Stable Solution of Inverse Problems

ISBN-13: 9783528089610 / Niemiecki / Miękka / 1987 / 264 str.

Johann Baumeister

Opis

These notes are intended to describe the basic concepts of solving inverse problems in a stable way. Since almost all in verse problems are ill-posed in its original formulation the discussion of methods to overcome difficulties which result from this fact is the main subject of this book. Over the past fifteen years, the number of publications on inverse problems has grown rapidly. Therefore, these notes can be neither a comprehensive introduction nor a complete mono graph on the topics considered; it is designed to provide the main ideas and methods. Throughout, we have not striven for the most general statement, but the clearest one which would cover the most situations. The presentation is intended to be accessible to students whose mathematical background includes basic courses in ad vanced calculus, linear algebra and functional analysis. Each chapter contains bibliographical comments. At the end of Chap ter 1 references are given which refer to topics which are not studied in this book. I am very grateful to Mrs. B. Brodt for typing and to W. Scondo and u. Schuch for inspecting the manuscript.

Spis treści

I Basic Concepts.- 1 Introduction.- 1.1 Inverse problems.- 1.2 Some examples of inverse problems.- 1.3 Analysis of inverse problems.- 2 Ill-posed problems.- 2.1 General properties.- 2.2 Restoration of continuity in the linear case.- 2.3 Stability estimates.- 3 Regularization.- 3.1 Reconstruction from non-exact data.- 3.2 Preliminary results on Tikhonov’s method.- 3.3 Regularizing schemes.- 3.4 A tutorial example: The reconstruction of a derivative.- 3.5 Optimal reconstruction of linear functionals.- II Regularization Methods.- 4 The singular value decomposition.- 4.1 Compact operators.- 4.2 The spectrum of compact selfadjoint operators.- 4.3 The singular value decomposition.- 4.4 The min-max principle.- 4.5 The asymptotics of singular values.- 4.6 Picard’s criterion.- 5 Applications of the singular value decomposition.- 5.1 Hilbert scales.- 5.2 Convergence of regularizing schemes.- 5.3 On the use of the conjugate gradient method.- 5.4 n-widths.- 6 The method of Tikhonov.- 6.1 The generalized inverse.- 6.2 The classical method of Tikhonov.- 6.3 Error bounds for Tikhonov regularization in Hilbert scales.- 6.4 On discrepancy principles.- 6.5 Discretization in Tikhonov’s method.- 7 Regularization by discretization.- 7.1 Discretization by projection methods.- 7.2 Quasioptimality and robustness.- 7.3 Specific methods.- 7.4 Asymptotic estimates.- III Least Squares Solutions of Systems of Linear Equations.- 8 Least squares problems.- 8.1 The singular value decomposition of a matrix.- 8.2 The pseudo-inverse.- 8.3 Least squares solutions.- 8.4 Perturbation results.- 8.5 Application: Fitting of data.- 9 Numerical aspects of least squares problems.- 9.1 Calculation of A+: The factorization approach.- 9.2 Rank decision.- 9.3 Cross-validation.- 9.4 Successive approximation.- 9.5 The ART-algorithm.- IV Specific Topics.- 10 Convolution equations.- 10.1 The Fourier transform.- 10.2 Regularization of convolution equations.- 10.3 On the discretization of convolution equations.- 10.4 Reconstruction by successive approximation.- 11 The final value problem.- 11.1 Introduction.- 11.2 The mild solution of the forward problem.- 11.3 The Hilbert scales Ea,t.- 11.4 Regularizing schemes.- 12 Parameter identification.- 12.1 Identifiability of parameters in dynamical systems.- 12.2 Identification in linear dynamic systems.- 12.3 Identification in bilinear structures.- 12.4 Adaptive identification.- References.- Notations.