Comptes Rendus
Numerical analysis
Parametric analytical preconditioning and its applications to the reduced collocation methods
Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 661-666.

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.

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.

Received:
Accepted:
Published online:
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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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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[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

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[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

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