A Static Parallel Multifrontal Solver for Finite Element Meshes

  • Alberto Bertoldo
  • Mauro Bianco
  • Geppino Pucci
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4330)


We present a static parallel implementation of the multifrontal method to solve unsymmetric sparse linear systems on distributed-memory architectures. We target Finite Element (FE) applications where numerical pivoting can be avoided, since an implicit minimum-degree ordering based on the FE mesh topology suffices to achieve numerical stability. Our strategy is static in the sense that work distribution and communication patterns are determined in a preprocessing phase preceding the actual numerical computation. To balance the load among the processors, we devise a simple model-driven partitioning strategy to precompute a high-quality balancing for a large family of structured meshes. The resulting approach is proved to be considerably more efficient than the strategies implemented by MUMPS and SuperLU_DIST, two state-of-the-art parallel multifrontal solvers.


Communication Pattern Finite Element Mesh Assembly Phase Sparsity Pattern Sparse Linear System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amestoy, P.R., Duff, I.S., Koster, J., L’Excellent, J.-Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM Journal of Matrix Analysis and Applications 23(1), 15–41 (2001)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Amestoy, P.R., Duff, I.S., L’Excellent, J.-Y., Li, X.S.: Analysis and comparison of two general sparse solvers for distributed memory computers. ACM Trans. Math. Softw. 27(4), 388–421 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Amestoy, P.R., Guermouche, A., L’Excellent, J.-Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Technical Report RR-5404, INRIA (2004)Google Scholar
  4. 4.
    Bertoldo, A., Bianco, M., Pucci, G.: A fast multifrontal solver for non-linear multi-physics problems. In: International Conference on Computational Science, pp. 614–617 (2004)Google Scholar
  5. 5.
    Bianco, M., Bilardi, G., Pesavento, F., Pucci, G., Schrefler, B.A.: An accurate and efficient frontal solver for fully-coupled hygro-thermo-mechanical problems. In: International Conference on Computational Science, vol. 1, pp. 733–742 (2002)Google Scholar
  6. 6.
    Bianco, M., Bilardi, G., Pesavento, F., Pucci, G., Schrefler, B.A.: A frontal solver tuned for fully-coupled non-linear hygro-thermo-mechanical problems. International Journal for Numerical Methods in Engineering 57(13), 1801–1818 (2003)MATHCrossRefGoogle Scholar
  7. 7.
    Cormen, T.H., Stein, C., Rivest, R.L., Leiserson, C.E.: Introduction to Algorithms. McGraw-Hill Higher Education (2001)Google Scholar
  8. 8.
    Js, W.D., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. SIAM J. Matrix Analysis and Applications 20(3), 720–755 (1999)CrossRefGoogle Scholar
  9. 9.
    Dongarra, J.J., Du Croz, J., Hammarling, S., Duff, I.: A set of level 3 Basic Linear Algebra Subprograms. ACM Transactions on Mathematical Software 16(1), 1–17 (1990)MATHCrossRefGoogle Scholar
  10. 10.
    Dongarra, J.J., Duff, I.S., Sorensen, D.C., van der Vorst, H.A.: Numerical Linear Algebra for High Performance Computers. In: Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (1998)Google Scholar
  11. 11.
    Duff, I.S., Reid, J.K.: The multifrontal solution of unsymmetric sets of linear systems. SIAM Journal on Scientific and Statistical Computing 5, 633–641 (1984)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gupta, A., Kumar, V.: Parallel algorithms for forward and back substitution in direct solution of sparse linear systems. In: Supercomputing 1995: Proceedings of the 1995 ACM/IEEE conference on Supercomputing (CDROM), p. 74 (1995)Google Scholar
  13. 13.
    Karypis, G., Kumar, V.: METIS: A Software Package for Partitioning Unstructured Graphs, Partitioning Meshes, and Computing Fill-Reducing Orderings of Sparse Matrices Version 4.0 (September 1998)Google Scholar
  14. 14.
    Li, X.S., Demmel, J.W.: SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Mathematical Software 29(2), 110–140 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Liu, J.W.H.: The multifrontal method for sparse matrix solution: theory and practice. SIAM Rev. 34(1), 82–109 (1992)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Zienkiewicz, O.C., Taylor, R.L.: The finite element method, 5th edn. Butterworth, Heinemann (2000)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Alberto Bertoldo
    • 1
  • Mauro Bianco
    • 1
  • Geppino Pucci
    • 1
  1. 1.Department of Information EngineeringUniversity of PadovaPadovaItaly

Personalised recommendations