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Convergence Rates For Variational Regularization Of Inverse Problems In Exponential Families
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Book Synopsis Convergence Rates for Variational Regularization of Inverse Problems in Exponential Families by : Yusufu Simayi
Download or read book Convergence Rates for Variational Regularization of Inverse Problems in Exponential Families written by Yusufu Simayi and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider statistical inverse problems with statistical noise. By using regularization methods one can approximate the true solution of the inverse problem by a regularized solution. The previous investigation of convergence rates for variational regularization with Poisson and empirical process data is shown to be suboptimal. In this thesis we obtain improved convergence rates for variational regularization methods of nonlinear ill-posed inverse problems with certain stochastic noise models described by exponential families and derive better reconstruction error bounds by applying deviat...
Book Synopsis Convergence Rates for Variational Regularization of Inverse Problems in Exponential Families by : Simayi Yusufu
Download or read book Convergence Rates for Variational Regularization of Inverse Problems in Exponential Families written by Simayi Yusufu and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider statistical inverse problems with statistical noise. By using regularization methods one can approximate the true solution of the inverse problem by a regularized solution. The previous investigation of convergence rates for variational regularization with Poisson and empirical process data is shown to be suboptimal. In this thesis we obtain improved convergence rates for variational regularization methods of nonlinear ill-posed inverse problems with certain stochastic noise models described by exponential families and derive better reconstruction error bounds by applying deviat...
Book Synopsis Convergence Rates for Variational Regularization of Statistical Inverse Problems by : Benjamin Sprung
Download or read book Convergence Rates for Variational Regularization of Statistical Inverse Problems written by Benjamin Sprung and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider inverse problems with statistical (noisy) data. By applying regularization methods one can approximate the true solution of the inverse problem by a regularized solution. In this thesis we show convergence rates of the regularized solution to the true solution as the noise tends to zero under so called source conditions on the true solution. Recently variational source conditions (VSCs) have become increasingly popular, due to their generality. However, they have the disadvantage that they only give optimal rates for low smoothness of the true solution. For this reason a second ...
Book Synopsis Inverse Problems: Tikhonov Theory And Algorithms by : Kazufumi Ito
Download or read book Inverse Problems: Tikhonov Theory And Algorithms written by Kazufumi Ito and published by World Scientific. This book was released on 2014-08-28 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems arise in practical applications whenever one needs to deduce unknowns from observables. This monograph is a valuable contribution to the highly topical field of computational inverse problems. Both mathematical theory and numerical algorithms for model-based inverse problems are discussed in detail. The mathematical theory focuses on nonsmooth Tikhonov regularization for linear and nonlinear inverse problems. The computational methods include nonsmooth optimization algorithms, direct inversion methods and uncertainty quantification via Bayesian inference.The book offers a comprehensive treatment of modern techniques, and seamlessly blends regularization theory with computational methods, which is essential for developing accurate and efficient inversion algorithms for many practical inverse problems.It demonstrates many current developments in the field of computational inversion, such as value function calculus, augmented Tikhonov regularization, multi-parameter Tikhonov regularization, semismooth Newton method, direct sampling method, uncertainty quantification and approximate Bayesian inference. It is written for graduate students and researchers in mathematics, natural science and engineering.
Book Synopsis Regularization of Inverse Problems by : Heinz Werner Engl
Download or read book Regularization of Inverse Problems written by Heinz Werner Engl and published by Springer Science & Business Media. This book was released on 2000-03-31 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the mathematical theory of regularization methods and gives an account of the currently available results about regularization methods for linear and nonlinear ill-posed problems. Both continuous and iterative regularization methods are considered in detail with special emphasis on the development of parameter choice and stopping rules which lead to optimal convergence rates.
Book Synopsis Variational Regularization for Systems of Inverse Problems by : Richard Huber
Download or read book Variational Regularization for Systems of Inverse Problems written by Richard Huber and published by Springer. This book was released on 2019-02-14 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Regularization of Ill-posed Inverse Problems with Tolerances and Sparsity in the Parameter Space by : Georgia Sfakianaki
Download or read book Regularization of Ill-posed Inverse Problems with Tolerances and Sparsity in the Parameter Space written by Georgia Sfakianaki and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the solution of ill-posed inverse problems using regularization with tolerances. In particular, we are interested in the reconstruction of solutions that lie within or close to an area outlined by a tolerance measure. To approximate the true solution of the problem in a stable way, we propose a Tikhonov functional with a tolerance function in the regularization term. The tolerances allow us to neglect errors in the penalty term up to a certain threshold. Our theoretical analysis proves that the proposed method complies with all the requirements of variational regularization methods. In addition, we establish convergence rates for the convergence of minimizers to the true solution. Moreover, we are interested in obtaining sparse solutions. For this purpose, we extend the proposed approach with the idea of elastic net regularization by introducing an additional penalty term that promotes the sparsity of the solution. We establish theoretical results for this elastic net approach and give a convergence rate analysis for the minimizers. To confirm our analytical findings, we illustrate the effect of tolerances in the computed regularized solutions on some numerical examples.
