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Inhaltsangabe
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.
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.
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