The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.
Mots-clés : greedy algorithm, reduced basis approximations, a priori analysis, best fit analysis
@article{M2AN_2012__46_3_595_0,
author = {Buffa, Annalisa and Maday, Yvon and Patera, Anthony T. and Prud{\textquoteright}homme, Christophe and Turinici, Gabriel},
title = {\protect\emph{A priori }convergence of the greedy algorithm for the parametrized reduced basis method},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {595--603},
publisher = {EDP-Sciences},
volume = {46},
number = {3},
year = {2012},
doi = {10.1051/m2an/2011056},
zbl = {1272.65084},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011056/}
}
TY - JOUR AU - Buffa, Annalisa AU - Maday, Yvon AU - Patera, Anthony T. AU - Prud’homme, Christophe AU - Turinici, Gabriel TI - A priori convergence of the greedy algorithm for the parametrized reduced basis method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 595 EP - 603 VL - 46 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011056/ DO - 10.1051/m2an/2011056 LA - en ID - M2AN_2012__46_3_595_0 ER -
%0 Journal Article %A Buffa, Annalisa %A Maday, Yvon %A Patera, Anthony T. %A Prud’homme, Christophe %A Turinici, Gabriel %T A priori convergence of the greedy algorithm for the parametrized reduced basis method %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 595-603 %V 46 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011056/ %R 10.1051/m2an/2011056 %G en %F M2AN_2012__46_3_595_0
Buffa, Annalisa; Maday, Yvon; Patera, Anthony T.; Prud’homme, Christophe; Turinici, Gabriel. A priori convergence of the greedy algorithm for the parametrized reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 595-603. doi : 10.1051/m2an/2011056. http://www.numdam.org/articles/10.1051/m2an/2011056/
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