The Factorization Method for Inverse Problems

by Kirsch, Andreas; Grinberg, Natalia

• ISBN13: 9780199213535
• ISBN10: 0199213534
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-02-09
• Publisher: Oxford University Press

Summary

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

Preface	p. v
The simplest cases: Dirichlet and Neumann boundary conditions	p. 1
The Helmholtz equation in acoustics	p. 2
The direct scattering problem	p. 4
The far field patterns and the inverse problem	p. 7
Factorization methods	p. 13
Factorization of the far field operator	p. 15
The inf-criterion	p. 19
The (F*F)[superscript 1/4]-method	p. 22
An explicit example	p. 29
The Neumann boundary condition	p. 31
Additional remarks and numerical examples	p. 35
The factorization method for other types of inverse obstacle scattering problems	p. 40
The direct scattering problem with impedance boundary conditions	p. 40
The obstacle reconstruction by the inf-criterion	p. 49
Reconstruction from limited data	p. 52
Reconstruction from near field data	p. 54
The F[subscript #] - factorization method	p. 57
The functional analytic background	p. 57
Applications to some inverse scattering problems	p. 62
Obstacle scattering in a half-space	p. 63
The direct scattering problem	p. 65
The factorization method for the inverse problem	p. 67
The mixed boundary value problem	p. 70
The direct scattering problem	p. 70
Factorization of the far field operator	p. 76
Application of the F[subscript #] - factorization method	p. 79
The MUSIC algorithm and scattering by an inhomogeneous medium	p. 86
The MUSIC algorithm	p. 86
Scattering by an inhomogeneous medium	p. 91
Factorization of the far field operators	p. 95
Localization of the support of the contrast	p. 97
The interior transmission eigenvalue problem	p. 102
The factorization method for Maxwell's equations	p. 109
Maxwell's equations	p. 109
The direct scattering problem	p. 111
Factorization of the far field operator	p. 123
Localization of the support of the contrast	p. 125
The interior transmission eigenvalue problem	p. 133
The factorization method in impedance tomography	p. 141
Derivation of the models	p. 141
The Neumann-to-Dirichlet operator and the inverse problem	p. 142
Factorization of the Neumann-to-Dirichlet operator	p. 148
Characterization of the inclusion	p. 150
Alternative sampling and probe methods	p. 159
Two approximation results	p. 159
The dual space method and the linear sampling method	p. 163
The singular sources method	p. 171
The probe method	p. 176
The probe method in impedance tomography	p. 176
The probe method for the inverse scattering problem with mixed boundary conditions	p. 183
Bibliography	p. 189
Index	p. 199
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