Advertisement

Scaling Algebraic Multigrid Solvers: On the Road to Exascale

  • Allison H. Baker
  • Robert D. Falgout
  • Todd Gamblin
  • Tzanio V. Kolev
  • Martin Schulz
  • Ulrike Meier Yang
Conference paper

Abstract

Algebraic Multigrid (AMG) solvers are an essential component of many large-scale scientific simulation codes. Their continued numerical scalability and efficient implementation is critical for preparing these codes for exascale. Our experiences on modern multi-core machines show that significant challenges must be addressed for AMG to perform well on such machines. We discuss our experiences and describe the techniques we have used to overcome scalability challenges for AMG on hybrid architectures in preparation for exascale.

Notes

Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. It also used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract DE-AC02-06CH11357, as well as resources of the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. These resources were made available via the Performance Evaluation and Analysis Consortium End Station, a Department of Energy INCITE project. Neither Contractor, DOE, or the U.S. Government, nor any person acting on their behalf: (a) makes any warranty or representation, express or implied, with respect to the information contained in this document; or (b) assumes any liabilities with respect to the use of, or damages resulting from the use of any information contained in the document.

References

  1. 1.
    Adams, M., Brezina, M., Hu, J., Tuminaro, R.: Parallel multigrid smoothing: Polynomial versus Gauss-Seidel. J. Comput. Phys. 188, 593–610 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baker, A.H., Falgout, R.D., Kolev, T.V., Yang, U.M.: Multigrid smoothers for ultra-parallel computing. 2010. (submitted). Also available as a Lawrence Livermore National Laboratory technical report LLNL-JRNL-435315Google Scholar
  3. 3.
    Baker, A.H., Gamblin, T., Schulz, M., Yang, U.M.: Challenges of scaling algebraic multigrid across modern multicore architectures. In: Proceedings of the 25th IEEE International Parallel and Distributed Processing Symposium (IPDPS 2011) (2011) To appear. Also available as LLNL Tech. Report LLNL-CONF-458074.Google Scholar
  4. 4.
    Baker, A.H., Schulz, M., Yang, U.M.: On the performance of an algebraic multigrid solver on multicore clusters. In: J.M.L.M. Palma et al, editor, VECPAR 2010, Lecture Notes in Computer Science 6449, pp. 102–115. Springer (2010) Berkeley, CA, June 2010. http://vecpar.fe.up.pt/2010/papers/24.php
  5. 5.
    Chow, E., Falgout, R., Hu, J., Tuminaro, R., Yang, U.: A survey of parallelization techniques for multigrid solvers. In: Heroux, M., Raghavan, P., Simon, H. (eds.) Parallel Processing for Scientific Computing. SIAM Series on Software, Environments, and Tools (2006)Google Scholar
  6. 6.
    De Sterck, H., Falgout, R.D., Nolting, J., Yang, U.M.: Distance-two interpolation for parallel algebraic multigrid. Num. Lin. Alg. Appl. 15, 115–139 (2008)zbMATHCrossRefGoogle Scholar
  7. 7.
    Falgout, R., Jones, J., Yang, U.M.: Pursuing scalability for hypre’s conceptual interfaces. ACM ToMS 31, 326–350 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    Falgout, R.D.: An introduction to algebraic multigrid. Comput. Sci. Eng. 8(6), 24–33 (2006)CrossRefGoogle Scholar
  9. 9.
    Falgout, R.D., Vassilevski, P.S.: On generalizing the algebraic multigrid framework. SIAM J. Numer. Anal. 42(4), 1669–1693 (2004) UCRL-JC-150807.Google Scholar
  10. 10.
    Falgout, R.D., Vassilevski, P.S., Zikatanov, L.T.: On two-grid convergence estimates. Numer. Linear Algebra Appl. 12(5–6), 471–494 (2005); UCRL-JRNL-203843.Google Scholar
  11. 11.
    Henson, V.E., Yang, U.M.: BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41, 155–177 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    hypre. High performance preconditioners. http://www.llnl.gov/CASC/linear\_solvers/
  13. 13.
    Kolev, T., Vassilevski, P.: Parallel auxiliary space AMG for H(curl) problems. J. Comput. Math. 27, 604–623 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Stüben, K.: An introduction to algebraic multigrid. In: Trottenberg, U., Oosterlee, C., Schüller, A. (eds.) Multigrid, pp. 413–532. Academic Press, London (2001)Google Scholar
  15. 15.
    Yang, U.M.: On the use of relaxation parameters in hybrid smoothers. Numer. Linear Algebra Appl. 11, 155–172 (2004); UCRL-JC-151575Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Allison H. Baker
    • 1
  • Robert D. Falgout
    • 1
  • Todd Gamblin
    • 1
  • Tzanio V. Kolev
    • 1
  • Martin Schulz
    • 1
  • Ulrike Meier Yang
    • 1
  1. 1.Lawrence Livermore National LaboratoryCenter for Applied Scientific ComputingLivermoreUSA

Personalised recommendations