A priori convergence of the Greedy algorithm for the parametrized reduced basis method

¹ Instituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia, Italy
² UPMC Univ Paris VI, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
maday@ann.jussieu.fr
³ Division of Applied Mathematics, Brown University, Providence, RI, USA
⁴ Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-266, 77 Mass. Ave., Cambridge, 02139-4307 MA, USA
⁵ Université de Grenoble 1-Joseph Fourier, Laboratoire Jean Kuntzmann, 51 rue des Mathèmatiques, BP 53, 38041 Grenoble Cedex 9, France
⁶ Université Paris Dauphine, CEREMADE, Place du Marechal de Lattre de Tassigny, 75016 Paris, France

Received: 9 November 2009

Abstract

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.