Regularization Methods In Banach Spaces

Author : Thomas Schuster,Barbara Kaltenbacher,Bernd Hofmann,Kamil S. Kazimierski
Publisher : Walter de Gruyter
Page : 296 pages
File Size : 49,8 Mb
Release : 2012-07-30
Category : Mathematics
ISBN : 9783110255720

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Regularization Methods in Banach Spaces by Thomas Schuster,Barbara Kaltenbacher,Bernd Hofmann,Kamil S. Kazimierski Pdf

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.

Author : Torsten Hein
Publisher : Logos Verlag Berlin GmbH
Page : 174 pages
File Size : 49,7 Mb
Release : 2010
Category : Mathematics
ISBN : 9783832527457

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Regularization in Banach Spaces - Convergence Rates Theory by Torsten Hein Pdf

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.

Author : Kamil S. Kazimierski
Publisher : Logos Verlag Berlin GmbH
Page : 149 pages
File Size : 49,6 Mb
Release : 2010
Category : Mathematics
ISBN : 9783832527310

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Aspects of Regularization in Banach Spaces by Kamil S. Kazimierski Pdf

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.

Author : Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer
Publisher : Walter de Gruyter
Page : 205 pages
File Size : 54,8 Mb
Release : 2008-09-25
Category : Mathematics
ISBN : 9783110208276

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Iterative Regularization Methods for Nonlinear Ill-Posed Problems by Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer Pdf

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.

Author : Ismael Rodrigo Bleyer,Antonio Leitão
Publisher : Unknown
Page : 121 pages
File Size : 53,6 Mb
Release : 2015
Category : Banach spaces
ISBN : 8524404108

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Novel Regularization Methods for Ill-posed Problems in Hilbert and Banach Spaces by Ismael Rodrigo Bleyer,Antonio Leitão Pdf

Author : Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 342 pages
File Size : 49,8 Mb
Release : 2018-02-05
Category : Mathematics
ISBN : 9783110556384

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Regularization Algorithms for Ill-Posed Problems by Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin Pdf

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

Author : Jens Flemming
Publisher : Springer
Page : 182 pages
File Size : 49,7 Mb
Release : 2018-09-08
Category : Mathematics
ISBN : 9783319952642

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Variational Source Conditions, Quadratic Inverse Problems, Sparsity Promoting Regularization by Jens Flemming Pdf

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.

Author : Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 342 pages
File Size : 41,8 Mb
Release : 2018-02-05
Category : Mathematics
ISBN : 9783110557350

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Regularization Algorithms for Ill-Posed Problems by Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin Pdf

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

Author : Frank Pörner
Publisher : BoD – Books on Demand
Page : 181 pages
File Size : 53,8 Mb
Release : 2018-10-04
Category : Mathematics
ISBN : 9783958260863

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Regularization Methods for Ill-Posed Optimal Control Problems by Frank Pörner Pdf

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.

Author : Yakov Alber,Irina Ryazantseva
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 51,5 Mb
Release : 2006-02-23
Category : Mathematics
ISBN : 9781402043963

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Nonlinear Ill-posed Problems of Monotone Type by Yakov Alber,Irina Ryazantseva Pdf

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.

Author : Otmar Scherzer,Markus Grasmair,Harald Grossauer,Markus Haltmeier,Frank Lenzen
Publisher : Springer Science & Business Media
Page : 323 pages
File Size : 55,5 Mb
Release : 2008-09-26
Category : Mathematics
ISBN : 9780387692777

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Variational Methods in Imaging by Otmar Scherzer,Markus Grasmair,Harald Grossauer,Markus Haltmeier,Frank Lenzen Pdf

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.

Author : Anonim
Publisher : Springer-Verlag
Page : 199 pages
File Size : 40,8 Mb
Release : 2013-11-22
Category : Technology & Engineering
ISBN : 9783322930347

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Regularization for Applied Inverse and Ill-Posed Problems by Anonim Pdf

Author : Ito Kazufumi,Jin Bangti
Publisher : World Scientific
Page : 332 pages
File Size : 45,6 Mb
Release : 2014-08-28
Category : Mathematics
ISBN : 9789814596213

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Inverse Problems: Tikhonov Theory And Algorithms by Ito Kazufumi,Jin Bangti Pdf

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.

Author : Willi Freeden,M. Zuhair Nashed
Publisher : American Mathematical Society
Page : 505 pages
File Size : 54,5 Mb
Release : 2023-08-21
Category : Mathematics
ISBN : 9781470473457

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Recovery Methodologies: Regularization and Sampling by Willi Freeden,M. Zuhair Nashed Pdf

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 : Carolin Homann
Publisher : Göttingen University Press
Page : 126 pages
File Size : 49,7 Mb
Release : 2015
Category : Electronic
ISBN : 9783863952105

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Phase retrieval problems in x-ray physics by Carolin Homann Pdf

In phase retrieval problems that occur in imaging by coherent x-ray diffraction, one tries to reconstruct information about a sample of interest from possibly noisy intensity measurements of the wave fi eld traversing the sample. The mathematical formulation of these problems bases on some assumptions. Usually one of them is that the x-ray wave fi eld is generated by a point source. In order to address this very idealized assumption, it is common to perform a data preprocessing step, the so-called empty beam correction. Within this work, we study the validity of this approach by presenting a quantitative error estimate. Moreover, in order to solve these phase retrieval problems, we want to incorporate a priori knowledge about the structure of the noise and the solution into the reconstruction process. For this reason, the application of a problem adapted iteratively regularized Newton-type method becomes particularly attractive. This method includes the solution of a convex minimization problem in each iteration step. We present a method for solving general optimization problems of this form. Our method is a generalization of a commonly used algorithm which makes it efficiently applicable to a wide class of problems. We also proof convergence results and show the performance of our method by numerical examples.