Comptes Rendus
no. 3
Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations
[Résultats globaux a priori pour l'approximation d'équations aux dérivées partielles coercives symétriques elliptiques dépendant d'un paramètre]
Yvon Maday1 ; Anthony T. Patera2 ; G. Turinici3
1 Laboratoire Jacques-Louis Lions, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
2 Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, MA 02139, USA
3 INRIA Rocquencourt, BP 105, 78153 Le Chesnay cedex, France
Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 289-294.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02466-4
Yvon Maday 1 ; Anthony T. Patera 2 ; G. Turinici 3

1 Laboratoire Jacques-Louis Lions, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
2 Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, MA 02139, USA
3 INRIA Rocquencourt, BP 105, 78153 Le Chesnay cedex, France 
@article{CRMATH_2002__335_3_289_0,
     author = {Yvon Maday and Anthony T. Patera and G. Turinici},
     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},
}
TY  - JOUR
AU  - Yvon Maday
AU  - Anthony T. Patera
AU  - G. Turinici
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
DO  - 10.1016/S1631-073X(02)02466-4
LA  - en
ID  - CRMATH_2002__335_3_289_0
ER  - 
%0 Journal Article
%A Yvon Maday
%A Anthony T. Patera
%A G. Turinici
%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
%R 10.1016/S1631-073X(02)02466-4
%G en
%F CRMATH_2002__335_3_289_0
Yvon Maday; Anthony T. Patera; G. Turinici. Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 289-294. doi : 10.1016/S1631-073X(02)02466-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02466-4/

[1] B.O. Almroth; P. Stern; F.A. Brogan Automatic choice of global shape functions in structural analysis, AIAA J., Volume 16 (1978), pp. 525-528

[2] A. Barrett; G. Reddien On the reduced basis method, Math. Mech., Volume 7 (1995) no. 75, pp. 543-549

[3] G. Dahlquist; Å. Björck Numerical Methods, Prentice-Hall, 1974 (p. 100)

[4] J.P. Fink; W.C. Rheinboldt 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] L. Machiels; Y. Maday; I.B. Oliveira; A.T. Patera; D.V. Rovas 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] Y. Maday; L. Machiels; A.T. Patera; D.V. Rovas Blackbox reduced-basis output bound methods for shape optimization, Proceedings 12th International Domain Decomposition Conference, Japan, 2000

[7] Y. Maday; A.T. Patera; D.V. Rovas A blackbox reduced-basis output bound method for noncoercive linear problems (D. Cioranescu; J.-L. Lions, 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] A.K. Noor; J.M. Peters Reduced basis technique for nonlinear analysis of structures, AIAA J., Volume 18 (1980) no. 4, pp. 455-462

[10] J.S. Peterson The reduced basis method for incompressible viscous flow calculations, SIAM J. Sci. Statist. Comput., Volume 10 (1989) no. 4, pp. 777-786

[11] C. Prud'homme; D.V. Rovas; K. Veroy; L. Machiels; Y. Maday; A.T. Patera; G. Turinici 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] W.C. Rheinboldt 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] W.G. Strang; G.J. Fix 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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations

Maxime Barrault; Yvon Maday; Ngoc Cuong Nguyen; ...

C. R. Math (2004)

Reduced-basis approximation of the viscous Burgers equation: rigorous a posteriori error bounds

Karen Veroy; Christophe Prud'homme; Anthony T. Patera

C. R. Math (2003)

A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations

Yanlai Chen; Jan S. Hesthaven; Yvon Maday; ...

C. R. Math (2008)