Greedy algorithms for reduced bases in banach spaces, Arxiv preprint arXiv:1204.2290

Abstract

Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space Xn ⊂ X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width dn(F)X. 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[2] in thecase X = H is aHilbert space. Theresults of [2] were significantly improved on in [1]. 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. Key words and phrases: greedy algorithms, convergence rates, reduced basis, genera

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