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Numerical Methods for the Solution of Ill-Posed Problems

2013-03-09 Mathematics
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Release : 2013-03-09
Genre : Mathematics
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Book Rating : 80X/5 ( reviews)

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Book Synopsis Numerical Methods for the Solution of Ill-Posed Problems by : A.N. Tikhonov

Download or read book Numerical Methods for the Solution of Ill-Posed Problems written by A.N. Tikhonov. This book was released on 2013-03-09. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Numerical Methods for the Solution of Ill-posed Problems

1977
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Release : 1977
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Book Synopsis Numerical Methods for the Solution of Ill-posed Problems by :

Download or read book Numerical Methods for the Solution of Ill-posed Problems written by . This book was released on 1977. Available in PDF, EPUB and Kindle. Book excerpt:

Rank-Deficient and Discrete Ill-Posed Problems

2005-01-01 Mathematics
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Release : 2005-01-01
Genre : Mathematics
Kind : eBook
Book Rating : 036/5 ( reviews)

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Book Synopsis Rank-Deficient and Discrete Ill-Posed Problems by : Per Christian Hansen

Download or read book Rank-Deficient and Discrete Ill-Posed Problems written by Per Christian Hansen. This book was released on 2005-01-01. 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.

Solutions of Ill-posed Problems

1977 Mathematics
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Release : 1977
Genre : Mathematics
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Book Synopsis Solutions of Ill-posed Problems by : Andreĭ Nikolaevich Tikhonov

Download or read book Solutions of Ill-posed Problems written by Andreĭ Nikolaevich Tikhonov. This book was released on 1977. Available in PDF, EPUB and Kindle. Book excerpt:

Well-posed, Ill-posed, and Intermediate Problems with Applications

2011-12-22 Mathematics
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Release : 2011-12-22
Genre : Mathematics
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Book Rating : 305/5 ( reviews)

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Book Synopsis Well-posed, Ill-posed, and Intermediate Problems with Applications by : Petrov Yuri P.

Download or read book Well-posed, Ill-posed, and Intermediate Problems with Applications written by Petrov Yuri P.. This book was released on 2011-12-22. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with one of the key problems in applied mathematics, namely the investigation into and providing for solution stability in solving equations with due allowance for inaccuracies in set initial data, parameters and coefficients of a mathematical model for an object under study, instrumental function, initial conditions, etc., and also with allowance for miscalculations, including roundoff errors. Until recently, all problems in mathematics, physics and engineering were divided into two classes: well-posed problems and ill-posed problems. The authors introduce a third class of problems: intermediate ones, which are problems that change their property of being well- or ill-posed on equivalent transformations of governing equations, and also problems that display the property of being either well- or ill-posed depending on the type of the functional space used. The book is divided into two parts: Part one deals with general properties of all three classes of mathematical, physical and engineering problems with approaches to solve them; Part two deals with several stable models for solving inverse ill-posed problems, illustrated with numerical examples.

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