Partitioning and mapping for parallel nested dissection on distributed memory architectures

  • Pierre Charrier
  • Jean Roman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 634)


In this paper, we consider the parallel implementation of a block Cholesky factorization based on a nested dissection ordering for unstructured problems. We focus on loosely coupled networks of many processors with local memory and message passing mechanism. More precisely, we study a parallel block solver associated with refined partitions from the separator partition; the aim is to find the partition corresponding to the correct granularity leading to a high quality mapping (in terms of load balancing for the processors, of average length for the routing paths, and of average edge contention on the network). Then, we propose a refinement algorithm leading to this good granularity, and we provide some numerical measurements using the mapping tool included in the ADAM environment.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. E. Bal, J. G. Steiner, and A. S. Tanembaum. Programming languages for distributed computing systems. ACM Computing Surveys, 21(3):261–322, 1989.Google Scholar
  2. 2.
    M. C. Counilh and J. Roman. Expression for massively parallel algorithms — description and illustrative example. Parallel Computing, 16:239–251, 1990.Google Scholar
  3. 3.
    F. André and J. L. Pazat. Le placement de tâches sur des architectures parallèles. TSI, 7(4):385–401, 1988.Google Scholar
  4. 4.
    T. Muntean and E. G. Talbi. Méthodes de placement statique des processus sur architectures parallèles. TSI, 10:355–374, 1991.Google Scholar
  5. 5.
    S. H. Bokhari. Communication overhead on the INTEL iPSC-860 hypercube. Interim Report 10, ICASE, May 1990.Google Scholar
  6. 6.
    S. H. Bokhari. Complete exchange on the INTEL iPSC-860. Technical Report 91–4, ICASE, January 1991.Google Scholar
  7. 7.
    J. A. George and J. W. H. Liu. Computer solution of large sparse positive definite systems. Prentice Hall, 1981.Google Scholar
  8. 8.
    P. Charrier and J. Roman. Etude de la séparation et de l'élimination sur une famille de graphes quotients déduite d'une méthode de dissections emboîtée. RAIRO Informatique théorique et Application, 22(2):245–265, 1988.Google Scholar
  9. 9.
    P. Charrier and J. Roman. Study of the parallelism induced by a nested dissection method and of its implementation on a message passing multiprocessor computer. Rapport interne I-8722, Université Bordeaux 1, July 1987.Google Scholar
  10. 10.
    P. Charrier and J. Roman. Parallel implementation of block cholesky method in the programming environment of the distributed memory multiprocessor CHEOPS. In Proceedings of the fifth International Symposium on Numerical Methods in Engineering, Vol. 2. Springer-Verlag, 1989.Google Scholar
  11. 11.
    J. R. Gilbert and R. E. Tarjan. The analysis of a nested dissection algorithm. Numerische Mathematik, 50:377–404, 1987.Google Scholar
  12. 12.
    J. Roman. Calcul de complexité relatifs à une méthode de dissection emboîtée. Numerische Mathematik, 47:175–190, 1985.Google Scholar
  13. 13.
    C. Ashcraft and R. Grimes. The influence of relaxed supernode partitions on the multifrontal method. ACM Trans. on Math. Software, 15:291–309, 1989.Google Scholar
  14. 14.
    C. Ashcraft, R. Grimes, J. Lewis, B. Peyton, and H. Simon. Progress in sparse matrix methods for large linear systems on vector supercomputers. Intern. J. Supercomp. Appl., 1(4):10–29, 1987.Google Scholar
  15. 15.
    J. W. H. Liu, E. Ng, and B. W. Peyton. On finding supernodes for sparse matrix computation. Technical report ORNL/TM-11563, Oak Ridge National Laboratory, 1990.Google Scholar
  16. 16.
    E. Ng. Supernodal symbolic Cholesky factorization on a local-memory multiprocessor. Technical report ORNL/TM-11836, Oak Ridge National Laboratory, 1991.Google Scholar
  17. 17.
    P. Charrier, S. Chaumette, M.C. Counilh, J. Roman, and B. Vauquelin. A programming environment for distributed memory computers — application to scientific computing. In Proceedings of the 13th world congress on computation and applied mathematics — IMACS'91, Vol. 3, 1991.Google Scholar
  18. 18.
    P. Charrier and J. Roman. Partitioning and mapping for parallel nested dissection on distributed memory architectures. Rapport interne I–9212, Université Bordeaux 1, March 1992.Google Scholar
  19. 19.
    P. Charrier and J. Roman. Algorithmique et calculs de complexité pour un solveur de type dissections emboîtée. Numerische Mathematik, 55:463–476, 1989.Google Scholar
  20. 20.
    J. W. H. Liu. The role of elimination trees in sparse factorization. Siam J. Matrix Anal. Appl., 11:134–172, 1990.CrossRefGoogle Scholar
  21. 21.
    R. Schreiber. A new implementation of sparse Gaussian elimination. ACM Trans. Math. Software, 8:256–276, 1982.Google Scholar
  22. 22.
    E. Zmijewski and J. R. Gilbert. A parallel algorithm for sparse symbolic Cholesky factorization on a multiprocessor. Parallel Computing, 7:199–210, 1988.Google Scholar
  23. 23.
    R. J. Lipton and R. E. Tarjan. A separator theorem for planar graphs. Siam J. Appl. Math., 36:177–189, 1979.CrossRefGoogle Scholar
  24. 24.
    G. L. Miller and W. Thurston. Separators in two and three dimensions. In Proceedings of the 22th Annual Symposium on Theory of Computing. ACM, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Pierre Charrier
    • 1
  • Jean Roman
    • 2
  1. 1.CeReMaBUniversité Bordeaux ITalenceFrance
  2. 2.LaBRIENSERB et Université Bordeaux ITalenceFrance

Personalised recommendations