Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T07:50:05.987Z Has data issue: false hasContentIssue false
  • English
  • Français

A priori convergence of the Greedy algorithm for 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.fr Division 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
Get access

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.

Keywords

Greedy algorithmreduced basis approximationsa priori analysisbest fit analysis
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 3: Special volume in honor of Professor David Gottlieb , May 2012 , pp. 595 - 603
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P., Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. Google Scholar
Kolmogorov, A., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. (2) 37 (1936) 107–110. Google Scholar
Maday, Y., Patera, A.T. and Turinici, G., A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. Google Scholar
Maday, Y., Turinici, A.T. Patera and G., Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci., Paris, Sér. I Math. 335 (2002) 289–294. Google Scholar
Melenk, J.M., On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272–289. Google Scholar
A. Pinkus, n -widths in approximation theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 7. Springer-Verlag, Berlin (1985).
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations – application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15 (2008) 229–275. Google Scholar
Sen, S., Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numer. Heat Transfer Part B 54 (2008) 369–389. Google Scholar
K. Veroy, C. Prud’homme, D.V. Rovas and A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proceedings of the 16th AIAA Computational Fluid Dynamics Conference (2003) 2003–3847.