Minimax Models in the Theory of Numerical Methods

By: Olga Chuyan (Translator), A. Sukharev

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The efficiency of computational methods and the choice of the most efficient methods for solving a specific problem or a specific class of problems have always played an important role in numerical analysis. Optimization of the computerized solution process is now a major problem of applied mathematics, which stimulates the search for new computational methods and ways to implement them. In Minimax Models in the Theory of Numerical Methods, methods for estimating the efficiency of computational algorithms and problems of their optimality are studied within the framework of a general computation model. The subjects dealt with in this important book are very different from the traditional subjects of computational methods. Close attention is paid to adaptive (sequential) computational algorithms, the process of computation being regarded as a controlled process and the algorithm as a control strategy. This approach allows methods of game theory and othermethods of operations research and systems analysis to be widely used for constructing optimal algorithms. The goal underlying the study of the various computation models dealt with in this title is the construction of concrete numerical algorithms admitting program implementation. The central role belongs to the concept of a sequentially optimal algorithm, which in many cases reflects the characteristics of real-life computational processes more fully than the traditional optimality concepts.

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Preface to the English Edition
Preface
Glossary of Symbols
General Computation Model	p. 1
Basic concepts	p. 1
Functional classes under consideration	p. 4
Classes of deterministic algorithms	p. 12
Minimax concept of optimality and specific notions of optimality	p. 18
Comparison of the best guaranteed results for adaptive and nonadaptive algorithms	p. 35
Sequentially optimal algorithms	p. 40
Stochastic algorithms	p. 46
Numerical Integration	p. 54
Optimal quadratures for functional classes determined by quasi-metrics	p. 54
Optimal quadratures for functional classes determined by moduli of continuity	p. 68
Sequentially optimal and one-step optimal integration algorithms	p. 77
Numerical tests	p. 97
Optimal computation of iterated integrals	p. 103
Computation of multiple integrals using Peano type developments	p. 114
Recovery of Functions from Their Values	p. 121
Optimal nonadaptive algorithms	p. 121
Sequentially optimal and one-step optimal recovery algorithms	p. 128
Solution of a multistep antagonistic game related to the problem of optimal recovery	p. 137
Search for the Global Extremum	p. 152
On the choice of starting points for local optimization methods	p. 153
Optimal nonadaptive search for a functional class determined by a quasi-metric	p. 157
Reduction of the problem of constructing a sequentially optimal algorithm for a functional class determined by a quasi-metric to a series of problems of optimal covering	p. 163
Specific computational algorithms	p. 169
Case of approximate information	p. 178
On-step optimal stochastic algorithm	p. 183
Some Special Classes of Extremal Problems	p. 196
Solution of equations and systems of equations	p. 196
Maximization of a minimum function with coupled variables	p. 206
Optimization with several criteria	p. 211
Bibliography	p. 218
Author Index	p. 247
Subject Index	p. 251
Table of Contents provided by Blackwell. All Rights Reserved.

Minimax Models in the Theory of Numerical Methods

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