Book Synopsis Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications by :
Download or read book Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications written by and published by . This book was released on 2007 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications by : Nicolai Bissantz
Download or read book Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications written by Nicolai Bissantz and published by . This book was released on 2007 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Iterative Regularization Methods for Nonlinear Ill-Posed Problems by : Barbara Kaltenbacher
Download or read book Iterative Regularization Methods for Nonlinear Ill-Posed Problems written by Barbara Kaltenbacher and published by Walter de Gruyter. This book was released on 2008-09-25 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.
Book Synopsis Regularization Algorithms for Ill-Posed Problems by : Anatoly B. Bakushinsky
Download or read book Regularization Algorithms for Ill-Posed Problems written by Anatoly B. Bakushinsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-02-05 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This specialized and authoritative book contains an overview of modern approaches to constructing approximations to solutions of ill-posed operator equations, both linear and nonlinear. These approximation schemes form a basis for implementable numerical algorithms for the stable solution of operator equations arising in contemporary mathematical modeling, and in particular when solving inverse problems of mathematical physics. The book presents in detail stable solution methods for ill-posed problems using the methodology of iterative regularization of classical iterative schemes and the techniques of finite dimensional and finite difference approximations of the problems under study. Special attention is paid to ill-posed Cauchy problems for linear operator differential equations and to ill-posed variational inequalities and optimization problems. The readers are expected to have basic knowledge in functional analysis and differential equations. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems, and also to advanced students in these fields. Contents Introduction Regularization Methods For Linear Equations Finite Difference Methods Iterative Regularization Methods Finite-Dimensional Iterative Processes Variational Inequalities and Optimization Problems
Book Synopsis Sampling Theory, a Renaissance by : Götz E. Pfander
Download or read book Sampling Theory, a Renaissance written by Götz E. Pfander and published by Birkhäuser. This book was released on 2015-12-08 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: Reconstructing or approximating objects from seemingly incomplete information is a frequent challenge in mathematics, science, and engineering. A multitude of tools designed to recover hidden information are based on Shannon’s classical sampling theorem, a central pillar of Sampling Theory. The growing need to efficiently obtain precise and tailored digital representations of complex objects and phenomena requires the maturation of available tools in Sampling Theory as well as the development of complementary, novel mathematical theories. Today, research themes such as Compressed Sensing and Frame Theory re-energize the broad area of Sampling Theory. This volume illustrates the renaissance that the area of Sampling Theory is currently experiencing. It touches upon trendsetting areas such as Compressed Sensing, Finite Frames, Parametric Partial Differential Equations, Quantization, Finite Rate of Innovation, System Theory, as well as sampling in Geometry and Algebraic Topology.
Book Synopsis A Note on Convergence Rates for Variational Regularization with Non-convex Residual Term by : Bernd Hofmann
Download or read book A Note on Convergence Rates for Variational Regularization with Non-convex Residual Term written by Bernd Hofmann and published by . This book was released on 2009 with total page 7 pages. Available in PDF, EPUB and Kindle. Book excerpt: Literaturverz. S. 6 - 7.
Book Synopsis Convergence Rates for Tikhonov Regularization of Implicitly Defined Nonlinear Inverse Problems with an Application to Inverse Scattering by : Heinz W. Engl
Download or read book Convergence Rates for Tikhonov Regularization of Implicitly Defined Nonlinear Inverse Problems with an Application to Inverse Scattering written by Heinz W. Engl and published by . This book was released on 1992 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Regularization for Applied Inverse and Ill-Posed Problems by : Bernd Hofmann
Download or read book Regularization for Applied Inverse and Ill-Posed Problems written by Bernd Hofmann and published by Springer. This book was released on 1986-06 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On Convergence Rates of Some Iterative Regularization Methods for an Inverse Problem for a Nonlinear Parabolic Equation Connected with Continous Casting of Steel by : Barbara Kaltenbacher
Download or read book On Convergence Rates of Some Iterative Regularization Methods for an Inverse Problem for a Nonlinear Parabolic Equation Connected with Continous Casting of Steel written by Barbara Kaltenbacher and published by . This book was released on 1997 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Handbook of Mathematical Methods in Imaging by : Otmar Scherzer
Download or read book Handbook of Mathematical Methods in Imaging written by Otmar Scherzer and published by Springer Science & Business Media. This book was released on 2010-11-23 with total page 1626 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing. Each section within the themes covers applications (modeling), mathematics, numerical methods (using a case example) and open questions. Written by experts in the area, the presentation is mathematically rigorous. The entries are cross-referenced for easy navigation through connected topics. Available in both print and electronic forms, the handbook is enhanced by more than 150 illustrations and an extended bibliography. It will benefit students, scientists and researchers in applied mathematics. Engineers and computer scientists working in imaging will also find this handbook useful.
