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A Trade-off Analysis of the Parallel Hybrid SPIKE Preconditioner in a Unique Multi-core Computer

  • Leonardo Muniz de Lima
  • Lucia CatabrigaEmail author
  • Maria Cristina Rangel
  • Maria Claudia Silva Boeres
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10405)

Abstract

In this paper we apply the parallel hybrid SPIKE algorithm as a preconditioner for a nonstationary iterative method to solve large sparse linear systems. In order to obtain a good preconditioner, we employ several strategies solving combinatorial problems such as matching, reordering, partitioning, and quadratic knapsack. Our SPIKE implementation combines MPI and OpenMP paradigms in a unique multi-core computer. The computational experiments show the influence of each strategy evaluating the number of iterations and CPU time of the iterative solver in a set of large systems from miscellaneous application areas. The experiments suggest that the SPIKE preconditioner is very advantageous when a suitable set of parameters is chosen. The choice of the number of MPI ranks and OpenMP threads is not an easy task, because the SPIKE algorithm can increase the number of iterations when the number of MPI ranks grows. Moreover, the increase in the number of threads does not ensure a better performance.

Keywords

Parallel hybrid SPIKE preconditioner Combinatorial strategies Nonstationary iterative method 

Notes

Acknowledgments

This work has been supported in part by CNPq, CAPES, and FAPES.

References

  1. 1.
    Barnard, S.T., Pothen, A., Simon, H.: A spectral algorithm for envelope reduction of sparse matrices. Numer. Linear Algebra Appl. 2(4), 317–334 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182(2), 418–477 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th National Conference, pp. 157–172. ACM, New York (1969). http://doi.acm.org/10.1145/800195.805928
  4. 4.
    Davis, T.A., Hu, Y.: The university of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), 1:1–1:25 (2011). http://www.cise.ufl.edu/research/sparse/matrices
  5. 5.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  6. 6.
    Duff, I.S., Koster, J.: On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl. 22(4), 973–996 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems, vol. 49. NBS (1952)Google Scholar
  8. 8.
    Kouris, A., Sobczyk, A., Venetis, I., Gallopoulos, E., Sameh, A.: Revisiting the SPIKE-based framework for GPU banded solvers: a givens rotation approach for tridiagonal systems in CUDA (2014)Google Scholar
  9. 9.
    Li, A., Deshmukh, O., Serban, R., Negrut, D.: A comparison of the performance of SPIKE: GPU and intel’s math kernel library (MKL) for solving dense banded linear systems (2014)Google Scholar
  10. 10.
    Liu, W.H., Sherman, A.H.: Comparative analysis of the Cuthill-mckee and the reverse Cuthill-mckee ordering algorithms for sparse matrices. SIAM J. Numer. Anal. 13(2), 198–213 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Macintosh, H.J., Warne, D.J., Kelson, N.A., Banks, J.E., Farrell, T.W.: Implementation of parallel tridiagonal solvers for a heterogeneous computing environment. ANZIAM J. 56, 446–462 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Manguoglu, M., Koyutürk, M., Sameh, A.H., Grama, A.: Weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers. SIAM J. Sci. Comput. 32(3), 1201–1216 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Manne, F., Sørevik, T.: Optimal partitioning of sequences. J. Algorithms 19(2), 235–249 (1995). doi: 10.1006/jagm.1995.1035 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Polizzi, E., Sameh, A.: SPIKE: a parallel environment for solving banded linear systems. Comput. Fluids 36(1), 113–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Polizzi, E., Sameh, A.H.: A parallel hybrid banded system solver: the SPIKE algorithm. Parallel Comput. 32(2), 177–194 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. Siam, Philadelphia (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Sathe, M., Schenk, O., Uçar, B., Sameh, A.: A scalable hybrid linear solver based on combinatorial algorithms. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing, pp. 95–128. Taylor & Francis, Chapman-Hall/CRC Computational Science, Boca Raton (2012)CrossRefGoogle Scholar
  19. 19.
    Schenk, O., Gärtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numerical Anal. 23(1), 158–179 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Situ, Y., Martha, C.S., Louis, M.E., Li, Z., Sameh, A.H., Blaisdell, G.A., Lyrintzis, A.S.: Petascale large eddy simulation of jet engine noise based on the truncated SPIKE algorithm. Parallel Comput. 40(9), 496–511 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wathen, A.: Preconditioning. Acta Numerica 24, 329–376 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Leonardo Muniz de Lima
    • 1
  • Lucia Catabriga
    • 1
    • 2
    Email author
  • Maria Cristina Rangel
    • 2
  • Maria Claudia Silva Boeres
    • 2
  1. 1.High Performance Computing LabFederal University of Espírito SantoVitóriaBrazil
  2. 2.Optimization LabFederal University of Espírito SantoVitóriaBrazil

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