Iterative Methods For Ill Posed Problems

Author : Anatoly B. Bakushinsky,Mihail Yu. Kokurin,Alexandra Smirnova
Publisher : Walter de Gruyter
Page : 153 pages
File Size : 40,7 Mb
Release : 2010-12-23
Category : Mathematics
ISBN : 9783110250657

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Iterative Methods for Ill-Posed Problems by Anatoly B. Bakushinsky,Mihail Yu. Kokurin,Alexandra Smirnova Pdf

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

Author : Anatoly B. Bakushinsky,Александра Борисовна Смирнова
Publisher : Walter de Gruyter
Page : 153 pages
File Size : 54,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9783110250640

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Iterative Methods for Ill-posed Problems by Anatoly B. Bakushinsky,Александра Борисовна Смирнова Pdf

Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

Author : Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer
Publisher : Walter de Gruyter
Page : 205 pages
File Size : 55,6 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 : Anatolij Borisovič Bakušinskij
Publisher : Unknown
Page : 136 pages
File Size : 46,6 Mb
Release : 2011
Category : Electronic
ISBN : 9067643963

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Iterative Methods for Ill-posed Problems by Anatolij Borisovič Bakušinskij Pdf

Author : A.B. Bakushinsky,M.Yu. Kokurin
Publisher : Springer Science & Business Media
Page : 298 pages
File Size : 55,8 Mb
Release : 2007-09-28
Category : Mathematics
ISBN : 9781402031229

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Iterative Methods for Approximate Solution of Inverse Problems by A.B. Bakushinsky,M.Yu. Kokurin Pdf

This volume presents a unified approach to constructing iterative methods for solving irregular operator equations and provides rigorous theoretical analysis for several classes of these methods. The analysis of methods includes convergence theorems as well as necessary and sufficient conditions for their convergence at a given rate. The principal groups of methods studied in the book are iterative processes based on the technique of universal linear approximations, stable gradient-type processes, and methods of stable continuous approximations. Compared to existing monographs and textbooks on ill-posed problems, the main distinguishing feature of the presented approach is that it doesn’t require any structural conditions on equations under consideration, except for standard smoothness conditions. This allows to obtain in a uniform style stable iterative methods applicable to wide classes of nonlinear inverse problems. Practical efficiency of suggested algorithms is illustrated in application to inverse problems of potential theory and acoustic scattering. The volume can be read by anyone with a basic knowledge of functional analysis. The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems.

Author : A.N. Tikhonov,A. Goncharsky,V.V. Stepanov,Anatoly G. Yagola
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 45,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401584807

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Numerical Methods for the Solution of Ill-Posed Problems by A.N. Tikhonov,A. Goncharsky,V.V. Stepanov,Anatoly G. Yagola Pdf

Many problems in science, technology and engineering are posed in the form of operator equations of the first kind, with the operator and RHS approximately known. But such problems often turn out to be ill-posed, having no solution, or a non-unique solution, and/or an unstable solution. Non-existence and non-uniqueness can usually be overcome by settling for `generalised' solutions, leading to the need to develop regularising algorithms. The theory of ill-posed problems has advanced greatly since A. N. Tikhonov laid its foundations, the Russian original of this book (1990) rapidly becoming a classical monograph on the topic. The present edition has been completely updated to consider linear ill-posed problems with or without a priori constraints (non-negativity, monotonicity, convexity, etc.). Besides the theoretical material, the book also contains a FORTRAN program library. Audience: Postgraduate students of physics, mathematics, chemistry, economics, engineering. Engineers and scientists interested in data processing and the theory of ill-posed problems.

Author : A. Bakushinsky,A. Goncharsky
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 50,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401110266

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Ill-Posed Problems: Theory and Applications by A. Bakushinsky,A. Goncharsky Pdf

Recent years have been characterized by the increasing amountofpublications in the field ofso-called ill-posed problems. This is easilyunderstandable because we observe the rapid progress of a relatively young branch ofmathematics, ofwhich the first results date back to about 30 years ago. By now, impressive results have been achieved both in the theory ofsolving ill-posed problems and in the applicationsofalgorithms using modem computers. To mention just one field, one can name the computer tomography which could not possibly have been developed without modem tools for solving ill-posed problems. When writing this book, the authors tried to define the place and role of ill posed problems in modem mathematics. In a few words, we define the theory of ill-posed problems as the theory of approximating functions with approximately given arguments in functional spaces. The difference between well-posed and ill posed problems is concerned with the fact that the latter are associated with discontinuous functions. This approach is followed by the authors throughout the whole book. We hope that the theoretical results will be of interest to researchers working in approximation theory and functional analysis. As for particular algorithms for solving ill-posed problems, the authors paid general attention to the principles ofconstructing such algorithms as the methods for approximating discontinuous functions with approximately specified arguments. In this way it proved possible to define the limits of applicability of regularization techniques.

Author : Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 342 pages
File Size : 47,5 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 : Otmar Scherzer
Publisher : Springer Science & Business Media
Page : 1626 pages
File Size : 53,9 Mb
Release : 2010-11-23
Category : Mathematics
ISBN : 9780387929194

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Handbook of Mathematical Methods in Imaging by Otmar Scherzer Pdf

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.

