Improvements of a Fast Parallel Poisson Solver on Irregular Domains

  • Andreas Adelmann
  • Peter Arbenz
  • Yves Ineichen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7133)


We discuss the scalable parallel solution of the Poisson equation on irregularly shaped domains discretized by finite differences. The symmetric positive definite system is solved by the preconditioned conjugate gradient algorithm with smoothed aggregation (SA) based algebraic multigrid (AMG) preconditioning. We investigate variants of the implementation of SA-AMG that lead to considerable improvements in the execution times. The improvements are due to a better data partitioning and the iterative solution of the coarsest level system in AMG. We demonstrate good scalability of the solver on a distributed memory parallel computer with up to 2048 processors.


Poisson equation finite differences preconditioned conjugate gradient algorithm algebraic multigrid data partitioning 


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  1. 1.
    Adams, M., Brezina, M., Hu, J., Tuminaro, R.: Parallel multigrid smoothing: polynomial versus Gauss–Seidel. J. Comput. Phys. 188(2), 593–610 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adelmann, A., Arbenz, P., Ineichen, Y.: A fast parallel Poisson solver on irregular domains applied to beam dynamics simulations. J. Comput. Phys. 229(12), 4554–4566 (2010)CrossRefzbMATHGoogle Scholar
  3. 3.
    Adelmann, A., Kraus, C., Ineichen, Y., Yang, J.J.: The Object Oriented Parallel Accelerator Library Framework. Technical Report. PSI-PR-08-02, Paul Scherrer Institut (2008-2010),
  4. 4.
    Boman, E., Devine, K., Fisk, L.A., Heaphy, R., Hendrickson, B., Vaughan, C., Çatalyürek, Ü., Bozdag, D., Mitchell, W., Teresco, J.: Zoltan 3.0: Parallel Partitioning, Load-balancing, and Data Management Services; User’s Guide. Sandia National Laboratories, Albuquerque, NM (2007), Tech. Report SAND2007-4748W,
  5. 5.
    Cray XT5 Brochure. Cray Inc., Seattle (2009), (retrieved on July 13, 2010)
  6. 6.
    Devine, K., Boman, E., Heaphy, R., Hendrickson, B., Vaughan, C.: Zoltan data management services for parallel dynamic applications. Comput. Sci. Eng. 4(2), 90–97 (2002)CrossRefGoogle Scholar
  7. 7.
    Forsythe, G.E., Wasow, W.R.: Finite-difference methods for partial differential equations. Wiley, New York (1960)zbMATHGoogle Scholar
  8. 8.
    Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 smoothed aggregation user’s guide. Tech. Report SAND2006-2649, Sandia National Laboratories (May 2006)Google Scholar
  9. 9.
    Golub, G.H., van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  10. 10.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hackbusch, W.: Iterative solution of large sparse systems of equations. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hestenes, M.R., Stiefel, E.: Methods of conjugent gradients for solving linear systems. J. Res. Nat. Bur. Standards 49, 409–436 (1952)CrossRefzbMATHGoogle Scholar
  14. 14.
    McCorquodale, P., Colella, P., Grote, D.P., Vay, J.-L.: A node-centered local refinement algorithm for Poisson’s equation in complex geometries. J. Comput. Phys. 201(1), 34–60 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pöplau, G., van Rienen, U.: A self-adaptive multigrid technique for 3-D space charge calculations. IEEE Trans. Magn. 44(6), 1242–1245 (2008)CrossRefGoogle Scholar
  16. 16.
    Qiang, J., Gluckstern, R.L.: Three-dimensional Poisson solver for a charged beam with large aspect ratio in a conducting pipe. Comput. Phys. Commun. 160(2), 120–128 (2004)CrossRefzbMATHGoogle Scholar
  17. 17.
    Qiang, J., Ryne, R.D.: Parallel 3D Poisson solver for a charged beam in a conducting pipe. Comput. Phys. Commun. 138(1), 18–28 (2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    The Trilinos Project Home Page,
  19. 19.
    Tuminaro, R.S., Tong, C.: Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines. In: ACM/IEEE SC 2000 Conference, SC 2000, 21 pages (2000), doi:10.1109/SC.2000.10008Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Adelmann
    • 1
  • Peter Arbenz
    • 2
  • Yves Ineichen
    • 1
    • 2
  1. 1.Paul Scherrer InstitutVilligenSwitzerland
  2. 2.Computational ScienceETH ZürchZürichSwitzerland

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