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Linear Operator Equations Approximation And Regularization
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Book Synopsis Linear Operator Equations by : M. Thamban Nair
Download or read book Linear Operator Equations written by M. Thamban Nair and published by World Scientific. This book was released on 2009 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in science and engineering have their mathematical formulation as an operator equation Tx=y, where T is a linear or nonlinear operator between certain function spaces. In practice, such equations are solved approximately using numerical methods, as their exact solution may not often be possible or may not be worth looking for due to physical constraints. In such situations, it is desirable to know how the so-called approximate solution approximates the exact solution, and what the error involved in such procedures would be. This book is concerned with the investigation of the above theoretical issues related to approximately solving linear operator equations. The main tools used for this purpose are basic results from functional analysis and some rudimentary ideas from numerical analysis. To make this book more accessible to readers, no in-depth knowledge on these disciplines is assumed for reading this book.
Book Synopsis Linear Operator Equations: Approximation And Regularization by : M Thamban Nair
Download or read book Linear Operator Equations: Approximation And Regularization written by M Thamban Nair and published by World Scientific. This book was released on 2009-05-05 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many problems in science and engineering have their mathematical formulation as an operator equation Tx=y, where T is a linear or nonlinear operator between certain function spaces. In practice, such equations are solved approximately using numerical methods, as their exact solution may not often be possible or may not be worth looking for due to physical constraints. In such situations, it is desirable to know how the so-called approximate solution approximates the exact solution, and what the error involved in such procedures would be.This book is concerned with the investigation of the above theoretical issues related to approximately solving linear operator equations. The main tools used for this purpose are basic results from functional analysis and some rudimentary ideas from numerical analysis. To make this book more accessible to readers, no in-depth knowledge on these disciplines is assumed for reading this book.
Book Synopsis Regularization of Linear Operator Equations by : Mark Anthony Lukas
Download or read book Regularization of Linear Operator Equations written by Mark Anthony Lukas and published by . This book was released on 1981 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Approximate Regularized Solutions to Linear Operator Equations when the Data-vector is Not in the Range of the Operator by : M. Zuhair Nashed
Download or read book Approximate Regularized Solutions to Linear Operator Equations when the Data-vector is Not in the Range of the Operator written by M. Zuhair Nashed and published by . This book was released on 1973 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt: The paper addresses itself to the problem of approximate pseudo solutions of the linear operator equation Af = g, when g is not necessarily in the range of the operator A. (Modified author abstract).
Book Synopsis Ill-Posed Problems with A Priori Information by : V. V. Vasin
Download or read book Ill-Posed Problems with A Priori Information written by V. V. Vasin and published by Walter de Gruyter. This book was released on 2013-02-18 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Regularization Algorithms for Ill-Posed Problems by : Anatoly B. Bakushinsky
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 342 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
Book Synopsis Tikhonov-regularization of Ill-posed Linear Operator Equations on Closed Convex Sets by : Andreas Neubauer
Download or read book Tikhonov-regularization of Ill-posed Linear Operator Equations on Closed Convex Sets written by Andreas Neubauer and published by . This book was released on 1986 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Optimal Methods for Ill-Posed Problems by : Vitalii P. Tanana
Download or read book Optimal Methods for Ill-Posed Problems written by Vitalii P. Tanana and published by Walter de Gruyter GmbH & Co KG. This book was released on 2018-03-19 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Methods for Solving Operator Equations by : V. P. Tanana
Download or read book Methods for Solving Operator Equations written by V. P. Tanana and published by Walter de Gruyter. This book was released on 2012-02-13 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis REGULARIZATION-PROJECTION METHODS AND FINITE-ELEMENT APPROXIMATIONS FOR ILL-POSED LINEAR OPERATOR EQUATIONS.. by : MANSUK SONG
Download or read book REGULARIZATION-PROJECTION METHODS AND FINITE-ELEMENT APPROXIMATIONS FOR ILL-POSED LINEAR OPERATOR EQUATIONS.. written by MANSUK SONG and published by . This book was released on 1978 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Approximate Solutions Of Operator Equations by : Guanrong Chen
Download or read book Approximate Solutions Of Operator Equations written by Guanrong Chen and published by World Scientific. This book was released on 1997-07-01 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an elementary and self-contained introduction to many fundamental issues concerning approximate solutions of operator equations formulated in an abstract Banach space setting, including important topics such as solvability, computational schemes, convergence, stability and error estimates. The operator equations under investigation include various linear and nonlinear types of ordinary and partial differential equations, integral equations, and abstract evolution equations, which are frequently involved in applied mathematics and engineering applications.Each chapter contains well-selected examples and exercises, for the purposes of demonstrating the fundamental theories and methods developed in the text and familiarizing the reader with functional analysis techniques useful for numerical solutions of various operator equations.
