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A priori convergence of the Greedy algorithmfor the parametrized reduced basis method

Published online by Cambridge University Press:  11 January 2012

Annalisa Buffa,
Yvon Maday,
Anthony T. Patera,
Christophe Prud’homme and
Gabriel Turinici
Annalisa Buffa
Affiliation:
Instituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia, Italy
Yvon Maday
Affiliation:
UPMC Univ Paris VI, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France. maday@ann.jussieu.frDivision of Applied Mathematics, Brown University, Providence, RI, USA
Anthony T. Patera
Affiliation:
Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Mass. Ave., Cambridge, 02139-4307 MA, USA
Christophe Prud’homme
Affiliation:
Université de Grenoble 1-Joseph Fourier, Laboratoire Jean Kuntzmann, 51 rue des Mathèmatiques, BP 53, 38041 Grenoble Cedex 9, France
Gabriel Turinici
Affiliation:
Université Paris Dauphine, CEREMADE, Place du Marechal de Lattre de Tassigny, 75016 Paris, France

Abstract

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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.

Keywords

Greedy algorithmreduced basis approximationsa priori analysisbest fit analysis

References

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