An Introduction to the Mathematical Theory of Inverse Problems

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Springer Science & Business Media, Sep 26, 1996 - Science - 300 pages
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Following Keller [119] we call two problems inverse to each other if the for mulation of each of them requires full or partial knowledge of the other. By this definition, it is obviously arbitrary which of the two problems we call the direct and which we call the inverse problem. But usually, one of the problems has been studied earlier and, perhaps, in more detail. This one is usually called the direct problem, whereas the other is the inverse problem. However, there is often another, more important difference between these two problems. Hadamard (see [91]) introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem ill-posed. It turns out that many interesting and important inverse in science lead to ill-posed problems, while the corresponding di problems rect problems are well-posed. Often, existence and uniqueness can be forced by enlarging or reducing the solution space (the space of "models"). For restoring stability, however, one has to change the topology of the spaces, which is in many cases impossible because of the presence of measurement errors. At first glance, it seems to be impossible to compute the solution of a problem numerically if the solution of the problem does not depend continuously on the data, i. e. , for the case of ill-posed problems.
 

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Contents

Introduction and Basic Concepts
1
12 IllPosed Problems
9
13 The WorstCase Error
14
14 Problems
21
Regularization Theory for Equations of the First Kind
23
21 A General Regularization Theory
24
22 Tikhonov Regularization
37
23 Landweber Iteration
42
42 Construction of a Fundamental System
127
43 Asymptotics of the Eigenvalues and Eigenfunctions
135
44 Some Hyperbolic Problems
146
45 The Inverse Problem
154
46 A Parameter Identification Problem
160
47 Numerical Reconstruction Techniques
165
48 Problems
170
An Inverse Scattering Problem
173

24 A Numerical Example
45
25 The Discrepancy Principle of Morozov
48
26 Landwebers Iteration Method with Stopping Rule
53
27 The Conjugate Gradient Method
57
28 Problems
63
Regularization by Discretization
65
31 Projection Methods
66
32 Galerkin Methods
73
321 The Least Squares Method
76
322 The Dual Least Squares Method
78
323 The BubnovGalerkin Method for Coercive Operators
80
33 Application to Symms Integral Equation of the First Kind
85
34 Collocation Methods
94
341 Minimum Norm Collocation
95
342 Collocation of Symms Equation
99
35 Numerical Experiments for Symms Equation
107
36 The BackusGilbert Method
115
37 Problems
122
Inverse Eigenvalue Problems
125
52 The Direct Scattering Problem
177
53 Properties of the Far Field Patterns
187
54 Uniqueness of the Inverse Problem
196
55 Numerical Methods
204
551 A Simplified Newton Method
205
552 A Modified Gradient Method
209
553 The Dual Space Method
210
56 Problems
214
Basic Facts from Functional Analysis
215
A2 Orthonormal Systems
222
A3 Linear Bounded and Compact Operators
224
A4 Sobolev Spaces of Periodic Functions
232
A5 Spectral Theory for Compact Operators in Hilbert Spaces
239
A6 The Frechet Derivative
243
Proofs of the Results of Section 27
249
References
261
Index
279
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Page 263 - On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980).

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