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逆问题的数学理论导论【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

Andreas Kirsch著著
出版社：北京：世界图书出版公司
ISBN：9787519202675
出版时间：2016
标注页数：307页
文件大小：39MB
文件页数：322页
主题词：逆问题－数学理论－研究－英文

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图书目录

1 Introduction and Basic Concepts1

1.1 Examples of Inverse Problems1

1.2 Ill-Posed Problems9

1.3 The Worst-Case Error13

1.4 Problems20

2 Regularization Theory for Equations of the First Kind23

2.1 A General Regularization Theory24

2.2 Tikhonov Regularization36

2.3 Landweber Iteration41

2.4 A Numerical Example43

2.5 The Discrepancy Principle of Morozov46

2.6 Landweber's Iteration Method with Stopping Rule51

2.7 The Conjugate Gradient Method55

2.8 Problems60

3 Regularization by Discretization63

3.1 Projection Methods63

3.2 Galerkin Methods70

3.2.1 The Least Squares Method73

3.2.2 The Dual Least Squares Method75

3.2.3 The Bubnov-Galerkin Method for Coercive Operators77

3.3 Application to Symm's Integral Equation of the First Kind81

3.4 Collocation Methods90

3.4.1 Minimum Norm Collocation91

3.4.2 Collocation of Symm's Equation95

3.5 Numerical Experiments for Symm's Equation103

3.6 The Backus-Gilbert Method110

3.7 Problems118

4 Inverse Eigenvalue Problems121

4.1 Introduction121

4.2 Construction of a Fundamental System123

4.3 Asymptotics of the Eigenvalues and Eigenfunctions130

4.4 Some Hyperbolic Problems140

4.5 The Inverse Problem148

4.6 A Parameter Identification Problem154

4.7 Numerical Reconstruction Techniques158

4.8 Problems164

5 An Inverse Problem in Electrical Impedance Tomography167

5.1 Introduction167

5.2 The Direct Problem and the Neumann-Dirichlet Operator169

5.3 The Inverse Problem172

5.4 The Factorization Method177

5.5 Problems188

6 An Inverse Scattering Problem191

6.1 Introduction191

6.2 The Direct Scattering Problem195

6.3 Properties of the Far Field Patterns206

6.4 Uniqueness of the Inverse Problem218

6.5 The Factorization Method225

6.6 Numerical Methods235

6.6.1 A Simplified Newton Method236

6.6.2 A Modified Gradient Method240

6.6.3 The Dual Space Method241

6.7 Problems244

A Basic Facts from Functional Analysis247

A.1 Normed Spaces and Hilbert Spaces247

A.2 Orthonormal Systems253

A.3 Linear Bounded and Compact Operators255

A.4 Sobolev Spaces of Periodic Functions261

A.5 Sobolev Spaces on the Unit Disc268

A.6 Spectral Theory for Compact Operators in Hilbert Spaces273

A.7 The Fréchet Derivative277

B Proofs of the Results of Section 2.7283

References295

Index305