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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 Sobol sequence ： ウィキペディア英語版 Sobol sequence Sobol sequences (also called LPτ sequences or (''t'', ''s'') sequences in base 2) are an example of quasi-random low-discrepancy sequences. They were first introduced by the Russian mathematician I. M. Sobol (Илья Меерович Соболь) in 1967.〔Sobol,I.M. (1967), "Distribution of points in a cube and approximate evaluation of integrals". ''Zh. Vych. Mat. Mat. Fiz.'' 7: 784–802 (in Russian); ''U.S.S.R Comput. Maths. Math. Phys.'' 7: 86–112 (in English).〕 These sequences use a base of two to form successively finer uniform partitions of the unit interval, and then reorder the coordinates in each dimension. == Good distributions in the s-dimensional unit hypercube == Let ''Is = ()s'' be the s-dimensional unit hypercube and ''f'' a real integrable function over ''Is''. The original motivation of Sobol was to construct a sequence ''xn'' in ''Is'' so that :$\backslash lim\_\backslash ;\; \backslash frac\backslash sum\_^n\; f(x\_i)\; \backslash ;\; =\backslash ;\; \backslash int\_\; f$ and the convergence be as fast as possible. It is more or less clear that for the sum to converge towards the integral, the points ''xn'' should fill ''Is'' minimizing the holes. Another good property would be that the projections of ''xn'' on a lower-dimensional face of ''Is'' leave very few holes as well. Hence the homogeneous filling of ''Is'' does not qualify ; because in lower-dimensions many points will be at the same place, therefore useless for the integral estimation. These good distributions are called (t,m,s)-nets and (t,s)-sequences in base b. To introduce them, define first an elementary s-interval in base b a subset of ''Is'' of the form :$\backslash prod\_^s\; \backslash left(\backslash frac$ for all j in Given 2 integers $0\backslash leq\; t\backslash leq\; m$, a (t,m,s)-net in base b is a sequence ''xn'' of bm points of ''Is'' such that $\backslash text\; \backslash ,\; P\; \backslash cap\; \backslash \; =\; b^t$ for all elementary interval ''P'' in base b of hypervolume ''λ(P) = bt-m''. Given a non-negative integer t, a (t,s)-sequence in base b is an infinite sequence of points ''xn'' such that for all integers $k\backslash geq0,\; \backslash ;\; m\backslash geq\; t$, the sequence $\backslash \backslash \}$ is a (t,m,s)-net in base b. In his article, Sobol described ''Πτ-meshes'' and ''LPτ sequences'', which are (t,m,s)-nets and (t,s)-sequences in base 2 respectively. The terms (t,m,s)-nets and (t,s)-sequences in base b (also called Niederreiter sequences) were coined in 1988 by Harald Niederreiter.〔Niederreiter, H. (1988). "Low-Discrepancy and Low-Dispersion Sequences", ''Journal of Number Theory'' 30: 51–70.〕 The term ''Sobol sequences'' was introduced in late English-speaking papers in comparison with Halton, Faure and other low-discrepancy sequences. 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「Sobol sequence」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.