PSPIKE: A Parallel Hybrid Sparse Linear System Solver

  • Murat Manguoglu
  • Ahmed H. Sameh
  • Olaf Schenk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5704)


The availability of large-scale computing platforms comprised of tens of thousands of multicore processors motivates the need for the next generation of highly scalable sparse linear system solvers. These solvers must optimize parallel performance, processor (serial) performance, as well as memory requirements, while being robust across broad classes of applications and systems. In this paper, we present a new parallel solver that combines the desirable characteristics of direct methods (robustness) and effective iterative solvers (low computational cost), while alleviating their drawbacks (memory requirements, lack of robustness). Our proposed hybrid solver is based on the general sparse solver PARDISO, and the “Spike” family of hybrid solvers. The resulting algorithm, called PSPIKE, is as robust as direct solvers, more reliable than classical preconditioned Krylov subspace methods, and much more scalable than direct sparse solvers. We support our performance and parallel scalability claims using detailed experimental studies and comparison with direct solvers, as well as classical preconditioned Krylov methods.


Hybrid Solvers Direct Solvers Krylov Subspace Methods Sparse Linear Systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen, S.C., Kuck, D.J., Sameh, A.H.: Practical parallel band triangular system solvers. ACM Transactions on Mathematical Software 4(3), 270–277 (1978)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Lawrie, D.H., Sameh, A.H.: The computation and communication complexity of a parallel banded system solver. ACM Trans. Math. Softw. 10(2), 185–195 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berry, M.W., Sameh, A.: Multiprocessor schemes for solving block tridiagonal linear systems. The International Journal of Supercomputer Applications 1(3), 37–57 (1988)CrossRefGoogle Scholar
  4. 4.
    Dongarra, J.J., Sameh, A.H.: On some parallel banded system solvers. Parallel Computing 1(3), 223–235 (1984)CrossRefMATHGoogle Scholar
  5. 5.
    Polizzi, E., Sameh, A.H.: A parallel hybrid banded system solver: the spike algorithm. Parallel Comput. 32(2), 177–194 (2006)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Polizzi, E., Sameh, A.H.: Spike: A parallel environment for solving banded linear systems. Computers & Fluids 36(1), 113–120 (2007)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Sameh, A.H., Sarin, V.: Hybrid parallel linear system solvers. Inter. J. of Comp. Fluid Dynamics 12, 213–223 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with pardiso. Future Generation Computer Systems 20(3), 475–487 (2004) Selected numerical algorithmsCrossRefMATHGoogle Scholar
  9. 9.
    Schenk, O., Gärtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. Electronic Transactions on Numerical Analysis 23, 158–179 (2006)MathSciNetMATHGoogle Scholar
  10. 10.
    van der Vorst, H.A.: Bi-cgstab: a fast and smoothly converging variant of bi-cg for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mikkelsen, C.C.K., Manguoglu, M.: Analysis of the truncated spike algorithm. SIAM Journal on Matrix Analysis and Applications 30(4), 1500–1519 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Duff, I.S.: Algorithm 575: Permutations for a zero-free diagonal [f1]. ACM Trans. Math. Softw. 7(3), 387–390 (1981)CrossRefGoogle Scholar
  13. 13.
    Duff, I.S., Koster, J.: The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. SIAM Journal on Matrix Analysis and Applications 20(4), 889–901 (1999)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    HSL: A collection of Fortran codes for large-scale scientific computation (2004),
  15. 15.
    Hu, Y., Scott, J.: HSL_MC73: a fast multilevel Fiedler and profile reduction code. Technical Report RAL-TR-2003-036 (2003)Google Scholar
  16. 16.
    Davis, T.A.: University of Florida sparse matrix collection. NA Digest (1997)Google Scholar
  17. 17.
    Benzi, M., Haws, J.C., Tuma, M.: Preconditioning highly indefinite and nonsymmetric matrices. SIAM J. Sci. Comput. 22(4), 1333–1353 (2000)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bollhöfer, M., Saad, Y., Schenk, O.: ILUPACK Volume 2.1—Preconditioning Software Package (May 2006),
  20. 20.
    Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Mathematical Programming 106(1), 25–57 (2006)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Amestoy, P.R., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.Y., Koster, J.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Amestoy, P.R., Duff, I.S.: Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184, 501–520 (2000)CrossRefMATHGoogle Scholar
  24. 24.
    Schenk, O., Manguoglu, M., Sameh, A., Christian, M., Sathe, M.: Parallel scalable PDE-constrained optimization: antenna identification in hyperthermia cancer treatment planning. Computer Science - Research and Development 23(3), 177–183 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Murat Manguoglu
    • 1
  • Ahmed H. Sameh
    • 1
  • Olaf Schenk
    • 2
  1. 1.Department of Computer SciencePurdue UniversityWest Lafayette
  2. 2.Computer Science DepartmentUniversity of BaselBasel

Personalised recommendations