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BL
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| Record Number
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881855
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| Main Entry
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Huber, Richard
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| Title & Author
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Variational regularization for systems of inverse problems : : Tikhonov regularization with multiple forward operators /\ Richard Huber.
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| Publication Statement
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Wiesbaden, Germany :: Springer Spektrum,, 2019.
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| Series Statement
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BestMasters,
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| Page. NO
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1 online resource (ix, 136 pages) :: illustrations
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| ISBN
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3658253908
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: 9783658253905
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3658253894
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9783658253899
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| Bibliographies/Indexes
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Includes bibliographical references.
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| Contents
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Intro; Acknowledgements; Contents; List of Figures; I. Introduction; I.1. Motivation; I.2. Mathematical Foundation; I.2.1 Topologies; I.2.2 Normed Vector Spaces; I.2.3 Measure Theory; I.2.4 Convex Analysis; II. General Tikhonov Regularisation; II. 1. Single-Data Tikhonov Regularisation; II. 1.1 Existence and Stability; II. 1.2 Convergence; II. 2. Multi-Data Tikhonov Regularisation; II. 2.1 Preliminaries; II. 2.2 Parameter Choices for Vanishing Noise; II. 2.3 Convergence rates; III. Specific Discrepancies; III. 1. Norm Discrepancies; III. 1.1 Classical Norms; III. 1.2 Subnorms
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III. 2. Kullback-Leibler DivergenceIII. 2.1 Motivation; III. 2.2 Basic Properties; III. 2.3 Continuity Results; III. 2.4 Applicability as a Discrepancy; IV. Regularisation Functionals; IV. 1. Regularisation with Norms and Closed Operators; IV. 2. Total Deformation; IV. 2.1 Symmetric Tensor Fields; IV. 2.2 Tensor Fields of Bounded Deformation; IV. 3. Total Generalised Variation; IV. 3.1 Basic Properties; IV. 3.2 Topological Properties; IV. 3.3 Total Generalised Variation of Vector-Valued Functions; IV. 4. TGV Regularisation in a Linear Setting; V. Application to STEM Tomography Reconstruction
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v. 1. The Radon TransformV. 1.1 Deriving the Radon Transform; V.1.2 Analytical Properties; V.1.3 Filtered Backprojection; V.2. Tikhonov Approach to Multi-Spectra STEM Tomography Reconstruction; V.2.1 Continuous Tikhonov Problem for STEM Tomography Reconstruction; V.2.2 Discretisation Scheme; V.2.3 Primal-Dual Optimisation Algorithm; V.2.4 STEM Tomography Reconstruction Algorithm; V.3. Discussion of Numerical Results; V.3.1 Preprocessing; V.3.2 Synthetic Experiments; V.3.3 Reconstruction of Single-Data HAADF Signals; V.3.4 STEM Multi-Spectral Reconstructions; Summary; Bibliography
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| Abstract
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Tikhonov regularization is a cornerstone technique in solving inverse problems with applications in countless scientific fields. Richard Huber discusses a multi-parameter Tikhonov approach for systems of inverse problems in order to take advantage of their specific structure. Such an approach allows to choose the regularization weights of each subproblem individually with respect to the corresponding noise levels and degrees of ill-posedness. Contents General Tikhonov Regularization Specific Discrepancies Regularization Functionals Application to STEM Tomography Reconstruction Target Groups Researchers and students in the field of mathematics Experts in the areas of mathematics, imaging, computer vision and nanotechnology The Author Richard Huber wrote his master?s thesis under the supervision of Prof. Dr. Kristian Bredies at the Institute for Mathematics and Scientific Computing at Graz University, Austria.
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| Subject
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Inverse problems (Differential equations)
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| Subject
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Inverse problems (Differential equations)
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| Dewey Classification
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515/.357
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| LC Classification
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QA371
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