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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Quasi-Monte Carlo method ： ウィキペディア英語版 Quasi-Monte Carlo method In numerical analysis, quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regular Monte Carlo method or Monte Carlo integration, which are based on sequences of pseudorandom numbers. Monte Carlo and quasi-Monte Carlo methods are stated in a similar way. The problem is to approximate the integral of a function ''f'' as the average of the function evaluated at a set of points ''x''1, ..., ''x''''N'': :$\backslash int\_\; f(u)\backslash ,u\; \backslash approx\; \backslash frac\backslash ,\backslash sum\_^N\; f(x\_i).$ Since we are integrating over the ''s''-dimensional unit cube, each ''x''''i'' is a vector of ''s'' elements. The difference between quasi-Monte Carlo and Monte Carlo is the way the xi are chosen. Quasi-Monte Carlo uses a low-discrepancy sequence such as the Halton sequence, the Sobol sequence, or the Faure sequence, whereas Monte Carlo uses a pseudorandom sequence. The advantage of using low-discrepancy sequences is a faster rate of convergence. Quasi-Monte Carlo has a rate of convergence close to O(1/N), whereas the rate for the Monte Carlo method is O(N−0.5).〔Søren Asmussen and Peter W. Glynn, ''Stochastic Simulation: Algorithms and Analysis'', Springer, 2007, 476 pages〕 The Quasi-Monte Carlo method recently became popular in the area of mathematical finance or computational finance.〔 In these areas, high-dimensional numerical integrals, where the integral should be evaluated within a threshold ε, occur frequently. Hence, the Monte Carlo method and the quasi-Monte Carlo method are beneficial in these situations. == Approximation error bounds of quasi-Monte Carlo == The approximation error of the quasi-Monte Carlo method is bounded by a term proportional to the discrepancy of the set ''x''1, ..., ''x''''N''. Specifically, the Koksma-Hlawka inequality states that the error :$\backslash epsilon\; =\; \backslash left|\; \backslash int\_\; f(u)\backslash ,u\; -\; \backslash frac\backslash ,\backslash sum\_^N\; f(x\_i)\; \backslash right|$ is bounded by :$|\backslash epsilon|\; \backslash leq\; V(f)\; D\_N$, where V(f) is the Hardy-Krause variation of the function f (see Morokoff and Caflisch (1995) 〔 for the detailed definitions). DN is the discrepancy of the set (x1,...,xN) and is defined as :$D\_N\; =\; \backslash sup\_\; \backslash left|\; \backslash frac\; -\; \backslash mbox(Q)\backslash right|$, where Q is a rectangular solid in ()s with sides parallel to the coordinate axes.〔 The inequality $|\backslash epsilon|\; \backslash leq\; V(f)\; D\_N$ can be used to show that the error of the approximation by the quasi-Monte Carlo method is $O\backslash left(\backslash frac\backslash right)$, whereas the Monte Carlo method has a probabilistic error of $O\backslash left(\backslash frac\{\backslash sqrt\{N\}\}\backslash right)$. Though we can only state the upper bound of the approximation error, the convergence rate of quasi-Monte Carlo method in practice is usually much faster than its theoretical bound.〔 Hence, in general, the accuracy of the quasi-Monte Carlo method increases faster than that of the Monte Carlo method. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Quasi-Monte Carlo method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.