Inverse problems and inverse scattering of plane waves Book

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