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1995 | Buch

Numerical Methods for the Solution of Ill-Posed Problems

verfasst von: A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

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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.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
From the point of view of modern mathematics, all problems can be classified as being either correctly posed or incorrectly posed.
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola
Chapter 1. Regularization methods
Abstract
In this Chapter we consider methods for solving ill-posed problems under the condition that the a priori information is, in general, insufficient in order to single out a compact set of well-posedness. The main ideas in this Chapter have been expressed in [165], [166]. We will consider the case when the operator is also given approximately, while the set of constraints of the problem is a closed convex set in a Hilbert space. The case when the operator is specified exactly and the case when constraints are absent (i.e. the set of constraints coincides with the whole space) are instances of the problem statement considered here.
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola
Chapter 2. Numerical methods for the approximate solution of ill-posed problems on compact sets
Abstract
In Chapter 1 we have discussed general questions concerning the construction of regularizing algorithms for solving a wide circle of ill-posed problems. In this Chapter we will consider in some detail the case when we a priori know that the exact solution of the problem belongs to a certain compact set. The idea of this approach was expressed already in 1943 [164].
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola
Chapter 3. Algorithms for the approximate solution of ill-posed problems on special sets
Abstract
In Chapter 2 we have succeeded in solving, in a number of cases, the first problem posed to us: starting from qualitative information regarding the unknown solution, how to find the compact set of well-posedness M containing the exact solution. It was shown that this can be readily done if the exact solution of the problem belongs to Z C , Ž C , Ž C . A uniform approximation to the exact solution of the problem can be constructed if the exact solution is a continuous function of bounded variation. We now turn to the second problem: how to construct an efficient numerical algorithm for solving ill-posed problems on the sets listed above?
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola
Chapter 4. Algorithms and programs for solving linear ill-posed problems
Abstract
In this chapter we give a description of the programs implementing the algorithms considered in this book. This description is accompanied by examples of computations of model problems. These examples are meant to serve two purposes. First, they have to illustrate the description of the program. Secondly, they have to convince a reader who wants to use our programs that these programs have been correctly inputted by him/her on his/her computer. In this case the model computations may serve as a test case for checking this.
A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola
Backmatter
Metadaten
Titel
Numerical Methods for the Solution of Ill-Posed Problems
verfasst von
A. N. Tikhonov
A. V. Goncharsky
V. V. Stepanov
A. G. Yagola
Copyright-Jahr
1995
Verlag
Springer Netherlands
Electronic ISBN
978-94-015-8480-7
Print ISBN
978-90-481-4583-6
DOI
https://doi.org/10.1007/978-94-015-8480-7