Inverse problems and inverse scattering of plane waves

Title:

Personal Author:

Ghosh Roy, D. N., 1939-

Publication Information:

San Diego : Academic Press, 2002

ISBN:

9780122818653

Subject Term:

Scattering (Physics)

Inverse scattering transform

Acoustics

Added Author:

Couchman, L. S.

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010046261	QC794.6.S3 G46 2002	Open Access Book	Book	Searching... Unknown

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.

Author Notes

D. N. Ghosh Roy is a staff scientist at Sachs and Freeman, Inc., in Maryland, USA, and is a consultant to the U.S. Naval Research Laboratory in Washington, DC, USA
L. S. Couchman is the head of the Theoretical Acoustics Branch of the U.S. Naval Research Laboratory in Washington, DC, USA

1 Introduction	p. 1
1.1 Direct and Inverse Problems	p. 1
1.1.1 Two Broad Divisions of Inverse Problems	p. 2
1.2 The Basic Concepts	p. 8
1.2.1 The Approximate Nature of an Inverse Solution	p. 8
1.2.2 The Smoothing Action Of An Integral Operator	p. 10
1.2.3 The Role of a priori Knowledge	p. 12
1.2.4 Ill- and Well-posed Problems	p. 13
2 Some Examples of Ill-posed Problems	p. 17
2.1 Introduction	p. 17
2.2 Examples	p. 17
2.2.1 Example 1. The Cauchy Problem for the Backward Heat Equation	p. 17
2.2.2 Example 2. The Cauchy Problem for the Laplace Equation	p. 21
2.2.3 Example 3. The Laplace Transform	p. 23
2.2.4 Example 4. Numerical Differentiation	p. 28
2.2.5 Example 5. Inverse Source Problem	p. 32
Inverse Diffraction and Near-Field Holography	p. 36
2.2.6 Example 6. An Example from Medical Diagnostics	p. 40
2.2.7 Example 7. A Nonlinear Problem	p. 44
3 Theory of Ill-posed Problems	p. 49
3.1 Introduction	p. 49
3.2 Tikhonov's Theorem	p. 51
3.3 Regularization on a Compactum: The Quasisolution	p. 54
3.4 Generalized Solutions	p. 56
3.4.1 Summary	p. 62
3.4.2 Connection with Quasisolution	p. 63
3.5 Singular Value Expansion	p. 63
3.6 Tikhonov's Theory of Regularization	p. 68
3.6.1 The Regularizing Operator	p. 70
The [epsilon]--[delta] Definition	p. 70
The Parametric Definition	p. 71
3.6.2 The Construction of Regularizers	p. 74
3.6.3 The Spectral or Filter Functions	p. 78
The Iterative Filters	p. 79
3.6.4 First-order regularization	p. 82
3.7 Convergence, Stability and Optimality	p. 85
3.7.1 Convergence and Stability Estimates	p. 85
3.7.2 The Optimality of a Regularization Strategy	p. 87
3.8 The Determination of [alpha]	p. 90
3.8.1 The Existence of an Optimal Value of [alpha]	p. 90
3.8.2 The Discrepancy Principle	p. 93
3.9 An Application	p. 96
3.10 The Method of Mollification	p. 97
3.10.1 The Method	p. 97
3.10.2 An Example: Numerical Differentiation	p. 102
4 Regularization by Projections	p. 105
4.1 Introduction	p. 105
4.2 The Basic Projection Methods	p. 105
4.3 The Method of Projections: General Framework	p. 108
4.4 The Method of Least-Square	p. 113
4.5 The Method of Collocation	p. 117
4.6 The Standard Galerkin Method	p. 120
4.6.1 The Galerkin Approximation in one Dimension	p. 120
4.6.2 The General Case	p. 125
4.6.3 The Galerkin Method and FEM	p. 128
4.6.4 The Perturbed Data	p. 130
4.6.5 The Petrov-Galerkin Method	p. 131
5 Discrete Ill-posed Problems	p. 133
5.1 Introduction	p. 133
5.2 Discrete Decompositions	p. 134
5.3 The Discrete Tikhonov Regularization	p. 145
5.4 An Example	p. 146
5.5 Discrete Solution of a Tikhonov Functional	p. 150
5.6 Appendix A.5.1	p. 154
5.7 Appendix A.5.2	p. 159
5.8 Appendix A.5.3	p. 164
6 The Helmholtz Scattering	p. 169
6.1 Introduction	p. 169
6.2 Gauss' or Divergence Theorem	p. 171
6.3 Green's Identities	p. 173
6.4 The Helmholtz Equation	p. 175
6.5 The Helmholtz Representation in the Interior	p. 178
6.6 The Radiation Condition	p. 180
6.7 The Helmholtz Representation in the Exterior	p. 188
6.8 Some Properties of the Scattering Solutions	p. 190
6.9 The Helmholtz Scattering from Inhomogeneities	p. 193
7 The Solutions	p. 203
7.1 Introduction	p. 203
7.2 The Layer Potentials	p. 204
7.3 Replacing G[superscript 0] (x, y; k) by g[superscript 0] (x, y)	p. 206
7.4 The Double-layer Potential	p. 211
7.5 The Single-layer Potential	p. 216
7.6 The Helmholtz Scattering Problems	p. 222
7.6.1 The Dirichet and Neumann Obstacle Scattering	p. 222
7.7 Unconditionally Unique Solution	p. 227
7.7.1 Combining Single and Double-layer Potentials	p. 228
7.8 The Transmission Problem	p. 234
7.9 Jones' Method	p. 237
7.10 Appendix A.7.1	p. 239
7.11 Appendix A.7.2	p. 241
7.12 Appendix A.7.3	p. 242
7.13 Appendix A.7.4	p. 243
7.14 Appendix A.7.5	p. 245
7.15 Appendix A.7.6	p. 245
8 Uniqueness Theorems in Inverse Problems	p. 247
8.1 Some Definitions	p. 247
8.2 Properties of the Total Fields	p. 249
8.2.1 Obstacle Scattering: Linear Independence of Total Fields	p. 249
8.2.2 Inhomogeneity Scattering	p. 250
8.3 The Dirichlet and Neumann Spectrum	p. 253
8.3.1 The Spectrum of the Negative of the Dirichlet Laplacian in a Bounded Domain	p. 253
8.3.2 The Analysis of the Neumann Laplacian	p. 256
8.4 The Uniqueness of the Inverse Dirichlet Obstacle Problem	p. 259
8.5 The Uniqueness of Inverse Neumann Obstacle	p. 261
8.6 Uniqueness in Inverse Transmission Obstacle	p. 267
8.7 Uniqueness of Inverse Inhomogeneity Scattering	p. 269
8.8 Appendix A.8.1	p. 277
8.9 Appendix A.8.2	p. 281
9 Some Algorithms	p. 283
9.1 Introduction	p. 283
9.2 The Method of Potentials	p. 286
9.3 The Method of Superposition of Incident Fields	p. 287
9.4 The Method of Wavefunction Expansion	p. 290
9.5 The Method of Boundary Variation	p. 292
9.5.1 Dirichlet and Neumann Problem In Two-dimension	p. 297
9.5.2 Transmission Problem in Two-dimensions	p. 298
9.6 Some Nonoptimizational Methods	p. 300
9.6.1 The Method of Colton and Kirsch	p. 301
9.6.2 The Method of Eigensystem of the Far-field Operator	p. 303
9.7 Appendix A.9.1	p. 305
9.8 Appendix A.9.2	p. 309
9.9 Appendix A.9.3	p. 311
10 Bibliography	p. 315

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