Greedy Algorithms for Reduced Bases in Banach Spaces
Greedy Algorithms for Reduced Bases in Banach Spaces
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DOI：
10.1007/s00365-013-9186-2
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发表时间：
2013-06-01
影响因子：
2.7
通讯作者：
Wojtaszczyk, Przemyslaw
中科院分区：
文献类型：
--
作者：
DeVore, Ronald;Petrova, Guergana;Wojtaszczyk, Przemyslaw
给定一个Banach空间X和它的一个紧集，我们考虑寻找一个好的n维空间X（n）aS，X，它可以用来逼近的元素的问题.对于这样的近似，我们可以实现的最佳可能误差由柯尔莫哥洛夫宽度给出。然而，找到提供这种性能的空间通常在数值上是难以处理的。最近，一个新的贪婪策略，以获得良好的空间的背景下，减少基方法求解参数族的偏微分方程。这种贪婪算法的性能最初在Buffa等人（Mod，l.数学。肛交。埃什特河46：595-603，2012）在这种情况下是希尔伯特空间。Buffa等人（Mod，l.数学。肛交。埃什特河46：595-603，2012）在Binev等人（SIAM J. Math. Anal. 43：1457-1472，2011）。本文的目的是给这样的贪婪算法的性能的一个新的分析。我们的分析不仅给出了改进的结果的Hilbert空间的情况下，但也可以适用于相同的贪婪程序在一般的Banach空间。
Given a Banach space X and one of its compact sets , we consider the problem of finding a good n-dimensional space X (n) aS,X which can be used to approximate the elements of . The best possible error we can achieve for such an approximation is given by the Kolmogorov width . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Mod,l. Math. Anal. Num,r. 46:595-603, 2012) in the case is a Hilbert space. The results of Buffa et al. (Mod,l. Math. Anal. Num,r. 46:595-603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457-1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.