The Journal of Supercomputing

, Volume 71, Issue 4, pp 1345–1356 | Cite as

A scalable multisplitting algorithm to solve large sparse linear systems

Article

Abstract

In this paper, we revisit the Krylov multisplitting algorithm presented in Huang and O’Leary (Linear Algebra Appl 194:9–29, 1993) which uses a sequential method to minimize the Krylov iterations computed by a multisplitting algorithm. Our new algorithm is based on a parallel multisplitting algorithm with few blocks of large size using a parallel GMRES method inside each block and on a parallel Krylov minimization to improve the convergence. Some large-scale experiments with a 3D Poisson problem are presented with up to 8,192 cores. They show the obtained improvements compared to a classical GMRES both in terms of number of iterations and in terms of execution times.

Keywords

Large sparse linear systems Multisplitting algorithm  3D Poisson problem 

References

  1. 1.
    Bahi JM (2000) Asynchronous iterative algorithms for nonexpansive linear systems. J Parallel Distrib Comput 60(1):92–112CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bahi JM, Contassot-Vivier S, Couturier R (2008) Parallel iterative algorithms: from sequential to grid computing, Chapman & Hall/CRC numerical analysis and scientific computing. Chapman & Hall/CRC, Boca Roton. ISBN 9781584888086Google Scholar
  3. 3.
    Bai Z-Z, Migallón V, Penadés J, Szyld DB (1999) Block and asynchronous two-stage methods for mildly nonlinear systems. Numer Math 82(1):1–20CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Brown N, Bull JM, Bethune I (2013) Solving large sparse linear systems using asynchronous multisplitting. Technical report, PRACE White paper number WP84Google Scholar
  5. 5.
    Bru R, Migallón V, Penadés J, Szyld DB (1995) Parallel, synchronous and asynchronous two-stage multisplitting methods. Electron Trans Numer Anal 3:24–38MATHMathSciNetGoogle Scholar
  6. 6.
    Couturier R, Denis C, Jézéquel F (2008) Gremlins: a large sparse linear solver for grid environment. Parallel Comput 34(6):380–391CrossRefMathSciNetGoogle Scholar
  7. 7.
    Frommer A, Szyld DB (1992) H-splittings and two-stage iterative methods. Numer Math 63(1):345–356CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    HECToR: UK National Supercomputing Service. http://www.hector.ac.uk
  9. 9.
    Huang C-M, O’Leary DP (1993) A krylov multisplitting algorithm for solving linear systems of equations. Linear Algebra Appl 194:9–29CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Li C (2001) An adaptive CGNR algorithm for solving large linear systems. Ann Oper Res 103(1–4):329–338CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    O’Leary DP, White RE (1985) Multi-splittings of matrices and parallel solution of linear systems. SIAM J Algebra Discrete Methods 6(4):630–640CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Saad Y (1996) Iterative methods for sparse linear systems. PWS Publishing, New YorkMATHGoogle Scholar
  13. 13.
    Saad Y, Schultz MH (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Ziane Khodja L, Couturier R, Giersch A, Bahi J (2014) Parallel sparse linear solver with GMRES method using minimization techniques of communications for GPU clusters. J Supercomput 69(1):200–224CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Femto-ST InstituteUniversity of Franche ComteBesançonFrance
  2. 2.Inria Bordeaux Sud-OuestTalenceFrance

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