The Factorization Method for Inverse Problems

✍ Scribed by Andreas Kirsch, Natalia Grinberg

• Publisher: Oxford Univ Pr
• Year: 2008
• Tongue: German
• Leaves: 216
• Series: Oxford Lecture Series in Mathematics and Its Applications
• Edition: 1
• Category: Library

✦ Synopsis

The factorization method is a relatively new method for solving certain types of inverse scattering 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 Method, and the Probe Method)

✦ Table of Contents

Contents......Page 14
Preface......Page 6
1 The simplest cases: Dirichlet and Neumann boundary conditions......Page 16
1.1 The Helmholtz equation in acoustics......Page 17
1.2 The direct scattering problem......Page 19
1.3 The far field patterns and the inverse problem......Page 22
1.4 Factorization methods......Page 28
1.4.1 Factorization of the far field operator......Page 30
1.4.2 The inf-criterion......Page 34
1.4.3 The (F*F)[sup(1/4)] -method......Page 37
1.5 An explicit example......Page 44
1.6 The Neumann boundary condition......Page 46
1.7 Additional remarks and numerical examples......Page 50
2.1 The direct scattering problem with impedance boundary conditions......Page 55
2.2 The obstacle reconstruction by the inf-criterion......Page 64
2.3 Reconstruction from limited data......Page 67
2.4 Reconstruction from near field data......Page 69
2.5.1 The functional analytic background......Page 72
2.5.2 Applications to some inverse scattering problems......Page 77
2.6 Obstacle scattering in a half-space......Page 78
2.6.1 The direct scattering problem......Page 80
2.6.2 The factorization method for the inverse problem......Page 82
3.1 The direct scattering problem......Page 85
3.2 Factorization of the far field operator......Page 91
3.3 Application of the F[sub(#)] – factorization method......Page 94
4.1 The MUSIC algorithm......Page 101
4.2 Scattering by an inhomogeneous medium......Page 106
4.3 Factorization of the far field operators......Page 110
4.4 Localization of the support of the contrast......Page 112
4.5 The interior transmission eigenvalue problem......Page 117
5.1 Maxwell’s equations......Page 124
5.2 The direct scattering problem......Page 126
5.3 Factorization of the far field operator......Page 138
5.4 Localization of the support of the contrast......Page 140
5.5 The interior transmission eigenvalue problem......Page 148
6.1 Derivation of the models......Page 156
6.2 The Neumann-to-Dirichlet operator and the inverse problem......Page 157
6.3 Factorization of the Neumann-to-Dirichlet operator......Page 163
6.4 Characterization of the inclusion......Page 165
7.1 Two approximation results......Page 174
7.2 The dual space method and the linear sampling method......Page 178
7.3 The singular sources method......Page 186
7.4.1 The probe method in impedance tomography......Page 191
7.4.2 The probe method for the inverse scattering problem with mixed boundary conditions......Page 198
Bibliography......Page 204
Index......Page 214
H......Page 215
W......Page 216