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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 Dykstra's projection algorithm ： ウィキペディア英語版 Dykstra's projection algorithm Dykstra's algorithm is a method that computes a point in the intersection of convex sets, and is a variant of the alternating projection method (also called the projections onto convex sets method). In its simplest form, the method finds a point in the intersection of two convex sets by iteratively projecting onto each of the convex set; it differs from the alternating projection method in that there are intermediate steps. A parallel version of the algorithm was developed by Gaffke and Mathar. The method is named after R. L. Dykstra who proposed it in the 1980s. A key difference between Dykstra's algorithm and the standard alternating projection method occurs when there is more than one point in the intersection of the two sets. In this case, the alternating projection method gives some arbitrary point in this intersection, whereas Dykstra's algorithm gives a specific point: the projection of ''r'' onto the intersection, where ''r'' is the initial point used in the algorithm, == Algorithm == Dykstra's algorithm finds for each $r$ the only $\backslash bar\; \backslash in\; C\backslash cap\; D$ such that: : $\backslash |\backslash bar-r\backslash |^2\backslash le\; \backslash |x-r\backslash |^2,\; \backslash text\; x\backslash in\; C\; \backslash cap\; D,$ where $C,D$ are convex sets. This problem is equivalent to finding the projection of $r$ onto the set $C\backslash cap\; D$, which we denote by $\backslash mathcal\_$. To use Dykstra's algorithm, one must know how to project onto the sets $C$ and $D$ separately. First, consider the basic alternating projection (aka POCS) method (first studied, in the case when the sets $C,D$ were linear subspaces, by John von Neumann〔J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50 (1949) 401–485 (a reprint of lecture notes first distributed in 1933).〕), which initializes $x\_0=r$ and then generates the sequence : $x\_\; =\; \backslash mathcal\_C\; \backslash left(\; \backslash mathcal\_D\; (\; x\_k\; )\; \backslash right)$. Dykstra's algorithm is of a similar form, but uses additional auxiliary variables. Start with $x\_0\; =r,\; p\_0=q\_0=0$ and update by : $y\_k\; =\; \backslash mathcal\_D(\; x\_k\; +\; p\_k\; )$ : $p\_\; =\; x\_k\; +\; p\_k\; -\; y\_k$ : $x\_\; =\; \backslash mathcal\_C(\; y\_k\; +\; q\_k\; )$ : $q\_\; =y\_k\; +\; q\_k\; -\; x\_.$ Then the sequence $(x\_k)$ converges to the solution of the original problem. For convergence results and a modern perspective on the literature, see 〔P. L. Combettes and J.-C. Pesquet, "Proximal splitting methods in signal processing," in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Editors), pp. 185–212. Springer, New York, 2011 ()〕 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「Dykstra's projection algorithm」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.