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E-raamat: Factorization Method for Inverse Problems

(University of Karlsruhe), (University of Karlsruhe)
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The factorization method is a relatively new method for solving certain types of inverse scattering problems and problems in tomography. Aimed at students and researchers in Applied Mathematics, Physics and Engineering, this text introduces the reader to this promising approach for solving important classes of inverse problems. The wide applicability of this method is discussed by choosing typical examples, such as inverse scattering problems for the scalar Helmholtz equation, a scattering problem for Maxwell's equation, and a problem in impedance and optical tomography. The last section of the book compares the Factorization Method to established sampling methods (the Linear Sampling Method, the Singular Source Method, and the Probe Method).
Preface v
1 The simplest cases: Dirichlet and Neumann boundary conditions 1
1.1 The Helmholtz equation in acoustics
2
1.2 The direct scattering problem
4
1.3 The far field patterns and the inverse problem
7
1.4 Factorization methods
13
1.4.1 Factorization of the far field operator
15
1.4.2 The inf-criterion
19
1.4.3 The (F*F)1/4-method
22
1.5 An explicit example
29
1.6 The Neumann boundary condition
31
1.7 Additional remarks and numerical examples
35
2 The factorization method for other types of inverse obstacle scattering problems 40
2.1 The direct scattering problem with impedance boundary conditions
40
2.2 The obstacle reconstruction by the inf-criterion
49
2.3 Reconstruction from limited data
52
2.4 Reconstruction from near field data
54
2.5 The F# – factorization method
57
2.5.1 The functional analytic background
57
2.5.2 Applications to some in verse scattering problems
62
2.6 Obstacle scattering in a half-space
63
2.6.1 The direct scattering problem
65
2.6.2 The factorization method for the inverse problem
67
3 The mixed boundary value problem 70
3.1 The direct scattering problem
70
3.2 Factorization of the far field operator
76
3.3 Application of the F# – factorization method
79
4 The MUSIC algorithm and scattering by an inhomogeneous medium 86
4.1 The MUSIC algorithm
86
4.2 Scattering by an inhomogeneous medium
91
4.3 Factorization of the far field operators
95
4.4 Localization of the support of the contrast
97
4.5 The interior transmission eigenvalue problem
102
5 The factorization method for Maxwell's equations 109
5.1 Maxwell's equations
109
5.2 The direct scattering problem
111
5.3 Factorization of the far field operator
123
5.4 Localization of the support of the contrast
125
5.5 The interior transmission eigenvalue problem
133
6 The factorization method in impedance tomography 141
6.1 Derivation of the models
141
6.2 The Neumann-to-Dirichlet operator and the inverse problem
142
6.3 Factorization of the Neumann-to-Dirichlet operator
148
6.4 Characterization of the inclusion
150
7 Alternative sampling and probe methods 159
7.1 Two approximation results
159
7.2 The dual space method and the linear sampling method
163
7.3 The singular sources method
171
7.4 The probe method
176
7.4.1 The probe method in impedance tomography
176
7.4.2 The probe method for the inverse scattering problem with mixed boundary conditions
183
Bibliography 189
Index 199


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