Comptes Rendus
no. 7-8
Numerical analysis
Parametric analytical preconditioning and its applications to the reduced collocation methods
[Préconditionnements analytiques en paramètres et applications aux méthodes de collocation réduites]

Présenté par : Olivier Pironneau

Yanlai Chen1 ; Sigal Gottlieb1 ; Yvon Maday2, 3
1 Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA
2 Sorbonne Universités, Université Paris-6 (UPMC), UMR 7598, Laboratoire Jacques-Louis-Lions & Institut universitaire de France, 75005 Paris, France
3 Division of Applied Mathematics, Brown University, 182 George St., Providence, RI 02912, USA
Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 661-666.

On étend dans cette note la méthode de collocation réduite récemment introduite dans [3] au cas non linéaire et on propose deux stratégies de préconditionnement dont une est indépendante des paramètres et facile a mettre en oeuvre et l'autre possède la propriété classique de décomposition affine en les paramètres qui permet une mise en oeuvre rapide en ligne/hors ligne. Ces stratégies améliorent la qualité de l'approximation et la vitesse de convergence.

In this paper, we extend the recently developed reduced collocation method [3] to the nonlinear case, and propose two analytical preconditioning strategies. One is parameter independent and easy to implement, the other one has the traditional affinity with respect to the parameters, which allows an efficient implementation through an offline–online decomposition. Overall, preconditioning improves the quality of the error estimation uniformly on the parameter domain, and speeds up the convergence of the reduced solution to the truth approximation.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2014.06.001
Yanlai Chen 1 ; Sigal Gottlieb 1 ; Yvon Maday 2, 3

1 Department of Mathematics, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, MA 02747, USA
2 Sorbonne Universités, Université Paris-6 (UPMC), UMR 7598, Laboratoire Jacques-Louis-Lions & Institut universitaire de France, 75005 Paris, France
3 Division of Applied Mathematics, Brown University, 182 George St., Providence, RI 02912, USA 
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     title = {Parametric analytical preconditioning and its applications to the reduced collocation methods},
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Yanlai Chen; Sigal Gottlieb; Yvon Maday. Parametric analytical preconditioning and its applications to the reduced collocation methods. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 661-666. doi : 10.1016/j.crma.2014.06.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.06.001/

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

[3] Y. Chen; S. Gottlieb Reduced collocation methods: reduced basis methods in the collocation framework, J. Sci. Comput., Volume 55 (2013) no. 3, pp. 718-737

[4] J.S. Hesthaven; S. Gottlieb; D. Gottlieb Spectral Methods for Time-Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, UK, 2007

[5] A.E. Løvgren; Y. Maday; E.M. Rønquist The reduced basis element method: offline–online decomposition in the nonconforming, nonaffine case, Spectral and High Order Methods for Partial Differential Equations, 2011, pp. 247-254

[6] A.K. Noor; J.M. Peters Reduced basis technique for nonlinear analysis of structures, AIAA J., Volume 18 (1980) no. 4, pp. 455-462

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

[8] C. Prud'homme; D. Rovas; K. Veroy; Y. Maday; A.T. Patera; G. Turinici Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. Fluids Eng., Volume 124 (2002) no. 1, pp. 70-80

[9] L.N. Trefethen Spectral Methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, 2000

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