[Résultats globaux a priori pour l'approximation d'équations aux dérivées partielles coercives symétriques elliptiques dépendant d'un paramètre]
On considère des méthodes de bases réduites de type Lagrange pour des équations aux dérivées partielles coercives symétriques elliptiques et dépendant d'un paramètre. On montre que, pour une répartition logarithmiquement quasi uniforme des points d'échantillonage, l'approximation en base réduite converge de façon exponentielle vers la solution exacte uniformément par rapport au paramètre. De plus la convergence ne dépend que faiblement du rapport entre les coefficients de coercivité et de continuité de l'opérateur : ainsi une approximation de très basse dimension procure une solution très précise même dans le cas d'un large eventail de paramètres. Des test numériques (présentés ailleurs) corroborent ces prédictions numériques.
We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced-basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity–coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.
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@article{CRMATH_2002__335_3_289_0,
author = {Maday, Yvon and Patera, Anthony T. and Turinici, G.},
title = {Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations},
journal = {Comptes Rendus. Math\'ematique},
pages = {289--294},
publisher = {Elsevier},
volume = {335},
number = {3},
year = {2002},
doi = {10.1016/S1631-073X(02)02466-4},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02466-4/}
}
TY - JOUR AU - Maday, Yvon AU - Patera, Anthony T. AU - Turinici, G. TI - Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations JO - Comptes Rendus. Mathématique PY - 2002 SP - 289 EP - 294 VL - 335 IS - 3 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02466-4/ DO - 10.1016/S1631-073X(02)02466-4 LA - en ID - CRMATH_2002__335_3_289_0 ER -
%0 Journal Article %A Maday, Yvon %A Patera, Anthony T. %A Turinici, G. %T Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations %J Comptes Rendus. Mathématique %D 2002 %P 289-294 %V 335 %N 3 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02466-4/ %R 10.1016/S1631-073X(02)02466-4 %G en %F CRMATH_2002__335_3_289_0
Maday, Yvon; Patera, Anthony T.; Turinici, G. Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus. Mathématique, Tome 335 (2002) no. 3, pp. 289-294. doi : 10.1016/S1631-073X(02)02466-4. http://www.numdam.org/articles/10.1016/S1631-073X(02)02466-4/
[1] Automatic choice of global shape functions in structural analysis, AIAA J., Volume 16 (1978), pp. 525-528
[2] On the reduced basis method, Math. Mech., Volume 7 (1995) no. 75, pp. 543-549
[3] Numerical Methods, Prentice-Hall, 1974 (p. 100)
[4] On the error behaviour of the reduced basis technique for nonlinear finite element approximations, Z. Angew. Math. Mech., Volume 63 (1983), pp. 21-28
[5] Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, C. R. Acad. Sci. Paris, Série I, Volume 331 (2000), pp. 153-158
[6] Blackbox reduced-basis output bound methods for shape optimization, Proceedings 12th International Domain Decomposition Conference, Japan, 2000
[7] A blackbox reduced-basis output bound method for noncoercive linear problems (Cioranescu, D.; Lions, J.-L., eds.), Studies in Mathematics and its Applications, Nonlinear Partial Differential Equations and Their Applications, College De France Seminar, XIV, North-Holland, 2002
[8] Y. Maday, A.T. Patera, G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, J. Sci. Comput. (December 2002)
[9] Reduced basis technique for nonlinear analysis of structures, AIAA J., Volume 18 (1980) no. 4, pp. 455-462
[10] The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Statist. Comput., Volume 10 (1989) no. 4, pp. 777-786
[11] Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods, J. Fluids Engrg. Trans. ASME, Volume 124 (2002) no. 1, pp. 70-80
[12] On the theory and error estimation of the reduced basis method for multi-parameter problems, Nonlinear Anal., Volume 21 (1993) no. 11, pp. 849-858
[13] An Analysis of the Finite Element Method, Wellesley–Cambridge Press, 1973
[14] K. Veroy, Reduced basis methods applied to problems in elasticity: Analysis and applications, Ph.D. thesis, Massachusetts Institute of Technology, 2003, in progress
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