Book Synopsis Regularization Methods in Banach Spaces by : Thomas Schuster
Download or read book Regularization Methods in Banach Spaces written by Thomas Schuster and published by Walter de Gruyter. This book was released on 2012-07-30 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Theory of Linear Ill-Posed Problems and its Applications by : Valentin K. Ivanov
Download or read book Theory of Linear Ill-Posed Problems and its Applications written by Valentin K. Ivanov and published by Walter de Gruyter. This book was released on 2013-02-18 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a revised and extended version of the Russian edition from 1978. It includes the general theory of linear ill-posed problems concerning e. g. the structure of sets of uniform regularization, the theory of error estimation, and the optimality method. As a distinguishing feature the book considers ill-posed problems not only in Hilbert but also in Banach spaces. It is natural that since the appearance of the first edition considerable progress has been made in the theory of inverse and ill-posed problems as wall as in ist applications. To reflect these accomplishments the authors included additional material e. g. comments to each chapter and a list of monographs with annotations.
Book Synopsis Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations by : Grace Wahba
Download or read book Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations written by Grace Wahba and published by . This book was released on 1973 with total page 40 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine by : Grace Wahba
Download or read book Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine written by Grace Wahba and published by . This book was released on 1981 with total page 45 pages. Available in PDF, EPUB and Kindle. Book excerpt: The relationship between certain regularization methods for solving ill posed linear operator equations and ridge methods in regression problems is described. The regularization estimates we describe may be viewed as ridge estimates in a (reproducing kernel) Hilbert space H. When the solution is known a priori to be in some closed, convex set in H, for example, the set of nonnegative functions, or the set of monotone functions, then one can propose regularized estimates subject to side conditions such as nonnegativity, monotonicity, etc. Some applications in medicine and meteorology are described. We describe the method of generalized cross validation for choosing the smoothing (or ridge) parameter in the presence of a family of linear inequality constraints. Some successful numerical examples, solving ill posed convolution equations with noisy data, subject to nonnegativity constraints, are presented. The technique appears to be quite successful in adding information, doing nearly the optimal amount of smoothing, and resolving distinct peaks in the solution which have been blurred by the convolution operation. (Author).
Book Synopsis Approximate Solution of Operator Equations by : Mark Aleksandrovich Krasnoselʹskiĭ
Download or read book Approximate Solution of Operator Equations written by Mark Aleksandrovich Krasnoselʹskiĭ and published by Springer. This book was released on 1972 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: One of the most important chapters in modern functional analysis is the theory of approximate methods for solution of various mathematical problems. Besides providing considerably simplified approaches to numerical methods, the ideas of functional analysis have also given rise to essentially new computation schemes in problems of linear algebra, differential and integral equations, nonlinear analysis, and so on. The general theory of approximate methods includes many known fundamental results. We refer to the classical work of Kantorovich; the investigations of projection methods by Bogolyubov, Krylov, Keldysh and Petrov, much furthered by Mikhlin and Pol'skii; Tikho nov's methods for approximate solution of ill-posed problems; the general theory of difference schemes; and so on. During the past decade, the Voronezh seminar on functional analysis has systematically discussed various questions related to numerical methods; several advanced courses have been held at Voronezh Uni versity on the application of functional analysis to numerical mathe matics. Some of this research is summarized in the present monograph. The authors' aim has not been to give an exhaustive account, even of the principal known results. The book consists of five chapters.
Book Synopsis Linear Inverse Problems and Tikhonov Regularization by : Mark S. Gockenbach
Download or read book Linear Inverse Problems and Tikhonov Regularization written by Mark S. Gockenbach and published by American Mathematical Soc.. This book was released on 2016-12-31 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Inverse problems occur frequently in science and technology, whenever we need to infer causes from effects that we can measure. Mathematically, they are difficult problems because they are unstable: small bits of noise in the measurement can completely throw off the solution. Nevertheless, there are methods for finding good approximate solutions. Linear Inverse Problems and Tikhonov Regularization examines one such method: Tikhonov regularization for linear inverse problems defined on Hilbert spaces. This is a clear example of the power of applying deep mathematical theory to solve practical problems. Beginning with a basic analysis of Tikhonov regularization, this book introduces the singular value expansion for compact operators, and uses it to explain why and how the method works. Tikhonov regularization with seminorms is also analyzed, which requires introducing densely defined unbounded operators and their basic properties. Some of the relevant background is included in appendices, making the book accessible to a wide range of readers.
