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.
Mots-clés : greedy approximation, proper orthogonal decomposition, convergence rates, reduced basis methods
@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 },
pages = {859--873},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/m2an/2012045},
mrnumber = {3056412},
zbl = {1277.65074},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012045/}
}
TY - JOUR AU - Haasdonk, Bernard TI - Convergence Rates of the POD-Greedy Method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 859 EP - 873 VL - 47 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012045/ DO - 10.1051/m2an/2012045 LA - en ID - M2AN_2013__47_3_859_0 ER -
%0 Journal Article %A Haasdonk, Bernard %T Convergence Rates of the POD-Greedy Method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 859-873 %V 47 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012045/ %R 10.1051/m2an/2012045 %G en %F M2AN_2013__47_3_859_0
Haasdonk, Bernard. Convergence Rates of the POD-Greedy Method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 859-873. doi : 10.1051/m2an/2012045. http://archive.numdam.org/articles/10.1051/m2an/2012045/
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