Regularization Of Linear Operator Equations
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Author | : M. Thamban Nair |
Publisher | : World Scientific |
Total Pages | : 264 |
Release | : 2009 |
Genre | : Mathematics |
ISBN | : 9812835652 |
Download Linear Operator Equations Book in PDF, Epub and Kindle
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.
Author | : Mark Anthony Lukas |
Publisher | : |
Total Pages | : 290 |
Release | : 1981 |
Genre | : Linear operators |
ISBN | : |
Download Regularization of Linear Operator Equations Book in PDF, Epub and Kindle
Author | : Grace Wahba |
Publisher | : |
Total Pages | : 40 |
Release | : 1973 |
Genre | : Convergence |
ISBN | : |
Download Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations Book in PDF, Epub and Kindle
Author | : Anatoly B. Bakushinsky |
Publisher | : Walter de Gruyter GmbH & Co KG |
Total Pages | : 447 |
Release | : 2018-02-05 |
Genre | : Mathematics |
ISBN | : 3110556383 |
Download Regularization Algorithms for Ill-Posed Problems Book in PDF, Epub and Kindle
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 | : Grace Wahba |
Publisher | : |
Total Pages | : 45 |
Release | : 1981 |
Genre | : |
ISBN | : |
Download Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine Book in PDF, Epub and Kindle
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).
Author | : C. W. Groetsch |
Publisher | : Pitman Advanced Publishing Program |
Total Pages | : 124 |
Release | : 1984 |
Genre | : Mathematics |
ISBN | : |
Download The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind Book in PDF, Epub and Kindle
Author | : V. P. Tanana |
Publisher | : Walter de Gruyter |
Total Pages | : 229 |
Release | : 2012-02-13 |
Genre | : Mathematics |
ISBN | : 3110900157 |
Download Methods for Solving Operator Equations Book in PDF, Epub and Kindle
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.
Author | : Heinz W. Engl |
Publisher | : |
Total Pages | : 15 |
Release | : 1982 |
Genre | : |
ISBN | : |
Download On the Convergence of Regularization Methods for Ill-posed Linear Operator Equations Book in PDF, Epub and Kindle
Author | : Thomas Schuster |
Publisher | : Walter de Gruyter |
Total Pages | : 296 |
Release | : 2012-07-30 |
Genre | : Mathematics |
ISBN | : 3110255723 |
Download Regularization Methods in Banach Spaces Book in PDF, Epub and Kindle
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.
Author | : M. Zuhair Nashed |
Publisher | : |
Total Pages | : 47 |
Release | : 1972 |
Genre | : Hilbert space |
ISBN | : |
Download Generalized Inverses in Reproducing Kernel Spaces Book in PDF, Epub and Kindle
In the paper a study of generalized inverses of linear operators in reproducing kernel Hilbert spaces (RKHS) is initiated. Explicit representations of generalized inverses in RKHS are obtained, and their relations to regularization of (ill-posed or poorly-conditioned) linear operator equations are explored. It is shown in particular that this approach provides a natural and effective setting for regularization problems when the operator maps one RKHS into another. Finally properties of the regularized pseudosolutions in this setting are studied. (Author).