Iterative Methods For Ill Posed Problems
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Iterative Methods For Ill Posed Problems
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Author : Anatoly B. Bakushinsky
language : en
Publisher: Walter de Gruyter
Release Date : 2011
Iterative Methods For Ill Posed Problems written by Anatoly B. Bakushinsky and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
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.
Iterative Regularization Methods For Nonlinear Ill Posed Problems
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Author : Barbara Kaltenbacher
language : en
Publisher: Walter de Gruyter
Release Date : 2008-09-25
Iterative Regularization Methods For Nonlinear Ill Posed Problems written by Barbara Kaltenbacher and has been published by Walter de Gruyter this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-25 with Mathematics categories.
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.
Iterative Methods For Ill Posed Problems
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Author : Anatolij Borisovič Bakušinskij
language : en
Publisher:
Release Date : 2011
Iterative Methods For Ill Posed Problems written by Anatolij Borisovič Bakušinskij and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.
Regularization Algorithms For Ill Posed Problems
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Author : Anatoly B. Bakushinsky
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2018-02-05
Regularization Algorithms For Ill Posed Problems written by Anatoly B. Bakushinsky and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-02-05 with Mathematics categories.
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
Iterative Methods Of Solving Inverse And Ill Posed Problems
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Author : S. I. Kabanikhin
language : en
Publisher: VSP Books
Release Date : 2007-03-01
Iterative Methods Of Solving Inverse And Ill Posed Problems written by S. I. Kabanikhin and has been published by VSP Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-03-01 with Inverse problems (Differential equations) categories.
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.
Regularization Of Ill Posed Problems By Iteration Methods
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Author : S.F. Gilyazov
language : en
Publisher: Springer
Release Date : 2014-03-14
Regularization Of Ill Posed Problems By Iteration Methods written by S.F. Gilyazov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-03-14 with Mathematics categories.
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.
Regularization Of Ill Posed Problems By Iteration Methods
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Author : S.F. Gilyazov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17
Regularization Of Ill Posed Problems By Iteration Methods written by S.F. Gilyazov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.
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.
Numerical Methods For The Solution Of Ill Posed Problems
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Author : A.N. Tikhonov
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09
Numerical Methods For The Solution Of Ill Posed Problems written by A.N. Tikhonov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-09 with Mathematics categories.
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.
Iterative Methods For Approximate Solution Of Inverse Problems
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Author : A.B. Bakushinsky
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-09-28
Iterative Methods For Approximate Solution Of Inverse Problems written by A.B. Bakushinsky and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-09-28 with Mathematics categories.
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.
Rank Deficient And Discrete Ill Posed Problems
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Author : Per Christian Hansen
language : en
Publisher: SIAM
Release Date : 2005-01-01
Rank Deficient And Discrete Ill Posed Problems written by Per Christian Hansen and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-01-01 with Mathematics categories.
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.