# An Introduction to the Mathematical Theory of Inverse Problems

Springer Science & Business Media, 26.09.1996 - 300 Seiten
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Following Keller  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 ) 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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### Inhalt

 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 27 247 References 259 Index 277 Urheberrecht

### Beliebte Passagen

Seite 261 - On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980).