Regularization Methods For Ill Posed Problems

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Iterative Regularization Methods for Nonlinear Ill-Posed Problems

Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer
Iterative Regularization Methods for Nonlinear Ill Posed Problems Book PDF

Author : Barbara Kaltenbacher,Andreas Neubauer,Otmar Scherzer
Publisher : Walter de Gruyter
Page : 205 pages
File Size : 44,6 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.

Regularization Algorithms for Ill-Posed Problems

Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Regularization Algorithms for Ill Posed Problems Book PDF

Author : Anatoly B. Bakushinsky,Mikhail M. Kokurin,Mikhail Yu. Kokurin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 342 pages
File Size : 52,8 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

Regularization Methods for Ill-posed Problems

Vladimir Alekseevich Morozov
Regularization Methods for Ill posed Problems Book PDF

Author : Vladimir Alekseevich Morozov
Publisher : CRC PressI Llc
Page : 257 pages
File Size : 45,6 Mb
Release : 1993
Category : Mathematics
ISBN : 0849393116

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Regularization Methods for Ill-posed Problems by Vladimir Alekseevich Morozov Pdf

Presents current theories and methods for obtaining approximate solutions of basic classes of incorrectly posed problems. The book provides simple conditions of optimality and the optimality of the order of regular methods for solving a wide class of unsteady problems.

Ill-Posed Problems: Theory and Applications

A. Bakushinsky,A. Goncharsky
Ill Posed Problems Theory and Applications Book PDF

Author : A. Bakushinsky,A. Goncharsky
Publisher : Springer Science & Business Media
Page : 268 pages
File Size : 53,9 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.

Regularization Theory for Ill-posed Problems

Shuai Lu,Sergei V. Pereverzev
Regularization Theory for Ill posed Problems Book PDF

Author : Shuai Lu,Sergei V. Pereverzev
Publisher : ISSN
Page : 0 pages
File Size : 47,5 Mb
Release : 2013
Category : Numerical analysis
ISBN : 3110286467

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

The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.

Regularization of Inverse Problems

Heinz Werner Engl,Martin Hanke,A. Neubauer
Regularization of Inverse Problems Book PDF

Author : Heinz Werner Engl,Martin Hanke,A. Neubauer
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 47,6 Mb
Release : 2000-03-31
Category : Mathematics
ISBN : 0792361407

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Regularization of Inverse Problems by Heinz Werner Engl,Martin Hanke,A. Neubauer Pdf

This book is devoted to the mathematical theory of regularization methods and gives an account of the currently available results about regularization methods for linear and nonlinear ill-posed problems. Both continuous and iterative regularization methods are considered in detail with special emphasis on the development of parameter choice and stopping rules which lead to optimal convergence rates.

Handbook of Mathematical Methods in Imaging

Otmar Scherzer
Handbook of Mathematical Methods in Imaging Book PDF

Author : Otmar Scherzer
Publisher : Springer Science & Business Media
Page : 1626 pages
File Size : 51,8 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.

Handbook of Mathematical Geodesy

Willi Freeden,M. Zuhair Nashed
Handbook of Mathematical Geodesy Book PDF

Author : Willi Freeden,M. Zuhair Nashed
Publisher : Birkhäuser
Page : 932 pages
File Size : 40,6 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.

Methods for Solving Incorrectly Posed Problems

V.A. Morozov
Methods for Solving Incorrectly Posed Problems Book PDF

Author : V.A. Morozov
Publisher : Springer Science & Business Media
Page : 275 pages
File Size : 54,9 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.

Iterative Methods for Ill-Posed Problems

Anatoly B. Bakushinsky,Mihail Yu. Kokurin,Alexandra Smirnova
Iterative Methods for Ill Posed Problems Book PDF

Author : Anatoly B. Bakushinsky,Mihail Yu. Kokurin,Alexandra Smirnova
Publisher : Walter de Gruyter
Page : 153 pages
File Size : 49,8 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.

Numerical Regularization for Atmospheric Inverse Problems

Adrian Doicu,Thomas Trautmann,Franz Schreier
Numerical Regularization for Atmospheric Inverse Problems Book PDF

Author : Adrian Doicu,Thomas Trautmann,Franz Schreier
Publisher : Springer Science & Business Media
Page : 432 pages
File Size : 49,8 Mb
Release : 2010-07-16
Category : Science
ISBN : 9783642054396

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Numerical Regularization for Atmospheric Inverse Problems by Adrian Doicu,Thomas Trautmann,Franz Schreier Pdf

The retrieval problems arising in atmospheric remote sensing belong to the class of the - called discrete ill-posed problems. These problems are unstable under data perturbations, and can be solved by numerical regularization methods, in which the solution is stabilized by taking additional information into account. The goal of this research monograph is to present and analyze numerical algorithms for atmospheric retrieval. The book is aimed at physicists and engineers with some ba- ground in numerical linear algebra and matrix computations. Although there are many practical details in this book, for a robust and ef?cient implementation of all numerical algorithms, the reader should consult the literature cited. The data model adopted in our analysis is semi-stochastic. From a practical point of view, there are no signi?cant differences between a semi-stochastic and a determin- tic framework; the differences are relevant from a theoretical point of view, e.g., in the convergence and convergence rates analysis. After an introductory chapter providing the state of the art in passive atmospheric remote sensing, Chapter 2 introduces the concept of ill-posedness for linear discrete eq- tions. To illustrate the dif?culties associated with the solution of discrete ill-posed pr- lems, we consider the temperature retrieval by nadir sounding and analyze the solvability of the discrete equation by using the singular value decomposition of the forward model matrix.

Regularization Methods in Banach Spaces

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

Author : Thomas Schuster,Barbara Kaltenbacher,Bernd Hofmann,Kamil S. Kazimierski
Publisher : Walter de Gruyter
Page : 296 pages
File Size : 49,7 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.

Regularization Theory for Ill-posed Problems

Shuai Lu,Sergei V. Pereverzev
Regularization Theory for Ill posed Problems Book PDF

Author : Shuai Lu,Sergei V. Pereverzev
Publisher : Walter de Gruyter
Page : 304 pages
File Size : 47,5 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.

Optimal Methods for Ill-Posed Problems

Vitalii P. Tanana,Anna I. Sidikova
Optimal Methods for Ill Posed Problems Book PDF

Author : Vitalii P. Tanana,Anna I. Sidikova
Publisher : Walter de Gruyter GmbH & Co KG
Page : 138 pages
File Size : 55,6 Mb
Release : 2018-03-19
Category : Mathematics
ISBN : 9783110577211

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Optimal Methods for Ill-Posed Problems by Vitalii P. Tanana,Anna I. Sidikova Pdf

The book covers fundamentals of the theory of optimal methods for solving ill-posed problems, as well as ways to obtain accurate and accurate-by-order error estimates for these methods. The methods described in the current book are used to solve a number of inverse problems in mathematical physics. Contents Modulus of continuity of the inverse operator and methods for solving ill-posed problems Lavrent’ev methods for constructing approximate solutions of linear operator equations of the first kind Tikhonov regularization method Projection-regularization method Inverse heat exchange problems

Numerical Methods for the Solution of Ill-Posed Problems

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

Author : A.N. Tikhonov,A. Goncharsky,V.V. Stepanov,Anatoly G. Yagola
Publisher : Springer Science & Business Media
Page : 257 pages
File Size : 51,6 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.