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Regularization Methods In Banach Spaces
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Book Synopsis Regularization Methods in Banach Spaces by : Thomas Schuster
Download or read book Regularization Methods in Banach Spaces written by Thomas Schuster and published by Walter de Gruyter. This book was released on 2012-07-30 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: Regularization methods aimed at finding stable approximate solutions are a necessary tool to tackle inverse and ill-posed problems. Inverse problems arise in a large variety of applications ranging from medical imaging and non-destructive testing via finance to systems biology. Many of these problems belong to the class of parameter identification problems in partial differential equations (PDEs) and thus are computationally demanding and mathematically challenging. Hence there is a substantial need for stable and efficient solvers for this kind of problems as well as for a rigorous convergence analysis of these methods. This monograph consists of five parts. Part I motivates the importance of developing and analyzing regularization methods in Banach spaces by presenting four applications which intrinsically demand for a Banach space setting and giving a brief glimpse of sparsity constraints. Part II summarizes all mathematical tools that are necessary to carry out an analysis in Banach spaces. Part III represents the current state-of-the-art concerning Tikhonov regularization in Banach spaces. Part IV about iterative regularization methods is concerned with linear operator equations and the iterative solution of nonlinear operator equations by gradient type methods and the iteratively regularized Gauß-Newton method. Part V finally outlines the method of approximate inverse which is based on the efficient evaluation of the measured data with reconstruction kernels.
Book Synopsis Regularization in Banach Spaces - Convergence Rates Theory by : Torsten Hein
Download or read book Regularization in Banach Spaces - Convergence Rates Theory written by Torsten Hein and published by Logos Verlag Berlin GmbH. This book was released on 2010 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motivated by their successful application in image restoring and sparsity reconstruction this manuscript deals with regularization theory of linear and nonlinear inverse and ill-posed problems in Banach space settings. Whereas regularization in Hilbert spaces has been widely studied in literature for a long period the developement and investigation of regularization methods in Banach spaces have become a field of modern research. The manuscript is twofolded. The first part deals with convergence rates theory for Tikhonov regularization as classical regularization method. In particular, generalizations of well-established results in Hilbert spaces are presented in the Banach space situation. Since the numerical effort of Tikhonov regularization in applications is rather high iterative approaches were considered as alternative regularization variants in the second part. In particular, two Gradient-type methods were presented and their behaviour concerning convergence and stability is investigated. For one of the methods, additionally, a convergence rates result is formulated. All the theoretical results are illustrated by some numerical examples.
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 Aspects of Regularization in Banach Spaces by : Kamil S. Kazimierski
Download or read book Aspects of Regularization in Banach Spaces written by Kamil S. Kazimierski and published by Logos Verlag Berlin GmbH. This book was released on 2010 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years there has been an increasing interest in the regularization of ill-posed inverse problems for operators mapping between two Banach spaces. This thesis focuses on the case of linear, continuous operators and Banach spaces, which are convex of power type and/or smooth of power type. The main aim is to present new results regarding the Tikhonov regularization and the Landweber regularization, some of which are: convexity and smoothness properties of the wavelet characterization of the norm of Besov spaces, generalization of the discrepancy principle of Engl to the setting of Banach spaces, convergence rates for two minimization methods for the Tikhonov functional, adaptation of the Landweber iteration to Banach spaces convex of power type and smooth of power type and introduction of a modified version of the Landweber iteration. The quality of the algorithms introduced in this thesis is discussed with help of several numerical examples.
Book Synopsis Novel Regularization Methods for Ill-posed Problems in Hilbert and Banach Spaces by : Ismael Rodrigo Bleyer
Download or read book Novel Regularization Methods for Ill-posed Problems in Hilbert and Banach Spaces written by Ismael Rodrigo Bleyer and published by . This book was released on 2015 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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 342 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 On quasioptional parameter choices and stopping rules for regularization methods in Banach spaces by : Robert Plato
Download or read book On quasioptional parameter choices and stopping rules for regularization methods in Banach spaces written by Robert Plato and published by . This book was released on 1994 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization by : Jens Flemming
Download or read book Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization written by Jens Flemming and published by Springer. This book was released on 2018-09-08 with total page 182 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book collects and contributes new results on the theory and practice of ill-posed inverse problems. Different notions of ill-posedness in Banach spaces for linear and nonlinear inverse problems are discussed not only in standard settings but also in situations up to now not covered by the literature. Especially, ill-posedness of linear operators with uncomplemented null spaces is examined.Tools for convergence rate analysis of regularization methods are extended to a wider field of applicability. It is shown that the tool known as variational source condition always yields convergence rate results. A theory for nonlinear inverse problems with quadratic structure is developed as well as corresponding regularization methods. The new methods are applied to a difficult inverse problem from laser optics.Sparsity promoting regularization is examined in detail from a Banach space point of view. Extensive convergence analysis reveals new insights into the behavior of Tikhonov-type regularization with sparsity enforcing penalty.
