A priori convergence of the greedy algorithm for the parametrized reduced basis method
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 3, pp. 595-603.

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.

DOI : 10.1051/m2an/2011056
Classification : 41A45, 41A65, 65N15
Mots clés : greedy algorithm, reduced basis approximations, a priori analysis, best fit analysis
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     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},
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     year = {2012},
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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 , Tome 46 (2012) no. 3, pp. 595-603. doi : 10.1051/m2an/2011056. http://www.numdam.org/articles/10.1051/m2an/2011056/

[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. | MR | Zbl

[2] A. Kolmogorov, Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. (2) 37 (1936) 107-110. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[5] J.M. Melenk, On n-widths for elliptic problems. J. Math. Anal. Appl. 247 (2000) 272-289. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[8] S. Sen, Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numer. Heat Transfer Part B 54 (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.

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