Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD-Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD-Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD-Greedy algorithm.
@article{M2AN_2013__47_3_859_0,
author = {Haasdonk, Bernard},
title = {Convergence Rates of the POD-Greedy Method},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {859-873},
doi = {10.1051/m2an/2012045},
mrnumber = {3056412},
zbl = {1277.65074},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_3_859_0}
}
Haasdonk, Bernard. Convergence Rates of the POD-Greedy Method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 859-873. doi : 10.1051/m2an/2012045. http://gdmltest.u-ga.fr/item/M2AN_2013__47_3_859_0/
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