Parallel Solutions of Linear Systems

Amine Loumi (1) , Sidi-Mahmoud Kaber (2) , Philippe Parnaudeau (2)

1 ENS Kouba-Alger - École normale supérieure - Kouba-Alger
2 LJLL - Laboratoire Jacques-Louis Lions

Optimizing solvers for linear systems is a major challenge in scientific computation. Indeed computational scientifics would like to have the more accurate result in a shorter time. To achieve this, a well-knowledge of the computer science is necessary, not only in term of programmation paradigms, but also in order to follow the evolution of hardware. With the advent of new computers, we have to rewrite the numerical algorithms or create new ones to fully use the parallel architecture of the computers. Parallel space discretisations of Partial Differential Equations (PDE), like Domain Decomposition Methods (DDM) are efficient and well documented methods. At a first glance, parallelisation seems to be inconsistent with the time evolution which is inherently sequential. In fact, parallelisation is not limited to the space direction. The time direction is also a candidate for parallelisation. In this paper we present a new and simple method for time parallelization. This solver is based on partial fraction decomposition of the inverse of some special matrices. We discuss its application for PDE as well as its limitations. Numerical experiments are given.

Domaines: Calcul parallèle, distribué et partagé [cs.DC]; Analyse numérique [math.NA]

Amine Loumi, Sidi-Mahmoud Kaber, Philippe Parnaudeau. Parallel Solutions of Linear Systems. East Asian journal on Applied Mathematics, 2016. ⟨hal-01418468⟩