Greedy Algorithms for Reduced Bases in Banach Spaces

Ronald DeVore, Guergana Petrova, Przemyslaw Wojtaszczyk

Research output: Contribution to journalArticlepeer-review

67 Scopus citations

Abstract

Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n-dimensional space X n⊂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 Buffa et al. (Modél. Math. Anal. Numér. 46:595-603, 2012) in the case X=H is a Hilbert space. The results of Buffa et al. (Modél. Math. Anal. Numér. 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. © 2013 Springer Science+Business Media New York.
Original languageEnglish (US)
Pages (from-to)455-466
Number of pages12
JournalConstructive Approximation
Volume37
Issue number3
DOIs
StatePublished - Feb 26 2013
Externally publishedYes

Fingerprint

Dive into the research topics of 'Greedy Algorithms for Reduced Bases in Banach Spaces'. Together they form a unique fingerprint.

Cite this