Inverse Problems and Inverse Scattering of Plane Waves
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Table of contents
- Inverse Problems and Inverse Scattering of Plane Wavesi
- Contentsv
- Chapter 1. Introduction1
- 1.1 Direct and Inverse Problems1
- 1.2 The Basic Concepts8
- Chapter 2. Some Examples of Ill-posed Problems17
- 2.1 Introduction17
- 2.2 Examples17
- Chapter 3. Theory of Ill-posed Problems49
- 3.1 Introduction49
- 3.2 Tikhonov’s Theorem51
- 3.3 Regularization on a Compactum: The Quasisolution54
- 3.4 Generalized Solutions56
- 3.5 Singular Value Expansion63
- 3.6 Tikhonov’s Theory of Regularization68
- 3.7 Convergence, Stability and Optimality85
- 3.8 The Determination of α90
- 3.9 An Application96
- 3.10 The Method of Mollification97
- Chapter 4. Regularization by Projections105
- 4.1 Introduction105
- 4.2 The Basic Projection Methods105
- 4.3 The Method of Projections: General Framework108
- 4.4 The Method of Least-Square113
- 4.5 The Method of Collocation117
- 4.6 The Standard Galerkin Method120
- Chapter 5. Discrete Ill-posed Problems133
- 5.1 Introduction133
- 5.2 Discrete Decompositions134
- 5.3 The Discrete Tikhonov Regularization145
- 5.4 An Example146
- 5.5 Discrete Solution of a Tikhonov Functional150
- 5.6 Appendix A.5.1154
- 5.7 Appendix A.5.2159
- 5.8 Appendix A.5.3164
- Chapter 6. The Helmholtz Scattering169
- 6.1 Introduction169
- 6.2 Gauss’ or Divergence Theorem171
- 6.3 Green’s Identities173
- 6.4 The Helmholtz Equation175
- 6.5 The Helmholtz Representation in the Interior178
- 6.6 The Radiation Condition180
- 6.7 The Helmholtz Representation in the Exterior188
- 6.8 Some Properties of the Scattering Solutions190
- 6.9 The Helmholtz Scattering from Inhomogeneities193
- Chapter 7. The Solutions203
- 7.1 Introduction203
- 7.2 The Layer Potentials204
- 7.3 Replacing G0(x, y; k) by g0(x, y).206
- 7.4 The Double-layer Potential211
- 7.5 The Single-layer Potential216
- 7.6 The Helmholtz Scattering Problems222
- 7.7 Unconditionally Unique Solution227
- 7.8 The Transmission Problem234
- 7.9 Jones’ Method237
- 7.10 Appendix A.7.1239
- 7.11 Appendix A.7.2241
- 7.12 Appendix A.7.3242
- 7.13 Appendix A.7.4243
- 7.14 Appendix A.7.5245
- 7.15 Appendix A.7.6245
- Chapter 8. Uniqueness Theorems in Inverse Problems247
- 8.1 Some Definitions247
- 8.2 Properties of the Total Fields249
- 8.3 The Dirichlet and Neumann Spectrum253
- 8.4 The Uniqueness of the Inverse Dirichlet Obstacle Problem259
- 8.5 The Uniqueness of Inverse Neumann Obstacle261
- 8.6 Uniqueness in Inverse Transmission Obstacle267
- 8.7 Uniqueness of Inverse Inhomogeneity Scattering269
- 8.8 Appendix A.8.1277
- 8.9 Appendix A.8.2281
- Chapter 9. Some Algorithms283
- 9.1 Introduction283
- 9.2 The Method of Potentials286
- 9.3 The Method of Superposition of Incident Fields287
- 9.4 The Method of Wavefunction Expansion290
- 9.5 The Method of Boundary Variation292
- 9.6 Some Nonoptimizational Methods300
- 9.7 Appendix A.9.1305
- 9.8 Appendix A.9.2309
- 9.9 Appendix A.9.3311
- Chapter 10. Bibliography315
The purpose of this text is to present the theory and mathematics of inverse scattering, in a simple way, to the many researchers and professionals who use it in their everyday research. While applications range across a broad spectrum of disciplines, examples in this text will focus primarly, but not exclusively, on acoustics. The text will be especially valuable for those applied workers who would like to delve more deeply into the fundamentally mathematical character of the subject matter.
Practitioners in this field comprise applied physicists, engineers, and technologists, whereas the theory is almost entirely in the domain of abstract mathematics. This gulf between the two, if bridged, can only lead to improvement in the level of scholarship in this highly important discipline. This is the book's primary focus.
Practitioners in this field comprise applied physicists, engineers, and technologists, whereas the theory is almost entirely in the domain of abstract mathematics. This gulf between the two, if bridged, can only lead to improvement in the level of scholarship in this highly important discipline. This is the book's primary focus.