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

Annalisa Buffa; Yvon Maday; Anthony T. Patera; Christophe Prud’homme; Gabriel Turinici

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 3, page 595-603
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Buffa, Annalisa, et al. "A priori convergence of the Greedy algorithm for the parametrized reduced basis method." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 595-603. <http://eudml.org/doc/277851>.

@article{Buffa2012,
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.},
author = {Buffa, Annalisa, Maday, Yvon, Patera, Anthony T., Prud’homme, Christophe, Turinici, Gabriel},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Greedy algorithm; reduced basis approximations; a priori analysis; best fit analysis; greedy algorithm; a priori error bounds; Galerkin method; convergence},
language = {eng},
month = {1},
number = {3},
pages = {595-603},
publisher = {EDP Sciences},
title = {A priori convergence of the Greedy algorithm for the parametrized reduced basis method},
url = {http://eudml.org/doc/277851},
volume = {46},
year = {2012},
}

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
DA - 2012/1//
PB - EDP Sciences
VL - 46
IS - 3
SP - 595
EP - 603
AB - 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.
LA - eng
KW - Greedy algorithm; reduced basis approximations; a priori analysis; best fit analysis; greedy algorithm; a priori error bounds; Galerkin method; convergence
UR - http://eudml.org/doc/277851
ER -

References

top
  1. P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal.43 (2011) 1457–1472.  Zbl1229.65193
  2. A. Kolmogorov, Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. (2)37 (1936) 107–110.  
  3. Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations. J. Sci. Comput.17 (2002) 437–446.  Zbl1014.65115
  4. Y. Maday, A.T. Patera and G. Turinici, 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.  Zbl1009.65066
  5. J.M. Melenk, On n-widths for elliptic problems. J. Math. Anal. Appl.247 (2000) 272–289.  Zbl0963.35047
  6. 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).  Zbl0551.41001
  7. G. Rozza, D.B.P. Huynh and A.T. Patera, 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.  Zbl1304.65251
  8. S. Sen, Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numer. Heat Transfer Part B54 (2008) 369–389.  
  9. 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.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.