Book Synopsis Nonlinear Ill-posed Problems of Monotone Type by : Yakov Alber
Download or read book Nonlinear Ill-posed Problems of Monotone Type written by Yakov Alber and published by Springer Science & Business Media. This book was released on 2006-02-23 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interest in regularization methods for ill-posed nonlinear operator equations and variational inequalities of monotone type in Hilbert and Banach spaces has grown rapidly over recent years. Results in the field over the last three decades, previously only available in journal articles, are comprehensively explored with particular attention given to applications of regularization methods as well as to practical methods used in computational analysis.
Book Synopsis Regularization Methods for Ill-Posed Optimal Control Problems by : Frank Pörner
Download or read book Regularization Methods for Ill-Posed Optimal Control Problems written by Frank Pörner and published by BoD – Books on Demand. This book was released on 2018-10-04 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ill-posed optimization problems appear in a wide range of mathematical applications, and their numerical solution requires the use of appropriate regularization techniques. In order to understand these techniques, a thorough analysis is inevitable. The main subject of this book are quadratic optimal control problems subject to elliptic linear or semi-linear partial differential equations. Depending on the structure of the differential equation, different regularization techniques are employed, and their analysis leads to novel results such as rate of convergence estimates.
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 342 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. ContentsIntroductionRegularization Methods For Linear EquationsFinite Difference MethodsIterative Regularization MethodsFinite-Dimensional Iterative ProcessesVariational Inequalities and Optimization Problems
Book Synopsis On quasioptimal parameter choices and stopping rules for regularization methods in Banach spaces by : Robert Plato
Download or read book On quasioptimal parameter choices and stopping rules for regularization methods in Banach spaces written by Robert Plato and published by . This book was released on 1994 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Regularization for Applied Inverse and Ill-Posed Problems by :
Download or read book Regularization for Applied Inverse and Ill-Posed Problems written by and published by Springer-Verlag. This book was released on 2013-11-22 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Recovery Methodologies: Regularization and Sampling by : Willi Freeden
Download or read book Recovery Methodologies: Regularization and Sampling written by Willi Freeden and published by American Mathematical Society. This book was released on 2023-08-21 with total page 505 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this book is to introduce the reader to methodologies in recovery problems for objects, such as functions and signals, from partial or indirect information. The recovery of objects from a set of data demands key solvers of inverse and sampling problems. Until recently, connections between the mathematical areas of inverse problems and sampling were rather tenuous. However, advances in several areas of mathematical research have revealed deep common threads between them, which proves that there is a serious need for a unifying description of the underlying mathematical ideas and concepts. Freeden and Nashed present an integrated approach to resolution methodologies from the perspective of both these areas. Researchers in sampling theory will benefit from learning about inverse problems and regularization methods, while specialists in inverse problems will gain a better understanding of the point of view of sampling concepts. This book requires some basic knowledge of functional analysis, Fourier theory, geometric number theory, constructive approximation, and special function theory. By avoiding extreme technicalities and elaborate proof techniques, it is an accessible resource for students and researchers not only from applied mathematics, but also from all branches of engineering and science.
Author :Stephan W. Anzengruber Publisher :Sudwestdeutscher Verlag Fur Hochschulschriften AG ISBN 13 :9783838131351 Total Pages :136 pages Book Rating :4.1/5 (313 download)
Book Synopsis The Discrepancy Principle for Tikhonov Regularization in Banach Spaces by : Stephan W. Anzengruber
Download or read book The Discrepancy Principle for Tikhonov Regularization in Banach Spaces written by Stephan W. Anzengruber and published by Sudwestdeutscher Verlag Fur Hochschulschriften AG. This book was released on 2012-04 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about inverse problems. A classical example of such an inverse problem is computerized tomography (CT), where one attempts to recover an image of, say, the human brain from measured x-ray intensities. Such measurements will in general be noisy, but even very little noise can have a severe impact on the quality of the reconstructed images. To overcome these difficulties regularization methods have been developed, which stabilize the reconstruction process. We study here one such method, known as Tikhonov-regularization, and prove general regularizing properties as well as rates on the speed of the convergence.
Book Synopsis Theory of Linear Ill-Posed Problems and its Applications by : Valentin K. Ivanov
Download or read book Theory of Linear Ill-Posed Problems and its Applications written by Valentin K. Ivanov and published by Walter de Gruyter. This book was released on 2013-02-18 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the book considers ill-posed problems not only in Hilbert but also in Banach spaces. It is natural that since the appearance of the first edition considerable progress has been made in the theory of inverse and ill-posed problems as wall as in ist applications. To reflect these accomplishments the authors included additional material e. g. comments to each chapter and a list of monographs with annotations.
Book Synopsis Variational Methods in Imaging by : Otmar Scherzer
Download or read book Variational Methods in Imaging written by Otmar Scherzer and published by Springer Science & Business Media. This book was released on 2008-09-26 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of variational methods in imaging. The presentation is mathematically rigorous and covers a detailed treatment of the approach from an inverse problems point of view. Many numerical examples accompany the theory throughout the text. It is geared towards graduate students and researchers in applied mathematics. Researchers in the area of imaging science will also find this book appealing. It can serve as a main text in courses in image processing or as a supplemental text for courses on regularization and inverse problems at the graduate level.
