Gebonden Engels 2016 9781848218185
Verwachte levertijd ongeveer 9 werkdagen
Samenvatting
This book studies methods to concretely address inverse problems. An inverse problem arises when the causes that produced a given effect must be determined or when one seeks to indirectly estimate the parameters of a physical system.
The author uses practical examples to illustrate inverse problems in physical sciences. He presents the techniques and specific methods chosen to solve inverse problems in a general domain of application, choosing to focus on a small number of methods that can be used in most applications.
This book is aimed at readers with a mathematical and scientific computing background. Despite this, it is a book with a practical perspective. The methods described are applicable, have been applied, and are often illustrated by numerical examples.
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Inhoudsopgave
<p>Preface ix</p>
<p>Part 1. Introduction and Examples 1</p>
<p>Chapter 1. Overview of Inverse Problems 3</p>
<p>1.1. Direct and inverse problems 3</p>
<p>1.2. Well–posed and ill–posed problems 4</p>
<p>Chapter 2. Examples of Inverse Problems 9</p>
<p>2.1. Inverse problems in heat transfer 10</p>
<p>2.2. Inverse problems in hydrogeology 13</p>
<p>2.3. Inverse problems in seismic exploration 16</p>
<p>2.4. Medical imaging 21</p>
<p>2.5. Other examples 25</p>
<p>Part 2. Linear Inverse Problems 29</p>
<p>Chapter 3. Integral Operators and Integral Equations 31</p>
<p>3.1. Definition and first properties 31</p>
<p>3.2. Discretization of integral equations 36</p>
<p>3.2.1. Discretization by quadrature collocation 36</p>
<p>3.2.2. Discretization by the Galerkin method 39</p>
<p>3.3. Exercises 42</p>
<p>Chapter 4. Linear Least Squares Problems Singular Value Decomposition 45</p>
<p>4.1. Mathematical properties of least squares problems 45</p>
<p>4.1.1. Finite dimensional case 50</p>
<p>4.2. Singular value decomposition for matrices 52</p>
<p>4.3. Singular value expansion for compact operators 57</p>
<p>4.4. Applications of the SVD to least squares problems 60</p>
<p>4.4.1. The matrix case 60</p>
<p>4.4.2. The operator case 63</p>
<p>4.5. Exercises 65</p>
<p>Chapter 5. Regularization of Linear Inverse Problems 71</p>
<p>5.1. Tikhonov s method 72</p>
<p>5.1.1. Presentation 72</p>
<p>5.1.2. Convergence 73</p>
<p>5.1.3. The L–curve 81</p>
<p>5.2. Applications of the SVE 83</p>
<p>5.2.1. SVE and Tikhonov s method 84</p>
<p>5.2.2. Regularization by truncated SVE 85</p>
<p>5.3. Choice of the regularization parameter 88</p>
<p>5.3.1. Morozov s discrepancy principle 88</p>
<p>5.3.2. The L–curve 91</p>
<p>5.3.3. Numerical methods 92</p>
<p>5.4. Iterative methods 94</p>
<p>5.5. Exercises 98</p>
<p>Part 3. Nonlinear Inverse Problems 103</p>
<p>Chapter 6. Nonlinear Inverse Problems Generalities 105</p>
<p>6.1. The three fundamental spaces 106</p>
<p>6.2. Least squares formulation 111</p>
<p>6.2.1. Difficulties of inverse problems 114</p>
<p>6.2.2. Optimization, parametrization, discretization 114</p>
<p>6.3. Methods for computing the gradient the adjoint state method 116</p>
<p>6.3.1. The finite difference method 116</p>
<p>6.3.2. Sensitivity functions 118</p>
<p>6.3.3. The adjoint state method 119</p>
<p>6.3.4. Computation of the adjoint state by the Lagrangian 120</p>
<p>6.3.5. The inner product test 123</p>
<p>6.4. Parametrization and general organization 123</p>
<p>6.5. Exercises 125</p>
<p>Chapter 7. Some Parameter Estimation Examples 127</p>
<p>7.1. Elliptic equation in one dimension 127</p>
<p>7.1.1. Computation of the gradient 128</p>
<p>7.2. Stationary diffusion: elliptic equation in two dimensions 129</p>
<p>7.2.1. Computation of the gradient: application of the general method 132</p>
<p>7.2.2. Computation of the gradient by the Lagrangian 134</p>
<p>7.2.3. The inner product test 135</p>
<p>7.2.4. Multiscale parametrization 135</p>
<p>7.2.5. Example 136</p>
<p>7.3. Ordinary differential equations 137</p>
<p>7.3.1. An application example 144</p>
<p>7.4. Transient diffusion: heat equation 147</p>
<p>7.5. Exercises 152</p>
<p>Chapter 8. Further Information 155</p>
<p>8.1. Regularization in other norms 155</p>
<p>8.1.1. Sobolev semi–norms 155</p>
<p>8.1.2. Bounded variation regularization norm 157</p>
<p>8.2. Statistical approach: Bayesian inversion 157</p>
