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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 Walk-on-spheres method ： ウィキペディア英語版 Walk-on-spheres method In mathematics, the walk-on-spheres method (WoS) is a numerical probabilistic algorithm, or Monte-Carlo method, used mainly in order to approximate the solutions of some specific boundary value problem for partial differential equations. The WoS method was first introduced by M. E. Muller in 1956 to solve Laplace's equation,〔 and was since then generalized to other problems. It relies on probabilistic interpretations of PDEs, by simulating paths of Brownian motion (or for some more general variants, diffusion processes), and it is today one of the most widely used "grid-free" algorithms for generating Brownian paths. ==Informal description== Let $\backslash Omega$ be a bounded domain in $\backslash mathbb^d$ with a sufficiently regular boundary $\backslash Gamma$, let ''h'' be a function on $\backslash Gamma$, and let $x$ be a point inside $\backslash Omega$. Let us consider the Dirichlet problem: : u(x) = h(x) & \mboxx \in \Gamma. \end It can be easily shown that when the solution $u$ exists, for $x\; \backslash in\; \backslash Omega$: :$u(x)\; =\; \backslash mathbb\_x(h(W\_))$ where is a -dimensional Wiener process, the expected value is taken conditionally on , and is the first-exit time out of . To compute a solution using this formula, we only have to simulate the first exit point of independent Brownian paths since with the law of large numbers: :$\backslash mathbb\_x(h(W\_\backslash tau))\; \backslash sim\; \backslash frac\; \backslash sum\_^n\; h(W^i\_\backslash tau)$ The WoS method provides an efficient way of sampling the first exit point of a Brownian motion from the domain, by remarking that for any -sphere centred on , the first point of exit of out of the sphere has a uniform distribution over its surface. Rather than simulating in detail the path of the process, it samples only the exit-points out of successive spheres, often making it less costly than "grid-based" algorithms. The WoS algorithm consists in drawing the largest sphere $\backslash mathcal\_0$ centered on and contained inside the domain. The first point of exit from $\backslash mathcal\_0$ is uniformly distributed on its surface. By repeating this step inductively, the WoS provides a sequence of positions of the Brownian Motion. According to intuition, the process will converge to the first exit point of the domain. However, this algorithm takes almost surely an infinite number of steps to end. For computational implementation, the process is usually stopped when it gets sufficiently close to the border, and returns the projection of the process on the border. This procedure is sometimes called introducing an $\backslash varepsilon$-shell, or $\backslash varepsilon$-layer.〔 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「Walk-on-spheres 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.