Design, Tuning and Evaluation of Parallel Multilevel ILU Preconditioners

  • José I. Aliaga
  • Matthias Bollhöfer
  • Alberto F. Martín
  • Enrique S. Quintana-Ortí
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5336)


In this paper, we present a parallel multilevel ILU preconditioner implemented with OpenMP. We employ METIS partitioning algorithms to decompose the computation into concurrent tasks, which are then scheduled to threads. Concretely, we combine decompositions which obtain significantly more tasks than processors, and the use of dynamic scheduling strategies in order to reduce the thread’s idle time, which it is shown to be the main source of overhead in our parallel algorithm. Experimental results on a shared-memory platform consisting of 16 processors report remarkable performance for our approach.


Sparse linear system incomplete LU factorization parallel algorithm OpenMP shared-memory multiprocessor 

Related conference topics

Parallel and Distributing Computing Numerical Algorithms for CS&E 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aliaga, J.I., Bollhoefer, M., Martín, A.F., Quintana-Ortí, E.S.: Parallelization of Multilevel Preconditioners Constructed from Inverse-Based ILUs on Shared-Memory Multiprocessors. In: Bischof, C., et al. (eds.) Parallel Computing: Architectures, Algorithms and Applications, pp. 287–294 (2007)Google Scholar
  2. 2.
    Aliaga, J.I., Bollhoefer, M., Martín, A.F., Quintana-Ortí, E.S.: Scheduling Strategies for Parallel Sparse Backward/Forward Substitution PARA 2008, Trondheim (2008) (under revision)Google Scholar
  3. 3.
    Bollhoefer, M.: A Robust ILU Based on Monitoring the Growth of the Inverse Factors. Linear Algebra Appl. 338(1-3), 201–218 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bollhoefer, M.: A robust and efficient ILU that incorporates the growth of the inverse triangular factors. SIAM J. Sci. Comput. 25(1), 86–103 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bollhoefer, M., Saad, Y.: Multilevel preconditioners constructed from inverse–based ILUs. SIAM J. Sci. Comput. 25(5), 1627–1650 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chan, E., Quintana-Ortí, E.S., Quintana-Ortí, G., van de Geijn, R.: SuperMatrix out-of-order scheduling of matrix operations for SMP and multi-core architectures. In: Proceed. 19th ACM SPAA 2007, pp. 116–125 (2007)Google Scholar
  7. 7.
    Davis, T.: Direct methods for sparse linear systems. SIAM Publications, Philadelphia (2006)CrossRefMATHGoogle Scholar
  8. 8.
    Demmel, W., Gilbert, J.R., Li, X.S.: An Asynchronous Parallel Supernodal Algorithm for Sparse Gaussian Elimination. SIAM J. Matrix. Anal. Appl. 20(4), 915–952 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Karypis, G., Kumar, V.: A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Leiserson, C., Plaat, A.: Programming parallel applications in Cilk. SIAM News, SINEWS (1998)Google Scholar
  11. 11.
    Strakos, Z., Tichy, P.: Error Estimation in Preconditioned Conjugate Gradients. BIT Numerical Mathematics 45, 789–817 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM Publications, Philadelphia (2003)CrossRefMATHGoogle Scholar
  13. 13.
    Saad, Y.: Multilevel ILU with reorderings for diagonal dominance. SIAM J. Sci. Comput. 27(3), 1032–1057 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Schenk, O., Bollhöfer, M., Römer, R.A.: On Large Scale Diagonalization Techniques for the Anderson Model of Localization. SIAM Review 50(1), 91–112 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • José I. Aliaga
    • 1
  • Matthias Bollhöfer
    • 2
  • Alberto F. Martín
    • 1
  • Enrique S. Quintana-Ortí
    • 1
  1. 1.Depto. de Ingeniería y Ciencia de ComputadoresUniversidad Jaume I12.071–CastellónSpain
  2. 2.Institute of Computational MathematicsTU-BraunschweigBraunschweigGermany

Personalised recommendations