no. 1-2
Numerical Analysis
Parallelization in time through tensor-product space–time solvers
[Algorithme de parallélisation en temps basé sur des techniques de produit tensoriel espace–temps]

Présenté par : Olivier Pironneau

Maday, Yvon1, 2 ; Rønquist, Einar M.3
1 Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire J.-L. Lions, 75252 Paris cedex 05, France
2 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
3 Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 113-118.

Dans cette Note on généralise à la simulation de phénomènes instationnaires l'algorithme de produit tensoriel. On utilise la tensorisation naturelle du domaine espace–temps pour proposer, après discrétisation un ensemble de problèmes indépendants, chacun d'eux ayant la complexité d'un simple problème stationnaire. Ceci permet une mise en œuvre parallèle déjà interessante sur des petites architectures mais elle peut être également combinée avec des techniques classiques de décomposition de domaine pour utiliser au mieux des architectures avec un nombre de processeurs plus important. Des premiers résultats sont présentés sur un problème de la chaleur instationnaire monodimensionnel.

In this Note, we extend the fast tensor-product algorithm for the simulation of time-dependent partial differential equations. We use the natural tensorization of the space–time domain to propose, after discretization, a set of independent problems, each one with the complexity of a single steady problem. This allows for an efficient parallel implementation that is already interesting on small architectures, but that can also be combined with standard domain-decomposition-based algorithms providing a further direction of parallelism on large computer platforms. Preliminary numerical simulations are presented for a one-dimensional unsteady heat equation.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.09.012
Maday, Yvon 1, 2 ; Rønquist, Einar M. 3

1 Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire J.-L. Lions, 75252 Paris cedex 05, France
2 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA
3 Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 
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Maday, Yvon; Rønquist, Einar M. Parallelization in time through tensor-product space–time solvers. Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 113-118. doi : 10.1016/j.crma.2007.09.012. http://www.numdam.org/articles/10.1016/j.crma.2007.09.012/

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[2] Hackbusch, W. Parabolic multi-grid methods (Glowinski, R.; Lions, J.-L., eds.), Computing Methods in Applied Sciences and Engineering, vol. IV, North-Holland, 1984, pp. 189-197

[3] Lions, J.-L.; Maday, Y.; Turinici, G. A parareal in time discretization of pde's, C. R. Acad. Sci. Paris, Ser. I, Volume 332 (2001), pp. 661-668

[4] Maday, Y.; Rønquist, E.M. Fast tensor-product solvers: Part II: Spectral discretization in space and time http://www.ann.jussieu.fr/publications/2007/R07038.html (submitted for publication, 2007 and)

[5] Nievergelt, J. Parallel methods for integration ordinary differential equations, Comm. ACM, Volume 7 (1964), pp. 731-733

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