Regularization Algorithms For Ill Posed Problems

Author : Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 342 pages
File Size : 49,9 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 : Shuai Lu,Sergei V. Pereverzev
Publisher : Walter de Gruyter
Page : 304 pages
File Size : 40,9 Mb
Release : 2013-07-31
Category : Mathematics
ISBN : 9783110286496

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Regularization Theory for Ill-posed Problems by Shuai Lu,Sergei V. Pereverzev Pdf

This monograph is a valuable contribution to the highly topical and extremly productive field of regularisation methods for inverse and ill-posed problems. The author is an internationally outstanding and accepted mathematician in this field. In his book he offers a well-balanced mixture of basic and innovative aspects. He demonstrates new, differentiated viewpoints, and important examples for applications. The book demontrates the current developments in the field of regularization theory, such as multiparameter regularization and regularization in learning theory. The book is written for graduate and PhD students and researchers in mathematics, natural sciences, engeneering, and medicine.

Author : Anatoliĭ Borisovich Bakushinskiĭ,Mikhail I︠U︡rʹevich Kokurin,Mikhail M. Kokurin
Publisher : Unknown
Page : 323 pages
File Size : 53,7 Mb
Release : 2018
Category : Differential equations, Partial
ISBN : 3110557363

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Regularization Algorithms for Ill-posed Problems by Anatoliĭ Borisovich Bakushinskiĭ,Mikhail I︠U︡rʹevich Kokurin,Mikhail M. Kokurin Pdf

Author : Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer
Publisher : Walter de Gruyter
Page : 205 pages
File Size : 52,8 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 : 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 : A.N. Tikhonov,A. Goncharsky,V.V. Stepanov,Anatoly G. Yagola
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 55,9 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 : S.F. Gilyazov,Nataliya Gol'dman
Publisher : Springer
Page : 342 pages
File Size : 51,5 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 : Anonim
Publisher : Springer-Verlag
Page : 199 pages
File Size : 48,7 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 : Willi Freeden,M. Zuhair Nashed
Publisher : Birkhäuser
Page : 932 pages
File Size : 42,7 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783319571812

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Handbook of Mathematical Geodesy by Willi Freeden,M. Zuhair Nashed Pdf

Written by leading experts, this book provides a clear and comprehensive survey of the “status quo” of the interrelating process and cross-fertilization of structures and methods in mathematical geodesy. Starting with a foundation of functional analysis, potential theory, constructive approximation, special function theory, and inverse problems, readers are subsequently introduced to today’s least squares approximation, spherical harmonics reflected spline and wavelet concepts, boundary value problems, Runge-Walsh framework, geodetic observables, geoidal modeling, ill-posed problems and regularizations, inverse gravimetry, and satellite gravity gradiometry. All chapters are self-contained and can be studied individually, making the book an ideal resource for both graduate students and active researchers who want to acquaint themselves with the mathematical aspects of modern geodesy.

Author : S.F. Gilyazov,N.L. Gol'dman
Publisher : Springer Science & Business Media
Page : 348 pages
File Size : 55,5 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 : V.A. Morozov
Publisher : Springer Science & Business Media
Page : 275 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461252801

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Methods for Solving Incorrectly Posed Problems by V.A. Morozov Pdf

Some problems of mathematical physics and analysis can be formulated as the problem of solving the equation f € F, (1) Au = f, where A: DA C U + F is an operator with a non-empty domain of definition D , in a metric space U, with range in a metric space F. The metrics A on U and F will be denoted by P and P ' respectively. Relative u F to the twin spaces U and F, J. Hadamard P-06] gave the following defini tion of correctness: the problem (1) is said to be well-posed (correct, properly posed) if the following conditions are satisfied: (1) The range of the value Q of the operator A coincides with A F ("sol vabi li ty" condition); (2) The equality AU = AU for any u ,u € DA implies the I 2 l 2 equality u = u ("uniqueness" condition); l 2 (3) The inverse operator A-I is continuous on F ("stability" condition). Any reasonable mathematical formulation of a physical problem requires that conditions (1)-(3) be satisfied. That is why Hadamard postulated that any "ill-posed" (improperly posed) problem, that is to say, one which does not satisfy conditions (1)-(3), is non-physical. Hadamard also gave the now classical example of an ill-posed problem, namely, the Cauchy problem for the Laplace equation.

Author : Otmar Scherzer
Publisher : Springer Science & Business Media
Page : 1626 pages
File Size : 46,8 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 : Frank Pörner
Publisher : BoD – Books on Demand
Page : 181 pages
File Size : 52,8 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 : Per Christian Hansen
Publisher : SIAM
Page : 220 pages
File Size : 41,6 Mb
Release : 2010-01-01
Category : Mathematics
ISBN : 9780898718836

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Discrete Inverse Problems by Per Christian Hansen Pdf

This book gives an introduction to the practical treatment of inverse problems by means of numerical methods, with a focus on basic mathematical and computational aspects. To solve inverse problems, we demonstrate that insight about them goes hand in hand with algorithms.

Author : Ito Kazufumi,Jin Bangti
Publisher : World Scientific
Page : 332 pages
File Size : 49,8 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.