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Numerical Algorithms

, Volume 61, Issue 2, pp 223–241 | Cite as

Banded target matrices and recursive FSAI for parallel preconditioning

  • Luca Bergamaschi
  • Ángeles Martínez
Original Paper

Abstract

In this paper we propose a parallel preconditioner for the CG solver based on successive applications of the FSAI preconditioner. We first compute an FSAI factor G out for coefficient matrix A, and then another FSAI preconditioner is computed for either the preconditioned matrix \(S = G_{\rm out} A G_{\rm out}^T\) or a sparse approximation of S. This process can be iterated to obtain a sequence of triangular factors whose product forms the final preconditioner. Numerical results onto large SPD matrices arising from geomechanical models account for the efficiency of the proposed preconditioner which provides a reduction of the iteration number and of the CPU time of the iterative phase with respect to the original FSAI preconditioner. The proposed preconditioner reveals particularly efficient for accelerating an iterative procedure to find the smallest eigenvalues of SPD matrices, where the increased setup cost of the RFSAI preconditioner does not affect the overall performance, being a small percentage of the total CPU time.

Keywords

Parallel computing Iterative methods Preconditioning Approximate inverse Scalability 

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References

  1. 1.
    Benzi, M., Cullum, J.K., Tůma, M.: Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput. 22, 1318–1332 (2000)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Benzi, M., Marín, J., Tůma, M.: A two-level parallel preconditioner based on sparse approximate inverses. In: Kincaid, D.R., Elster, A.C. (eds.) Iterative Methods in Scientific Computation IV. IMACS Series in Computational and Applied Mathematics, vol. 5, pp. 167–178. International Assoc. for Mathematics and Computers in Simulation, New Brunswick, New Jersey, USA (1999)Google Scholar
  3. 3.
    Benzi, M., Meyer, C.D., Tůma, M.: A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17, 1135–1149 (1996)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Benzi, M., Tůma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bergamaschi, L., Gambolati, G., Pini, G.: Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblem. Numer. Linear Algebr. Appl. 4, 69–84 (1997)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bergamaschi, L., Martínez, A.: Parallel acceleration of Krylov solvers by factorized approximate inverse preconditioners. In Daydè, M., et al. (eds.) VECPAR 2004. Lecture Notes in Computer Sciences, vol. 3402, pp. 623–636. Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Bergamaschi, L., Martínez, A.: Parallel inexact constraint preconditioners for saddle point problems. In: Jeannot, R.N.E., Roman, J. (eds.) Euro-Par 2011, Bordeaux (France). Lecture Notes in Computer Sciences, vol. 6853, part II, pp. 78–89. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Bergamaschi, L., Martínez, A., Pini, G.: An efficient parallel MLPG method for poroelastic models. Comput. Model. Eng. Sci. 49 191–216 (2009)MATHGoogle Scholar
  9. 9.
    Bergamaschi, L., Martínez, A., Pini, G.: Parallel Rayleigh quotient optimization with FSAI-based preconditioning. J. Appl. Math. 2012, article ID 872901, 14 pp. (2012)CrossRefGoogle Scholar
  10. 10.
    Bergamaschi, L., Putti, M.: Numerical comparison of iterative eigensolvers for large sparse symmetric matrices. Comput. Methods Appl. Mech. Eng. 191, 5233–5247 (2002)MATHCrossRefGoogle Scholar
  11. 11.
    Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000, electronic)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Holland, R.M., Wathen, A.J., Shaw, G.J.: Sparse approximate inverses and target matrices. SIAM J. Sci. Comput. 26, 1000–1011 (2005, electronic)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Huckle, T.: Factorized sparse approximate inverses for preconditioning. J. Supercomput. 25, 109–117 (2003)MATHCrossRefGoogle Scholar
  15. 15.
    Huckle, T., Kallischko, A., Roy, A., Sedlacek, M., Weinzierl, T.: An efficient parallel implementation of the MSPAI preconditioner. Parallel Comput. 36, 273–284 (2010)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Janna, C., Ferronato, M.: Adaptive pattern research for block FSAI preconditioning. SIAM J. Sci. Comput. 33, 3357–3380 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Janna, C., Ferronato, M., Gambolati, G.: A block FSAI-ILU parallel preconditioner for symmetric positive definite linear systems. SIAM J. Sci. Comput. 32, 2468–2484 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kaporin, I.E.: New convergence results and preconditioning strategies for the conjugate gradient method. Numer. Linear Algebra Appl. 1, 179–210 (1994)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Knyazev, A.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kolotilina, L.Yu., Nikhishin, A.A., Yeremin, A.Yu.: Factorized sparse approximate inverse preconditionings. IV: simple approaches to rising efficiency. Numer. Linear Algebr. Appl. 6, 515–531 (1999)MATHCrossRefGoogle Scholar
  21. 21.
    Kolotilina, L.Yu., Yeremin, A.Yu.: Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. 14, 45–58 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lazarov, B.S., Sigmund, O.: Factorized parallel preconditioner for the saddle point problem. Int. J. Numer. Methods Biomed. Eng. 27, 1398–1410 (2011)MathSciNetMATHGoogle Scholar
  23. 23.
    Martínez, A., Bergamaschi, L., Caliari, M., Vianello, M.: A massively parallel exponential integrator for advection-diffusion models. J. Comput. Appl. Math. 231, 82–91 (2009)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Wang, K., Zhang, J.: MSP: a class of parallel multistep successive sparse approximate inverse preconditioning strategies. SIAM J. Sci. Comput. 24, 1141–1156 (2003, electronic)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Dept. of Civil Environmental and Architectural EngineeringUniversity of PaduaPadovaItaly
  2. 2.Department of MathematicsUniversity of PadovaPadovaItaly

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