New Analytic and Geometric Methods in Inverse Problems: Lectures Given at the EMS Summer School and Conference Held in Edinburgh, Scotland 2000

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Springer Science & Business Media, Jan 12, 2004 - Mathematics - 381 pages
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In inverse problems, the aim is to obtain, via a mathematical model, information on quantities that are not directly observable but rather depend on other observable quantities. Inverse problems are encountered in such diverse areas of application as medical imaging, remote sensing, material testing, geosciences and financing. It has become evident that new ideas coming from differential geometry and modern analysis are needed to tackle even some of the most classical inverse problems. This book contains a collection of presentations, written by leading specialists, aiming to give the reader up-to-date tools for understanding the current developments in the field.

  

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Contents

Metric Geometry
3
Intrinsic Metrics
7
Angles
11
Riemannian Metrics
12
Covariant Derivatives
19
Curvature
23
Comparison Theorems
28
Alexandrov Spaces
31
The Ray Transform
211
Inversion of the Ray Transform
225
The Modified Horizontal Derivative
238
On the Local DirichlettoNeumann Map
261
The Cauchy Data for the Schrödinger Equation
269
Semiclassical Complex Geometrical Solutions
275
EMS Conference Recent Developments in the Wave Field and Diffuse Tomographic Inverse Problems
281
Remarks on the Inverse Scattering Problem for Acoustic Waves
283

Meaning of Sectional Curvature Size and Pinching
34
Lipschitz and GromovHausdorff Convergences
36
LargeScale Geometry
42
Intertwining Operators in Inverse Scattering
51
Direction Dependent Fundamental Solutions of the Operator ᐃx ᐃy
53
Construction of Intertwining Operators for Small Potentials
63
An Expression for the Scattering Matrix and the Inverse Problem
78
Carleman Type Estimates and Their Applications
93
Carleman Type Estimates and PseudoConvexity
94
Use of a Second Large Parameter
102
Uniqueness and Stability in the Cauchy Problem
104
Applications to Inverse Problems
115
Sharp Uniqueness of the Continuation Results
119
Gaussian Beams and Inverse Boundary Spectral Problems
127
Gauge Transformations
129
Boundary Spectral Data and Main Results
131
Spectral Representation of Waves
133
Gaussian Beams
135
The HamiltonJacobi Equation and Transport Equations
136
Riccati Equation
140
Gaussian Beams from the Boundary
148
Construction of Manifold and Boundary Distance Functions
151
Analytic Methods for Inverse Scattering Theory
165
Maximal Functions and Sobolev Spaces
167
Mapping Properties of ᐃ + k21
171
Faddeevs Greens Function
174
Uniqueness of the Inverse Scattering Problem
176
Born Approximation
181
Ray Transform on Riemannian Manifolds
187
Twodimensional Integral Geometry Problem
188
Some Questions of Tensor Analysis
195
The Linear Sampling Method
285
Piecewise Homogeneous Background Medium
287
Asymptotic Properties of Solutions to 3particle Schrodinger Equations
291
3body Schrödinger Operator
297
Analysis of Ha
301
Timedependent Scattering Theory and Generalized Fourier Transformation for H
302
Main Theorems
305
Stability and Reconstruction in Gelfand Inverse Boundary Spectral Problem
309
Main Results Stability
311
Main Results Approximate Reconstruction
314
Construction of a Finite enet
315
Concluding Remarks
320
Uniqueness in Inverse Obstacle Scattering
323
Global Uniqueness
325
Local Uniqueness
330
Geometric Methods for Anisotopic Inverse Boundary Value Problems
337
Scalar Anisotropic Inverse Conductivity Problem
339
Linearization
342
Constrained Anisotropic Problems
343
Laplacians on Forms
344
Linear Elasticity
345
Maxwells Equations
346
Symbols and PseudoDifferential Operators
347
Applications of the OscillatingDecaying Solutions to Inverse Problems
353
Pointwise Determination at the Boundary from the Localized DirichlettoNeumann Map
354
Identification of a Polygonal Cavity in a Conductive Medium
360
Identification of Inclusion in a Conductive Medium
363
TimeDependent Methods in Inverse Scattering Theory
367
The Proofs
373
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Page 336 - Frechet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain. J. Inverse Ill-Posed Probl. 4, 67-84 34.

About the author (2004)

Erkki Somersalo is a Professor at Case Western Reserve University. His areas of expertise are computational inverse problems, statistical scientific computing, and fields and waves, in particular with applications to medical imaging. His work also includes applications in life sciences and medicine. He has published two monographs and over a hundred research papers.

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