<p>8.2.1. Least squares and statistics 158</p>
<p>8.2.2. Bayesian inversion 160</p>
<p>8.3. Other topics 163</p>
<p>8.3.1. Theoretical aspects: identifiability 163</p>
<p>8.3.2. Algorithmic differentiation . 163</p>
<p>8.3.3. Iterative methods and large–scale problems 164</p>
<p>8.3.4. Software 164</p>
<p>Appendices 167</p>
<p>Appendix 1 169</p>
<p>Appendix 2 183</p>
<p>Appendix 3 193</p>
<p>Bibliography 205</p>
<p>Index 213</p>
<p>Part 1. Introduction and Examples 1</p>
<p>Chapter 1. Overview of Inverse Problems 3</p>
<p>1.1. Direct and inverse problems 3</p>
<p>1.2. Well–posed and ill–posed problems 4</p>
<p>Chapter 2. Examples of Inverse Problems 9</p>
<p>2.1. Inverse problems in heat transfer 10</p>
<p>2.2. Inverse problems in hydrogeology 13</p>
<p>2.3. Inverse problems in seismic exploration 16</p>
<p>2.4. Medical imaging 21</p>
<p>2.5. Other examples 25</p>
<p>Part 2. Linear Inverse Problems 29</p>
<p>Chapter 3. Integral Operators and Integral Equations 31</p>
<p>3.1. Definition and first properties 31</p>
<p>3.2. Discretization of integral equations 36</p>
<p>3.2.1. Discretization by quadrature collocation 36</p>
<p>3.2.2. Discretization by the Galerkin method 39</p>
<p>3.3. Exercises 42</p>
<p>Chapter 4. Linear Least Squares Problems Singular Value Decomposition 45</p>
<p>4.1. Mathematical properties of least squares problems 45</p>
<p>4.1.1. Finite dimensional case 50</p>
<p>4.2. Singular value decomposition for matrices 52</p>
<p>4.3. Singular value expansion for compact operators 57</p>
<p>4.4. Applications of the SVD to least squares problems 60</p>
<p>4.4.1. The matrix case 60</p>
<p>4.4.2. The operator case 63</p>
<p>4.5. Exercises 65</p>
<p>Chapter 5. Regularization of Linear Inverse Problems 71</p>
<p>5.1. Tikhonov s method 72</p>
<p>5.1.1. Presentation 72</p>
<p>5.1.2. Convergence 73</p>
<p>5.1.3. The L–curve 81</p>
<p>5.2. Applications of the SVE 83</p>
<p>5.2.1. SVE and Tikhonov s method 84</p>
<p>5.2.2. Regularization by truncated SVE 85</p>
<p>5.3. Choice of the regularization parameter 88</p>
<p>5.3.1. Morozov s discrepancy principle 88</p>
<p>5.3.2. The L–curve 91</p>
<p>5.3.3. Numerical methods 92</p>
<p>5.4. Iterative methods 94</p>
<p>5.5. Exercises 98</p>
<p>Part 3. Nonlinear Inverse Problems 103</p>
<p>Chapter 6. Nonlinear Inverse Problems Generalities 105</p>
<p>6.1. The three fundamental spaces 106</p>
<p>6.2. Least squares formulation 111</p>
<p>6.2.1. Difficulties of inverse problems 114</p>
<p>6.2.2. Optimization, parametrization, discretization 114</p>
<p>6.3. Methods for computing the gradient the adjoint state method 116</p>
<p>6.3.1. The finite difference method 116</p>
<p>6.3.2. Sensitivity functions 118</p>
<p>6.3.3. The adjoint state method 119</p>
<p>6.3.4. Computation of the adjoint state by the Lagrangian 120</p>
<p>6.3.5. The inner product test 123</p>
<p>6.4. Parametrization and general organization 123</p>
<p>6.5. Exercises 125</p>
<p>Chapter 7. Some Parameter Estimation Examples 127</p>
<p>7.1. Elliptic equation in one dimension 127</p>
<p>7.1.1. Computation of the gradient 128</p>
<p>7.2. Stationary diffusion: elliptic equation in two dimensions 129</p>
<p>7.2.1. Computation of the gradient: application of the general method 132</p>
<p>7.2.2. Computation of the gradient by the Lagrangian 134</p>
<p>7.2.3. The inner product test 135</p>
<p>7.2.4. Multiscale parametrization 135</p>
<p>7.2.5. Example 136</p>
<p>7.3. Ordinary differential equations 137</p>
<p>7.3.1. An application example 144</p>
<p>7.4. Transient diffusion: heat equation 147</p>
<p>7.5. Exercises 152</p>
<p>Chapter 8. Further Information 155</p>
<p>8.1. Regularization in other norms 155</p>
<p>8.1.1. Sobolev semi–norms 155</p>
<p>8.1.2. Bounded variation regularization norm 157</p>
<p>8.2. Statistical approach: Bayesian inversion 157</p>
<p>8.2.1. Least squares and statistics 158</p>
<p>8.2.2. Bayesian inversion 160</p>
<p>8.3. Other topics 163</p>
<p>8.3.1. Theoretical aspects: identifiability 163</p>
<p>8.3.2. Algorithmic differentiation . 163</p>
<p>8.3.3. Iterative methods and large–scale problems 164</p>
<p>8.3.4. Software 164</p>
<p>Appendices 167</p>
<p>Appendix 1 169</p>
<p>Appendix 2 183</p>
<p>Appendix 3 193</p>
<p>Bibliography 205</p>
<p>Index 213</p>
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