A Direct Solver for Time Parallelization

Un solveur direct pour la parallélisation en temps

Martin J. Gander (1) , Laurence Halpern (2) , Juliet Ryan (3) , Thuy Thi Bich Tran (2)

1 UNIGE - Université de Genève = University of Geneva
2 LAGA - Laboratoire Analyse, Géométrie et Applications
3 ONERA - The French Aerospace Lab [Palaiseau]

With the advent of very large scale parallel computer, having millions of processing cores, it has become important to also use the time direction for parallelization. Among the successful methods doing this are the parareal algorithm, the paraexp algorithm, PFASST and also waveform relaxation methods of Schwarz or Dirichlet-Neumann or Neumann type. We present here a mathematical analysis of a further method to parallelize in time, proposed by Maday and Ronquist in 2007. It is based on the diagonalization of the time stepping matrix. Like for many time domain parallelization methods, this seems at first not to be a very promising approach, since this matrix is essentially triangular, and for a fixed time step even a Jordan block, and thus not diagonalizable. If one however chooses different time steps, diagonalization is possible, and one has to trade of between the accuracy due to necessarily having different time steps, and numerical errors in the diagonalization process of these almost not diagonalizable matrices. We study this trade-off mathematically and propose an optimization strategy for the choice of the parameters, for a Backward Euler discretization of the heat equation in two dimensions.

Utiliser le temps pour la parallélisation dans des problèmes d'evolution est un thème de recherches en plein développement. La plupart des méthodes sont itératives , voir par exemple la méthode Pararéel de Maday et al ainsi qu'une variante nommée PFASST, et les méthodes de relaxation d'ondes basées sur des techniques de décomposition de domaines . Les solveurs directs parallèles en temps sont plus rares. Dans cet article, une analyse mathématique d'un produit tensoriel parallèle en temps est présentée.

Mots clés: PARALLÉLISATION EN TEMPS; SOLVEUR DIRECT

Domaines: Algorithme et structure de données [cs.DS]

Martin J. Gander, Laurence Halpern, Juliet Ryan, Thuy Thi Bich Tran. A Direct Solver for Time Parallelization. Domain Decomposition Methods in Science and Engineering XXII, 104, pp.491-499, 2016, ⟨10.1007/978-3-319-18827-0_50⟩. ⟨hal-01429894⟩