Author : S. I. Kabanikhin,M. F Bektemesov,A. T. Nurseitova
Publisher : VSP Books
Page : 450 pages
File Size : 50,7 Mb
Release : 2007-03-01
Category : Inverse problems (Differential equations)
ISBN : 9004155244

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Iterative Methods of Solving Inverse and Ill-Posed Problems by S. I. Kabanikhin,M. F Bektemesov,A. T. Nurseitova Pdf

In this book the iterative methods are applied to several inverse and ill-posed problems such as inverse problems of acoustics, seismics, electrodynamics, heat transfer, Cauchy problem for Laplace equation and some others.

Author : Per Christian Hansen
Publisher : SIAM
Page : 259 pages
File Size : 51,9 Mb
Release : 2005-01-01
Category : Mathematics
ISBN : 9780898714036

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Rank-Deficient and Discrete Ill-Posed Problems by Per Christian Hansen Pdf

Here is an overview of modern computational stabilization methods for linear inversion, with applications to a variety of problems in audio processing, medical imaging, tomography, seismology, astronomy, and other areas. Rank-deficient problems involve matrices that are either exactly or nearly rank deficient. Such problems often arise in connection with noise suppression and other problems where the goal is to suppress unwanted disturbances of the given measurements. Discrete ill-posed problems arise in connection with the numerical treatment of inverse problems, where one typically wants to compute information about some interior properties using exterior measurements. Examples of inverse problems are image restoration and tomography, where one needs to improve blurred images or reconstruct pictures from raw data. This book describes, in a common framework, new and existing numerical methods for the analysis and solution of rank-deficient and discrete ill-posed problems. The emphasis is on insight into the stabilizing properties of the algorithms and on the efficiency and reliability of the computations. The setting is that of numerical linear algebra rather than abstract functional analysis, and the theoretical development is complemented with numerical examples and figures that illustrate the features of the various algorithms.

Author : S.F. Gilyazov,N.L. Gol'dman
Publisher : Springer Science & Business Media
Page : 348 pages
File Size : 48,7 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401594820

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Regularization of Ill-Posed Problems by Iteration Methods by S.F. Gilyazov,N.L. Gol'dman Pdf

Iteration regularization, i.e., utilization of iteration methods of any form for the stable approximate solution of ill-posed problems, is one of the most important but still insufficiently developed topics of the new theory of ill-posed problems. In this monograph, a general approach to the justification of iteration regulari zation algorithms is developed, which allows us to consider linear and nonlinear methods from unified positions. Regularization algorithms are the 'classical' iterative methods (steepest descent methods, conjugate direction methods, gradient projection methods, etc.) complemented by the stopping rule depending on level of errors in input data. They are investigated for solving linear and nonlinear operator equations in Hilbert spaces. Great attention is given to the choice of iteration index as the regularization parameter and to estimates of errors of approximate solutions. Stabilizing properties such as smoothness and shape constraints imposed on the solution are used. On the basis of these investigations, we propose and establish efficient regularization algorithms for stable numerical solution of a wide class of ill-posed problems. In particular, descriptive regularization algorithms, utilizing a priori information about the qualitative behavior of the sought solution and ensuring a substantial saving in computational costs, are considered for model and applied problems in nonlinear thermophysics. The results of calculations for important applications in various technical fields (a continuous casting, the treatment of materials and perfection of heat-protective systems using laser and composite technologies) are given.

Author : S.F. Gilyazov,Nataliya Gol'dman
Publisher : Springer
Page : 342 pages
File Size : 45,6 Mb
Release : 2014-03-14
Category : Mathematics
ISBN : 940159483X

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Regularization of Ill-Posed Problems by Iteration Methods by S.F. Gilyazov,Nataliya Gol'dman Pdf

Iteration regularization, i.e., utilization of iteration methods of any form for the stable approximate solution of ill-posed problems, is one of the most important but still insufficiently developed topics of the new theory of ill-posed problems. In this monograph, a general approach to the justification of iteration regulari zation algorithms is developed, which allows us to consider linear and nonlinear methods from unified positions. Regularization algorithms are the 'classical' iterative methods (steepest descent methods, conjugate direction methods, gradient projection methods, etc.) complemented by the stopping rule depending on level of errors in input data. They are investigated for solving linear and nonlinear operator equations in Hilbert spaces. Great attention is given to the choice of iteration index as the regularization parameter and to estimates of errors of approximate solutions. Stabilizing properties such as smoothness and shape constraints imposed on the solution are used. On the basis of these investigations, we propose and establish efficient regularization algorithms for stable numerical solution of a wide class of ill-posed problems. In particular, descriptive regularization algorithms, utilizing a priori information about the qualitative behavior of the sought solution and ensuring a substantial saving in computational costs, are considered for model and applied problems in nonlinear thermophysics. The results of calculations for important applications in various technical fields (a continuous casting, the treatment of materials and perfection of heat-protective systems using laser and composite technologies) are given.

Author : Yousef Saad
Publisher : SIAM
Page : 537 pages
File Size : 41,5 Mb
Release : 2003-04-01
Category : Mathematics
ISBN : 9780898715347

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Iterative Methods for Sparse Linear Systems by Yousef Saad Pdf

Mathematics of Computing -- General.

Author : Andrzej Cegielski
Publisher : Springer
Page : 312 pages
File Size : 49,6 Mb
Release : 2012-09-14
Category : Mathematics
ISBN : 9783642309014

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Iterative Methods for Fixed Point Problems in Hilbert Spaces by Andrzej Cegielski Pdf

Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present general convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.