Download or read book Iterative Regularization Methods for Nonlinear Ill Posed Problems written by Barbara Kaltenbacher and published by Walter de Gruyter. This book was released on 2008-09-25 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization of Ill Posed Problems by Iteration Methods written by S.F. Gilyazov and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization of Ill Posed Problems by Iteration Methods written by S.F. Gilyazov and published by Springer. This book was released on 2014-03-14 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization Algorithms for Ill Posed Problems written by Anatoly B. Bakushinsky and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-02-05 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Download or read book Iterative Methods for Ill posed Problems written by Anatoly B. Bakushinsky and published by Walter de Gruyter. This book was released on 2011 with total page 153 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Rank Deficient and Discrete Ill Posed Problems written by Per Christian Hansen and published by SIAM. This book was released on 2005-01-01 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Handbook of Mathematical Methods in Imaging written by Otmar Scherzer and published by Springer Science & Business Media. This book was released on 2010-11-23 with total page 1626 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Ill Posed Problems Theory and Applications written by A. Bakushinsky and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization for Applied Inverse and Ill Posed Problems written by and published by Springer-Verlag. This book was released on 2013-11-22 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Regularization Methods for Ill Posed Optimal Control Problems written by Frank Pörner and published by BoD – Books on Demand. This book was released on 2018-10-04 with total page 181 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization Methods for Ill posed Problems written by Arthur Neuman and published by . This book was released on 2010 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis examines solution methods for large linear systems of equations with a matrix of ill-determined rank and an error-contaminated right-hand side. The numerical solution is delicate, because the matrix is very ill-conditioned and may be singular. To solve such systems, one replaces the system with one that is less sensitive to error a process known as regularization. This thesis focuses on the regularization method known as truncated iteration. A new algorithm is presented and compared to other existing methods.
Download or read book Regularization of Inverse Problems written by Heinz Werner Engl and published by Springer Science & Business Media. This book was released on 2000-03-31 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonlinear Ill Posed Problems written by A.N. Tikhonov and published by Springer. This book was released on 2014-08-23 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Numerical Regularization for Atmospheric Inverse Problems written by Adrian Doicu and published by Springer Science & Business Media. This book was released on 2010-07-16 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Regularization Theory for Ill posed Problems written by Shuai Lu and published by Walter de Gruyter. This book was released on 2013-07-31 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Discrete Inverse Problems written by Per Christian Hansen and published by SIAM. This book was released on 2010-01-01 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Iterative Methods for Approximate Solution of Inverse Problems written by A.B. Bakushinsky and published by Springer Science & Business Media. This book was released on 2007-09-28 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
