A Laplace transform certified reduced basis method; application to the heat equation and wave equation

D.B. Phuong Huynh; David J. Knezevic; Anthony T. Patera

doi:10.1016/j.crma.2011.02.003

Partial Differential Equations/Numerical Analysis

A Laplace transform certified reduced basis method; application to the heat equation and wave equation
[Une méthode de bases réduites « certifiée » utilisant la transformation de Laplace ; Application à lʼéquation de la chaleur et à lʼéquation des ondes]

Présenté par : Philippe G. Ciarlet

D.B. Phuong Huynh ¹ ; David J. Knezevic ¹ ; Anthony T. Patera ¹

¹ Massachusetts Institute of Technology, room 3-266, Cambridge, MA 02139, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 401-405.

Résumé
Abstract

On introduit une méthode de bases réduites « certifiée » pour lʼéquation de la chaleur et pour lʼéquation des ondes. Les outils sont les suivants : approximation en bases réduites « certifiée » de la transformée de Laplace, transformée de Laplace inverse pour lʼapproximation de la sortie en bases réduites pour la variable temps, estimations dʼerreurs rigoureuses, filtre en temps (de Butterworth) mettant en évidence la nécessité dʼune troncature « modale », décomposition en fonctions propres en bases réduites, intégrale de contour pour la décomposition « Offline–Online ». On donne des résultats numériques pour montrer lʼéfficacité et la précision de la méthode.

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accuracy and efficiency of the approach.

Reçu le : 2010-09-13
Accepté le : 2011-02-03
Publié le : 2011-03-08

DOI : 10.1016/j.crma.2011.02.003

Affiliations des auteurs :

D.B. Phuong Huynh ¹ ; David J. Knezevic ¹ ; Anthony T. Patera ¹

¹ Massachusetts Institute of Technology, room 3-266, Cambridge, MA 02139, USA

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     title = {A {Laplace} transform certified reduced basis method; application to the heat equation and wave equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {401--405},
     publisher = {Elsevier},
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     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.02.003},
     language = {en},
}

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D.B. Phuong Huynh; David J. Knezevic; Anthony T. Patera. A Laplace transform certified reduced basis method; application to the heat equation and wave equation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 7-8, pp. 401-405. doi : 10.1016/j.crma.2011.02.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.02.003/

Bibliographie
Cité par

[1] L. Daniel; O.C. Siong; L.S. Chay; K.H. Lee; J. White A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models, IEEE Trans. Computer-Aided Design Integr. Circ. Syst., Volume 23 (2004) no. 5, pp. 678-693

[2] B. Davies Integral Transforms and Their Applications, Springer, 2002

[3] B. Haasdonk; M. Ohlberger Reduced basis method for finite volume approximations of parametrized linear evolution equations, M2AN Math. Model. Numer. Anal., Volume 42 (2008), pp. 277-302

[4] 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, Ser. I, Volume 331 (2000), pp. 153-158

[5] H. Panzer; J. Mohring; R. Eid; B. Lohmann Parametric model order reduction by matrix interpolation, Automatisierungstechnik, Volume 58 (2010) no. 8, pp. 475-484

[6] M. Petyt Introduction to Finite Element Vibration Analysis, Cambridge University Press, 2010

[7] G. Pini; M. Putti Parallel finite element Laplace transform method for the non-equilibrium groundwater transport equation, Internat. J. Numer. Methods Engrg., Volume 40 (1997), pp. 2653-2664

[8] T.A. Porsching Estimation of the error in the reduced basis method solution of nonlinear equations, Math. Comp., Volume 45 (1985) no. 172, pp. 487-496

[9] G. Rozza; D.B.P. Huynh; 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., Volume 15 (2008) no. 3, pp. 229-275

[10] E.A. Sudicky The Laplace transform Galerkin technique: A time-continuous finite element theory and application to mass transport in groundwater, Water Resources Res., Volume 25 (1989), pp. 1833-1846

[11] A. Tan, Reduced basis methods for 2nd order wave equation: Application to one dimensional seismic problem, masters thesis, Singapore–MIT Alliance, National University of Singapore, 2006.

[12] K. Veroy, C. Prudʼhomme, D.V. Rovas, A.T. Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, AIAA paper 2003-3847, 2003.

[13] K. Willcox; J. Peraire Balance model reduction via the proper orthogonal decomposition, AIAA J., Volume 40 (2002) no. 11, pp. 2323-2330

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