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 | 9 |
Compact Operator Equations | 11 |
The HausdorffTikhonov Lemma | 17 |
Computerized XRay Tomography | 23 |
The Cauchy Problem for the Laplace Equation | 31 |
The Factorization Method in EIT | 39 |
Tikhonov Regularization | 47 |
The Discrepancy Principle | 91 |
Iterative Solution of the Cauchy Problem | 97 |
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 |
Tikhonovs Method | 49 |
The Discrepancy Principle | 55 |
Smoothing Splines | 61 |
Deconvolution and Imaging | 67 |
Evaluation of Unbounded Operators | 75 |
Iterative Regularization | 83 |
Landweber Iteration | 85 |
The Poisson Equation | 139 |
Sobolev Spaces cont | 143 |
E The Fourier Transform | 147 |
F The Conjugate Gradient Iteration | 151 |
| 155 | |
| 161 | |
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Sık kullanılan terimler ve kelime öbekleri
adjoint Algorithm 15.1 Appendix approximation error assume assumptions boundary conditions boundary value problem bounded Lipschitz domain Cauchy problem Cauchy sequence CGLS Chapter compact operator conclude conjugate gradient constant continuous converges corresponding data g defined denote dense range R(K density Dirichlet data discrepancy principle Example f₁ Figure follows Fourier transform function g given data Green's formula H¹(D hence Hilbert space ill-posed problem implies inequality injective with dense inner product inverse problem iteration error K-¹g L²(D L²(M L²(Rd Landweber iteration Lemma linear Lipschitz domain Neumann derivative noise level nonzero numerical differentiation problem perturbed Proof Proposition Radon transform refer the reader regularization parameter residuals result right-hand side satisfies singular value decomposition Sobolev space solves subspace Theorem Tikhonov regularization tion tomography vanishing vector virtue zero ΩΤ