Greedy Algorithms for Reduced Bases in Banach Spaces

Type
Article

Authors
DeVore, Ronald
Petrova, Guergana
Wojtaszczyk, Przemyslaw

KAUST Grant Number
KUS-C1-016-04

Online Publication Date
2013-02-26

Print Publication Date
2013-06

Date
2013-02-26

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.

Citation
DeVore R, Petrova G, Wojtaszczyk P (2013) Greedy Algorithms for Reduced Bases in Banach Spaces. Constructive Approximation 37: 455–466. Available: http://dx.doi.org/10.1007/s00365-013-9186-2.

Acknowledgements
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107, and ONR N00014-11-1-0712; the AFOSR Contract FA95500910500; the NSF Grants DMS-0810869, and DMS 0915231; and the EU Project POWIEW. This publication is based in part on work supported by Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).

Publisher
Springer Nature

Journal
Constructive Approximation

DOI
10.1007/s00365-013-9186-2

Permanent link to this record