Skip to main content
SpringerLink
Log in

The effect of graph partitioning techniques on parallel Block FSAI preconditioning: a computational study

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

Abstract

Adaptive Block FSAI (ABF) is a novel preconditioner which has proved efficient for the parallel solution of symmetric positive definite (SPD) linear systems and eigenproblems. A possible drawback stems from its reduced strong scalability, as the iteration count to converge for a given problem tends to grow with the number of processors used. The preliminary use of graph partitioning techniques can help improve the preconditioner quality and scalability. According to the specific theoretical properties of Block FSAI, different partitionings are selected and tested in a set of matrices arising from SPD engineering applications. The results show that using an appropriate graph partitioning technique with ABF may play an important role to increase the preconditioner efficiency and robustness, allowing for its effective use also in massively parallel simulations.

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. Cerdàn, J., Faraj, T., Malla, N., Marin, J., Mas, J.: Block approximate inverse preconditioners for sparse nonsymmetric linear systems. Electron. Trans. Numer. Anal. 37, 23–40 (2010)

    MATH  MathSciNet  Google Scholar 

  2. Bergamaschi, L., Martinez, A.: FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics. J. Comput. Appl. Math. 236, 308–318 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  4. Ferronato, M., Janna, C., Pini, G.: Parallel solution to ill-conditioned FE geomechanical problems. Int. J. Numer. Anal. Methods Geomech. 36, 422–437 (2012)

    Article  Google Scholar 

  5. Ferronato, M., Janna, C., Pini, G.: Shifted FSAI preconditioners for the efficient parallel solution of non-linear groundwater flow models. Int. J. Numer. Methods Eng. 89, 1707–1719 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Janna, C., Ferronato, M., Gambolati, G.: Parallel inexact constraint preconditioning for ill-conditioned consolidation problems. Comput. Geosci. 16, 661–675 (2012)

    Article  MathSciNet  Google Scholar 

  7. Ferronato, M.: Preconditioning for sparse linear systems at the dawn of the 21st century: history, current developments, and future perspectives. ISRN Appl. Math. (2012). doi:10.5402/2012/127647

  8. 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  MATH  MathSciNet  Google Scholar 

  9. Janna, C., Ferronato, M.: Adaptive pattern research for Block FSAI preconditioning. SIAM J. Sci. Comput. 33, 3357–3380 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ferronato, M., Janna, C., Pini, G.: Efficient parallel solution to large size sparse eigenproblems with Block FSAI preconditioning. Numer. Linear Algebra Appl. 19, 797–815 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Janna, C., Ferronato, M., Gambolati, G.: Enhanced Block FSAI preconditioning using domain decomposition. SIAM J. Sci. Comput. 35, S229–S249 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ferronato, M., Janna, C., Pini, G.: A generalized Block FSAI preconditioner for nonsymmetric linear systems. J. Comput. Appl. Math. 256, 230–241 (2014)

    Article  MathSciNet  Google Scholar 

  13. Kolotilina, L.Y., Yeremin, A.Y.: Factorized sparse approximate inverse preconditioning. I. Theory. SIAM J. Matrix Anal. Appl. 14, 45–58 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Benzi, M., Szyld, D.B., Duin, A.: Orderings for incomplete factorization preconditioning of nonsymmetric problems. SIAM J. Sci. Comput. 20, 1652–1670 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Benzi, M., Tůma, M.: Orderings for factorized sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1851–1868 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cullum, J.K., Johnson, K., Tůma, M.: Effects of problem decomposition (partitioning) on the rate of convergence of parallel numerical algorithms. Numer. Linear Algebra Appl. 10, 445–465 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th National Conference, pp. 157–172 (1969)

  18. George, A.: Computer Implementation of the Finite Element Method. Tech. Rep. STAN-CS-208. Department of Computer Science, Stanford University, Stanford (1971)

    Google Scholar 

  19. Georges, J.A., Liu, J.W.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliffs (1981)

    Google Scholar 

  20. Tinney, W.F., Walker, J.W.: Direct solutions of sparse network equations by optimally ordered triangular factorization. Proc. IEEE 55, 1801–1809 (1967)

    Article  Google Scholar 

  21. George, A., Liu, J.W.: The evolution of the minimum degree ordering algorithm. SIAM Rev. 31, 1–19 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Amestoy, P., Davis, T., Duff, I.: An approximate minimum degree ordering algorithm. SIAM J. Matrix Anal. Appl. 17, 886–905 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Barnard, S., Simon, H.D.: A fast multilevel implementation of recursive spectral bisection for partitioning unstructured problems. Concurr Pract Experience 6, 101–117 (1994)

