A Taste of Inverse Problems: Basic Theory and ExamplesSIAM, 1 Oca 2017 - 170 sayfa Inverse problems need to be solved in order to properly interpret indirect measurements. Often, inverse problems are ill-posed and sensitive to data errors. Therefore one has to incorporate some sort of regularization to reconstruct significant information from the given data. A Taste of Inverse Problems: Basic Theory and Examples?presents the main achievements that have emerged in regularization theory over the past 50 years, focusing on linear ill-posed problems and the development of methods that can be applied to them. Some of this material has previously appeared only in journal articles. This book rigorously discusses state-of-the-art inverse problems theory, focusing on numerically relevant aspects and omitting subordinate generalizations; presents diverse real-world applications, important test cases, and possible pitfalls; and treats these applications with the same rigor and depth as the theory. |
İçindekiler
Numerical Differentiation | 1 |
IllPosed Problems | 4 |
Compact Operator Equations | 11 |
The HausdorffTikhonov Lemma | 17 |
Computerized XRay Tomography | 23 |
The Cauchy Problem for the Laplace Equation | 31 |
The Factorization Method in | 39 |
Tikhonov Regularization | 47 |
The Conjugate Gradient Method | 103 |
The GerchbergPapoulis Algorithm | 111 |
Inverse Source Problems | 119 |
Appendices | 127 |
A The Singular Value Decomposition | 129 |
B Sobolev Spaces | 133 |
The Poisson Equation | 139 |
Sobolev Spaces cont | 143 |
Tikhonovs Method | 49 |
The Discrepancy Principle | 55 |
Smoothing Splines | 61 |
Deconvolution and Imaging | 67 |
Evaluation of Unbounded Operators | 75 |
Iterative Regularization | 83 |
Landweber Iteration | 92 |
The Discrepancy Principle | 95 |
Iterative Solution of the Cauchy Problem | 97 |
E The Fourier Transform | 147 |
F The Conjugate Gradient Iteration | 151 |
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according Algorithm Appendix application approximation assume assumptions belongs boundary value problem bounded Cauchy problem CGLS Chapter choice choosing closed compact conclude condition consider constant continuous converges corresponding data g defined denote dense range depends derivative determined differentiation Dirichlet domain element equation error estimate exact Example exists fact Figure follows Fourier transform function further given harmonic hence Hilbert space holds ill-posed implies inequality injective inner product integral inverse iteration known Landweber iteration Lemma linear means measured method minimal Neumann noise norm Note numerical obtain operator parameter particular problem Proof Proposition prove refer regularization residuals respectively result satisfies sequence shows side singular values solution solves space Theorem tion trace true unique vanishing virtue weak yields zero