Skip to main content
SpringerLink
Log in

Banded target matrices and recursive FSAI for parallel preconditioning

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benzi, M., Tůma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Bergamaschi, L., Martínez, A., Pini, G.: An efficient parallel MLPG method for poroelastic models. Comput. Model. Eng. Sci. 49 191–216 (2009)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  10. Bergamaschi, L., Putti, M.: Numerical comparison of iterative eigensolvers for large sparse symmetric matrices. Comput. Methods Appl. Mech. Eng. 191, 5233–5247 (2002)

    Article  MATH  Google Scholar 

  11. Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000, electronic)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huckle, T.: Factorized sparse approximate inverses for preconditioning. J. Supercomput. 25, 109–117 (2003)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Janna, C., Ferronato, M.: Adaptive pattern research for block FSAI preconditioning. SIAM J. Sci. Comput. 33, 3357–3380 (2012)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaporin, I.E.: New convergence results and preconditioning strategies for the conjugate gradient method. Numer. Linear Algebra Appl. 1, 179–210 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Knyazev, A.: Toward the optimal preconditioned eigensolver: locally optimal block preconditioned conjugate gradient method. SIAM J. Sci. Comput. 23, 517–541 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  21. Kolotilina, L.Yu., Yeremin, A.Yu.: Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. 14, 45–58 (1993)

    Article  MathSciNet  Google Scholar 

  22. Lazarov, B.S., Sigmund, O.: Factorized parallel preconditioner for the saddle point problem. Int. J. Numer. Methods Biomed. Eng. 27, 1398–1410 (2011)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Civil Environmental and Architectural Engineering, University of Padua, Torre Archimede via Trieste 63, 35121, Padova, Italy

    Luca Bergamaschi

  2. Department of Mathematics, University of Padova, via Trieste 63, 35121, Padova, Italy

    Ángeles Martínez

Authors
  1. Luca Bergamaschi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ángeles Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luca Bergamaschi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bergamaschi, L., Martínez, Á. Banded target matrices and recursive FSAI for parallel preconditioning. Numer Algor 61, 223–241 (2012). https://doi.org/10.1007/s11075-012-9605-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-012-9605-7

Keywords

Search

Navigation