Greedy Algorithms for Reduced Bases in Banach Spaces

TL;DR

This paper analyzes greedy algorithms for constructing reduced bases in Banach spaces, providing improved performance guarantees that extend beyond Hilbert spaces to general Banach spaces.

Contribution

It offers a new analysis of greedy algorithms for reduced basis construction, improving existing results and extending applicability to Banach spaces.

Findings

Improved convergence rates for greedy algorithms in Hilbert spaces.

Extension of analysis to general Banach spaces.

Enhanced understanding of approximation errors in reduced basis methods.

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 \subset 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 d_n(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 A. Buffa, Y. Maday, A.T. Patera, C. Prud'homme, and G. Turinici, "A Priori convergence of the greedy algorithm for the parameterized reduced basis", M2AN Math. Model. Numer. Anal., 46(2012), 595-603 in the case X = H is a Hilbert space. The results there were significantly improved on in P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, "Convergence rates for greedy algorithms in reduced bases Methods", SIAM J. Math. Anal., 43 (2011), 1457-1472. 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.