    Article  Google Scholar 

  24. Hendrickson, B., Leland, R.: A multilevel algorithm for partitioning graphs. In: Proceedings of the ACM/IEEE Conference on Supercomputing (1995)

  25. Karypis, G., Kumar, V.: Multilevel k-way partitioning scheme for irregular graphs. J. Parallel Distrib. Comput. 48, 96–129 (1998)

    Article  Google Scholar 

  26. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359–392 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Chevalier, C., Pellegrini, F.: Improvement of the efficiency of genetic algorithms for scalable parallel graph partitioning in a multi-level framework. In: Proceedings of EuroPar 2006. Lecture Notes on Computer Science, no. 4128, pp. 243–252 (2006)

  28. Pellegrini, F.: A parallelisable multi-level banded diffusion scheme for computing balanced partitions with smooth boundaries. In: Proceedings of EuroPar 2007. Lecture Notes on Computer Science, no. 4641, pp. 191–200 (2007)

  29. Pellegrini, F.: SCOTCH, software package and libraries for sequential and parallel graph partitioning, static mapping, and sparse matrix block ordering, and sequential mesh and hypergraph partitioning (version 5.1.10). Electronic document available at http://www.labri.fr/perso/pelegrin/scotch (2010)

  30. Karypis, G., Kumar, V.: METIS - A software package for partitioning unstructured graphs, partitioning meshes and computing fill-reducing orderings of sparse matrices - Version 5.0. Electronic document available at http://glaros.dtc.umn.edu/gkhome/metis/metis/overview (2011)

  31. Holland, R.M., Wathen, A.J., Shaw, G.J.: Sparse approximate inverses and target matrices. SIAM J. Sci. Comput. 26, 1000–1011 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  32. Saad, Y.: ILUT: a dual threshold incomplete ILU factorization. Numer. Linear Algebra Appl. 1, 387–402 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  33. Li, N., Saad, Y., Chow, E.: Crout version of ILU for general sparse matrices. SIAM J. Sci. Comput. 25, 716–728 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  34. Janna, C., Comerlati, A., Gambolati, G.: A comparison of projective and direct solvers for finite elements in elastostatics. Adv. Eng. Softw. 40, 675–685 (2009)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Huckle, T.: Approximate sparsity patterns for the inverse of a matrix and preconditioning. Appl. Numer. Math. 30, 291–303 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  40. Anderson, G., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Kenney, A.M., Ostrouchov, S., Sorensen, D.: LAPACK User’s Guide. SIAM, Philadelphia (1992)

    Google Scholar 

  41. Bridson, R., Tang, W.P.: Ordering, anisotropy and factored sparse approximate inverses. SIAM J. Sci. Comput. 21, 867–882 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  43. Saad, Y.: SPARSKIT, a basic tool-kit for sparse matrix computations (Version 2). Electronic document available at http://www-users.cs.umn.edu/~saad/software/SPARSKIT (1988)

  44. Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Technol. J. 49, 291–307 (1970)

    Article  MATH  Google Scholar 

  45. Duff, I.S., Kaya, K.: Preconditioners based on strong subgraphs. Electron. Trans. Numer. Anal. 40, 225–249 (2013)

    MATH  MathSciNet  Google Scholar 

  46. Vecharinsky, E., Saad, Y., Sosonika, M.: Graph partitioning using matrix values for preconditioning symmetric positive definite systems. SIAM J. Sci. Comput. 36, A63–A87 (2014)

    Article  Google Scholar 

  47. Teatini, P., Ferronato, M., Gambolati, G., Baù, D., Putti, M.: Anthropogenic Venice uplift by seawater pumping into a heterogeneous aquifer system. Water Resour. Res. 46, W11547 (2010). doi:10.1029/2010WR009161

    Google Scholar 

  48. Ferronato, M., Gambolati, G., Janna, C., Teatini, P.: Geomechanical issues of anthropogenic CO2 sequestration in exploited gas fields. Energy Convers. Manag. 51, 1918–1928 (2010)

    Article  Google Scholar 

  49. Davis, T.A., Hu, Y.: The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38, 1–25 (2011)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department ICEA, University of Padova, via Trieste 63, 35121, Padova, Italy

    Carlo Janna & Massimiliano Ferronato

  2. Department Energy Resources Engineering, Stanford University, 367 Panama St., Stanford, CA, 94305-2220, USA

    Nicola Castelletto

Authors
  1. Carlo Janna
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicola Castelletto
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Massimiliano Ferronato
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimiliano Ferronato.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Janna, C., Castelletto, N. & Ferronato, M. The effect of graph partitioning techniques on parallel Block FSAI preconditioning: a computational study. Numer Algor 68, 813–836 (2015). https://doi.org/10.1007/s11075-014-9873-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-014-9873-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation