Advertisement

Parallel Geometric Multigrid

  • Frank Hülsemann
  • Markus Kowarschik
  • Marcus Mohr
  • Ulrich Rüde
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 51)

Summary

Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computation-to-communication ratio on the coarse grids. Our contribution discusses parallelisation techniques for geometric multigrid methods. It covers both theoretical approaches as well as practical implementation issues that may guide code development.

Keywords

Coarse Grid Multigrid Method Unstructured Grid Grid Level Discrete Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter. The multi-grid methods for the diffusion equation with strongly discontinuous coefficients. SIAM J. Sci. Stat. Comput., 2:430–454, 1981.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R. Allen and K. Kennedy. Optimizing Compilers for Modern Architectures. Morgan Kaufmann Publishers, San Francisco, California, USA, 2001.Google Scholar
  3. 3.
    R. E. Bank and M. Holst. A new paradigm for parallel adaptive meshing algorithms. SIAM J. Sci. Comput., 22(4):1411–1443, 2000.MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Bastian, W. Hackbusch, and G. Wittum. Additive and multiplicative multi-grid — a comparison. Computing, 60(4):345–364, 1998.MathSciNetGoogle Scholar
  5. 5.
    B. Bergen and F. Hülsemann. Hierarchical hybrid grids: data structures and core algorithms for multigrid. Numerical Linear Algebra with Applications, 11:279–291, 2004.MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. H. Bramble, J. E. Pasciak, and J. Xu. Parallel multilevel preconditioners. Math. Comp., 55:1–22, 1990.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Brandt. Multi-level adaptive solutions to boundary-value problems. Math. Comp., 31:333–390, 1977.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Brandt. Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studien Nr. 85. Gesellschaft für Mathematik und Datenverarbeitung, St. Augustin, 1984.Google Scholar
  9. 9.
    A. Brandt and B. Diskin. Multigrid solvers on decomposed domains. In Domain Decomposition Methods in Science and Engineering: The Sixth International Conference on Domain Decomposition, volume 157 of Contemporary Mathematics, pp. 135–155, Providence, Rhode Island, 1994. American Mathematical Society.MathSciNetGoogle Scholar
  10. 10.
    W. Briggs, V. Henson, and S. McCormick. A Multigrid Tutorial. SIAM, 2. edition, 2000.Google Scholar
  11. 11.
    T. F. Chan and R. S. Tuminaro. Analysis of a parallel multigrid algorithm. In J. Mandel, S. F. McCormick, J. E. Dendy, C. Farhat, G. Lonsdale, S. V. Parter, J. W. Ruge, and K. Stüben, editors, Proceedings of the Fourth Copper Mountain Conference on Multigrid Methods, pp. 66–86, Philadelphia, 1989. SIAM.Google Scholar
  12. 12.
    R. Chandra, L. Dagum, D. Kohr, D. Maydan, J. McDonald, and R. Menon. Parallel Programming in OpenMP. Morgan Kaufmann, 2001.Google Scholar
  13. 13.
    L. Dagum and R. Menon. OpenMP: An industry-standard API for shared-memory programming. IEEE Comp. Science and Engineering, 5(1):46–55, 1998.CrossRefGoogle Scholar
  14. 14.
    B. Diskin. Multigrid Solvers on Decomposed Domains. Master’s thesis, Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, 1993.Google Scholar
  15. 15.
    C. C. Douglas. A review of numerous parallel multigrid methods. In G. Astfalk, editor, Applications on Advanced Architecture Computers, pp. 187–202. SIAM, Philadelphia, 1996.Google Scholar
  16. 16.
    C. C. Douglas and M. B. Douglas. MGNet Bibliography. Department of Computer Science and the Center for Computational Sciences, University of Kentucky, Lexington, KY, USA and Department of Computer Science, Yale University, New Haven, CT, USA, 1991–2002 (last modified on September 28, 2002); see http://www.mgnet.org/mgnet-bib.html.Google Scholar
  17. 17.
    C. C. Douglas, G. Haase, and U. Langer. A Tutorial on Elliptic PDE Solvers and their Parallelization. SIAM, 2003.Google Scholar
  18. 18.
    C. C. Douglas, J. Hu, M. Kowarschik, U. Rüde, and C. Weiß. Weimization for structured and unstructured grid multigrid. Elect. Trans. Numer. Anal., 10:21–40, 2000.Google Scholar
  19. 19.
    L. Formaggia, M. Sala, and F. Saleri. Domain decomposition techniques. In A. M. Bruaset and A. Tveito, editors, Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pp. 135–163. Springer-Verlag, 2005.Google Scholar
  20. 20.
    P. O. Frederickson and O. A. McBryan. Parallel superconvergent multigrid. In S. F. Mc-Cormick, editor, Multigrid Methods: Theory, Applications, and Supercomputing, volume 110 of Lecture Notes in Pure and Applied Mathematics, pp. 195–210. Marcel Dekker, New York, 1988.Google Scholar
  21. 21.
    T. L. Freeman and C. Phillips. Parallel numerical algorithms. Prentice Hall, New York, 1992.Google Scholar
  22. 22.
    D. B. Gannon and J. R. Rosendale. On the structure of parallelism in a highly concurrent pde solver. J. Parallel Distrib. Comput., 3:106–135, 1986.CrossRefGoogle Scholar
  23. 23.
    A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, 1997.Google Scholar
  24. 24.
    M. Griebel. Grid-and point-oriented multilevel algorithms. In W. Hackbusch and G. Wittum, editors, Incomplete Decompositions (ILU) — Algorithms, Theory, and Applications, Notes on Numerical Fluid Mechanics, pp. 32–46. Vieweg, Braunschweig, 1993.Google Scholar
  25. 25.
    M. Griebel. Multilevel algorithms considered as iterative methods on semidefinite systems. SIAM J. Sci. Stat. Comput., 15:547–565, 1994.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Griebel. Parallel point-oriented multilevel methods. In Multigrid Methods IV, Proceedings of the Fourth European Multigrid Conference, Amsterdam, July 6–9, 1993, volume 116 of ISNM, pp. 215–232, Basel, 1994. Birkhäuser.zbMATHMathSciNetGoogle Scholar
  27. 27.
    M. Griebel and P. Oswald. On the abstract theory of additive and multiplicative Schwarz algorithms. Numer. Math., 70:163–180, 1995.MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Griebel and G.W. Zumbusch. Hash-storage techniques for adaptive multilevel solvers and their domain decomposition parallelization. In J. Mandel, C. Farhat, and X.-C. Cai, editors, Proceedings of Domain Decomposition Methods 10, DD10, number 218 in Contemporary Mathematics, pp. 279–286, Providence, 1998. AMS.Google Scholar
  29. 29.
    W. Gropp, E. Lusk, N. Doss, and A. Skjellum. A high-performance, portable implementation of the MPI message passing interface standard. Parallel Computing, 22(6):789–828, Sept. 1996.CrossRefGoogle Scholar
  30. 30.
    W. Gropp, E. Lusk, and A. Skjellum. Using MPI, Portable Parallel Programming with the Mesage-Passing Interface. MIT Press, second edition, 1999.Google Scholar
  31. 31.
    G. Haase. Parallelisierung numerischer Algorithmen für partielle Differentialgleichungen. B. G. Teubner Stuttgart — Leipzig, 1999.Google Scholar
  32. 32.
    W. Hackbusch. Multigrid Methods and Applications, volume 4 of Computational Mathematics. Springer-Verlag, Berlin, 1985.Google Scholar
  33. 33.
    W. Hackbusch. Iterative Solution of Large Sparse Systems of Equations, volume 95 of Applied Mathematical Sciences. Springer, 1993.Google Scholar
  34. 34.
    J. Hennessy and D. Patterson. Computer Architecture: A Quantitative Approach. Morgan Kaufmann Publisher, Inc., San Francisco, California, USA, 3. edition, 2003.Google Scholar
  35. 35.
    F. Hülsemann, B. Bergen, and U. Rüde. Hierarchical hybrid grids as basis for parallel numerical solution of PDE. In H. Kosch, L. Böszörményi, and H. Hellwagner, editors, Euro-Par 2003 Parallel Processing, volume 2790 of Lecture Notes in Computer Science, pp. 840–843, Berlin, 2003. Springer.Google Scholar
  36. 36.
    F. Hülsemann, S. Meinlschmidt, B. Bergen, G. Greiner, and U. Rüde. gridlib — a parallel, object-oriented framework for hierarchical-hybrid grid structures in technical simulation and scientific visualization. In High Performance Computing in Science and Engineering, Munich 2004. Transactions of the Second Joint HLRB and KONWIHR Result and Reviewing Workshop, pp. 37–50, Berlin, 2004. Springer.Google Scholar
  37. 37.
    M. Jung. On the parallelization of multi-grid methods using a non-overlapping domain decomposition data structure. Appl. Numer. Math., 23(1):119–137, 1997.zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    M. Jung. Parallel multiplicative and additive multilevel methods for elliptic problems in three-dimensional domains. In B. H. V. Topping, editor, Advances in Computational Mechanics with Parallel and Distributed Processing, pp. 171–177, Edinburgh, 1997. Civil-Comp Press. Proceedings of the EURO-CM-PAR97, Lochinver, April 28–May 1, 1997.Google Scholar
  39. 39.
    G. Karypis and V. Kumar. A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput., 20(1):359–392, 1999.MathSciNetCrossRefGoogle Scholar
  40. 40.
    G. Karypis and V. Kumar. Parallel multilevel k-way partition scheme for irregular graphs. SIAM Review, 41(2):278–300, 1999.MathSciNetCrossRefGoogle Scholar
  41. 41.
    C. Körner, T. Pohl, U. Rüde, N. Thürey, and T. Zeiser. Parallel lattice boltzmann methods for cfd applications. In A. M. Bruaset and A. Tveito, editors, Numerical Solution. of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pp. 439–466. Springer-Verlag, 2005.Google Scholar
  42. 42.
    M. Kowarschik. Data Locality Optimizations for Iterative Numerical Algorithms and Cellular Automata on Hierarchical Memory Architectures. PhD thesis, Lehrstuhl für Informatik 10 (Systemsimulation), Institut für Informatik, Universität Erlangen-Nürnberg, Erlangen, Germany, July 2004. SCS Publishing House.Google Scholar
  43. 43.
    H. Lötzbeyer and U. Rüde. Patch-adaptive multilevel iteration. BIT, 37:739–758, 1997.MathSciNetGoogle Scholar
  44. 44.
    L. R. Matheson and R. E. Tarjan. Parallelism in multigrid methods: how much is too much? Int. J. Paral. Prog., 24:397–432, 1996.Google Scholar
  45. 45.
    O. A. McBryan, P. O. Frederickson, J. Linden, A. Schuller, K. Solchenbach, K. Stuben, C.-A. Thole, and U. Trottenberg. Multigrid methods on parallel computers — a survey of recent developments. Impact Comput. Sci. Eng., 3:1–75, 1991.CrossRefGoogle Scholar
  46. 46.
    W. F. Mitchell. A parallel multigrid method using the full domain partition. Elect. Trans Numer. Anal., 6:224–233, 1997.zbMATHGoogle Scholar
  47. 47.
    W. F. Mitchell. The full domain partition approach to distributing adaptive grids. Appl Numer. Math., 26:265–275, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  48. 48.
    W. F. Mitchell. Parallel adaptive multilevel methods with full domain partitions. App Num. Anal. and Comp. Math., 1:36–48, 2004.zbMATHCrossRefGoogle Scholar
  49. 49.
    M. Mohr. Low Communication Parallel Multigrid: A Fine Level Approach. In A. Bode, T. Ludwig, W. Karl, and R. Wismüller, editors, Proceedings of Euro-Par 2000: Parallel. Processing, volume 1900 of Lecture Notes in Computer Science, pp. 806–814. Springer, 2000.Google Scholar
  50. 50.
    M. Mohr and U. Rüde. Communication Reduced Parallel Multigrid: Analysis and Experiments. Technical Report 394, Institut für Mathematik, Universität Augsburg, 1998.Google Scholar
  51. 51.
    P. Oswald. Multilevel Finite Element Approximation, Theory and Applications. Teubner Skripten zur Numerik. Teubner Verlag, Stuttgart, 1994.Google Scholar
  52. 52.
    A. Pothen. Graph partitioning algorithms with applications to scientific computing. In D. E. Keyes, A. H. Sameh, and V. Venkatakrishnan, editors, Parallel Numerical Algorithms, volume 4 of ICASE/LaRC Interdisciplinary Series in Science and Engineering. Kluwer Academic Press, 1997.Google Scholar
  53. 53.
    M. Prieto, I. Llorente, and F. Tirado. A Review of Regular Domain Partitioning. SIAM News, 33(1), 2000.Google Scholar
  54. 54.
    G. Rivera and C.-W. Tseng. Data Transformations for Eliminating Conflict Misses. In Proc. of the ACM SIGPLAN Conf. on Programming Language Design and Implementation, Montreal, Canada, 1998.Google Scholar
  55. 55.
    U. Rüde. Mathematical and Computational Techniques for Multilevel Adaptive Methods, volume 13 of Frontiers in Applied Mathematics. SIAM, Philadelphia, 1993.Google Scholar
  56. 56.
    Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, 2nd edition, 2003.Google Scholar
  57. 57.
    J. Stoer and R. Bulirsch. Numerische Mathematik 2. Springer, 4. edition, 2000.Google Scholar
  58. 58.
    J. D. Teresco, K. D. Devine, and J. E. Flaherty. Partitioning and dynamic load balancing for the numerical solution of partial differential equations. In A. M. Bruaset and A. Tveito, editors, Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pp. 55–88. Springer-Verlag, 2005.Google Scholar
  59. 59.
    C.-A. Thole and U. Trottenberg. Basic smoothing procedures for the multigrid treatment of elliptic 3D-operators. Appl. Math. Comput., 19:333–345, 1986.MathSciNetCrossRefGoogle Scholar
  60. 60.
    U. Trottenberg, C. W. Oosterlee, and A. Schüller. Multigrid. Academic Press, London, 2000.Google Scholar
  61. 61.
    C. Weiß Data Locality Optimizations for Multigrid Methods on Structured Grids. PhD thesis, Lehrstuhl für Rechnertechnik und Rechnerorganisation, Institut für Informatik, Technische Universität München, Munich, Germany, Dec. 2001.Google Scholar
  62. 62.
    R. Wienands and C. W. Oosterlee. On three-grid fourier analysis for multigrid. SIAM J Sci. Comput., 23(2):651–671, 2001.MathSciNetCrossRefGoogle Scholar
  63. 63.
    G. Wittum. On the robustness of ILU-smoothing. SIAM J. Sci. Stat. Comput., 10:699–717, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    D. Xie and L. Scott. The Parallel U-Cycle Multigrid Method. In Virtual Proceedings. of the 8th Copper Mountain Conference on Multigrid Methods, 1997. Available at http://www.mgnet.org.Google Scholar
  65. 65.
    U. M. Yang. Parallel algebraic multigrid methods-high performance preconditioners. In A. M. Bruaset and A. Tveito, editors, Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of Lecture Notes in Computational Science and Engineering, pp. 209–236. Springer-Verlag, 2005.Google Scholar
  66. 66.
    I. Yavneh. On red-black SOR smoothing in multigrid. SIAM J. Sci. Comput., 17:180–192, 1996.zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    G. Zumbusch. Parallel Multilevel Methods — Adaptive Mesh Refinement and Loadbalancing. Advances in Numerical Mathematics. Teubner, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Frank Hülsemann
    • 1
  • Markus Kowarschik
    • 1
  • Marcus Mohr
    • 2
  • Ulrich Rüde
    • 1
  1. 1.System Simulation GroupUniversity of ErlangenGermany
  2. 2.Department for Sensor TechnologyUniversity of ErlangenGermany

Personalised recommendations