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Parallel Solution of Linear Systems

Part of: Numerical linear algebra Computer aspects of numerical algorithms

Published online by Cambridge University Press:  20 July 2016

Sidi-Mahmoud Kaber ,
Amine Loumi  and
Philippe Parnaudeau
Sidi-Mahmoud Kaber*
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France
Amine Loumi
Affiliation:
Laboratoire de Mathématiques et Applications, Université Hassiba Benbouali, Chlef, Algeria and Laboratoire d’Équations aux Dérivées Partielles non linéaires et d’Histoire des Mathématiques, ENS-Kouba, Algers, Algeria
Philippe Parnaudeau
Affiliation:
Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France
*
*Corresponding author. Email address:kaber@ljll.math.upmc.fr (S.-M. Kaber)
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Abstract

Computational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.

Keywords

Parallel computationnumerical linear algebra

MSC classification

Secondary: 65Y05: Parallel computation 65F05: Direct methods for linear systems and matrix inversion 65F10: Iterative methods for linear systems
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 3 , August 2016 , pp. 278 - 289
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Nievergelt, J., Parallel methods for integrating ordinary differential equations, Commun. ACMachinery 7, 731–733 (1964).Google Scholar
[2]Miranker, W.L. and Liniger, W., Parallel methods for the numerical integration of ordinary differential equations, Math. Comput. 21, 303–320 (1967).CrossRefGoogle Scholar
[3]Lions, J.-L., Maday, Y. and Turinici, G., Résolution d’EDP par un schéma en temps “pararéel”, C. R. Acad. Sci. Paris Sér. I Math. 7, 332, 661–668 (2001).CrossRefGoogle Scholar
[4] Message Passing Interface Forum, http://www.mpi-forum.org.Google Scholar
[5]Gander, M.J., Halpern, L. and Tran, T.T.B., A direct solver for time parallelization, Domain Decomposition Methods in Science and Engineering XXII, Lecture Notes in Comp. Sci. and Eng., Springer (2015).Google Scholar
[6]Maday, Y. and Rønquist, E.M., Parallelization in time through tensor-product space-time solvers, C. R. Math. Acad. Sci. Paris 1-2, 346, 113–118, (2008).CrossRefGoogle Scholar
[7]Gander, M.J., 50 Years of Time Parallel Time Integration, in Multiple Shooting and Time Domain Decomposition, pp. 69–114, Carraro, T., Geiger, M., Körkel, S. and Rannacher, R. (Eds.), Springer (2015).CrossRefGoogle Scholar