逆问题的数学理论导论PDF电子书下载

1 Introduction and Basic Concepts 1

1.1 Examples of Inverse Problems 1

1.2 Ill-Posed Problems 9

1.3 The Worst-Case Error 13

1.4 Problems 20

2 Regularization Theory for Equations of the First Kind 23

2.1 A General Regularization Theory 24

2.2 Tikhonov Regularization 36

2.3 Landweber Iteration 41

2.4 A Numerical Example 43

2.5 The Discrepancy Principle of Morozov 46

2.6 Landweber's Iteration Method with Stopping Rule 51

2.7 The Conjugate Gradient Method 55

2.8 Problems 60

3 Regularization by Discretization 63

3.1 Projection Methods 63

3.2 Galerkin Methods 70

3.2.1 The Least Squares Method 73

3.2.2 The Dual Least Squares Method 75

3.2.3 The Bubnov-Galerkin Method for Coercive Operators 77

3.3 Application to Symm's Integral Equation of the First Kind 81

3.4 Collocation Methods 90

3.4.1 Minimum Norm Collocation 91

3.4.2 Collocation of Symm's Equation 95

3.5 Numerical Experiments for Symm's Equation 103

3.6 The Backus-Gilbert Method 110

3.7 Problems 118

4 Inverse Eigenvalue Problems 121

4.1 Introduction 121

4.2 Construction of a Fundamental System 123

4.3 Asymptotics of the Eigenvalues and Eigenfunctions 130

4.4 Some Hyperbolic Problems 140

4.5 The Inverse Problem 148

4.6 A Parameter Identification Problem 154

4.7 Numerical Reconstruction Techniques 158

4.8 Problems 164

5 An Inverse Problem in Electrical Impedance Tomography 167

5.1 Introduction 167

5.2 The Direct Problem and the Neumann-Dirichlet Operator 169

5.3 The Inverse Problem 172

5.4 The Factorization Method 177

5.5 Problems 188

6 An Inverse Scattering Problem 191

6.1 Introduction 191

6.2 The Direct Scattering Problem 195

6.3 Properties of the Far Field Patterns 206

6.4 Uniqueness of the Inverse Problem 218

6.5 The Factorization Method 225

6.6 Numerical Methods 235

6.6.1 A Simplified Newton Method 236

6.6.2 A Modified Gradient Method 240

6.6.3 The Dual Space Method 241

6.7 Problems 244

A Basic Facts from Functional Analysis 247

A.1 Normed Spaces and Hilbert Spaces 247

A.2 Orthonormal Systems 253

A.3 Linear Bounded and Compact Operators 255

A.4 Sobolev Spaces of Periodic Functions 261

A.5 Sobolev Spaces on the Unit Disc 268

A.6 Spectral Theory for Compact Operators in Hilbert Spaces 273

A.7 The Fréchet Derivative 277

B Proofs of the Results of Section 2.7 283

References 295

Index 305