## An Introduction to the Mathematical Theory of Inverse ProblemsFollowing 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 |

261 | |

279 | |

### Other editions - View all

An Introduction to the Mathematical Theory of Inverse Problems Andreas Kirsch No preview available - 2011 |

An Introduction to the Mathematical Theory of Inverse Problems Andreas Kirsch No preview available - 2011 |

An Introduction to the Mathematical Theory of Inverse Problems Andreas Kirsch No preview available - 1996 |

### Common terms and phrases

algorithm Appendix apply approximation assume assumptions asymptotic Banach spaces boundary conditions boundary value problem bounded operator Cauchy Cauchy-Schwarz inequality collocation method compact operator compute conclude conjugate gradient method corresponding defined Definition denotes dense derivative differential equation direct problem discrepancy principle discrete eigenfunctions eigenvalue problem equation Kx example exists field pattern finite finite-dimensional following theorem formulate Furthermore Galerkin method given Helmholtz equation Hilbert spaces ill-posed implies inequality inner product integral operator inverse problem kernel least squares method Lemma mapping minimize nonlinear normed space numerical one-to-one optimal orthogonal projection orthonormal system parameter perturbation polynomial Proof properties regularization strategy respect right-hand side satisfies Section self-adjoint sequence singular system singular values Sobolev spaces solves spectral subspace tend to infinity theory Tikhonov Tikhonov regularization uniformly unique solution uniquely solvable vanishes vector Volterra worst-case error yields

### Popular passages

Page 263 - On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980).