Convergence Rates of the POD-Greedy Method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 3, p. 859-873

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.

DOI : https://doi.org/10.1051/m2an/2012045
Classification:  65D15,  35B30,  58B99,  37C50
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {3},
     year = {2013},
     pages = {859-873},
     doi = {10.1051/m2an/2012045},
     zbl = {1277.65074},
     mrnumber = {3056412},
     language = {en},
     url = {http://www.numdam.org/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, Volume 47 (2013) no. 3, pp. 859-873. doi : 10.1051/m2an/2012045. http://www.numdam.org/item/M2AN_2013__47_3_859_0/

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