Parallel multigrid methods: Implementation on SUPRENUM-like architectures and applications

  • Karl Solchenbach
  • Clemens-August Thole
  • Ulrich Trottenberg
Session 2: Parallel Architectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 297)


Multigrid (MG) methods for partial differential equations (and for other important mathematical models in scientific computing) have turned out to be optimal on sequential computers. Clearly, one wants to apply them also on vector and parallel computers in order to exploit both, the high MG-efficiency (compared to classical methods) and the full computational power of modern supercomputers. For this purpose, parallel MG methods are needed. It turns out that certain well-known standard MG methods (with RB and zebra-type relaxation, as described in [25]) already contain a sufficiently high degree of parallelism.

Among innovative supercomputer architectures, MIMD multiprocessor computers with local memory and a vector unit in each processor are particularly promising. A software approach that corresponds to such architectures in a natural way is the abstract SUPRENUM concept. It is characterized by a dynamical process system, where each process has its own data space and communicates with other processes by message-passing.

In this paper, we show how such architectures and software concepts are used for the solution of large scale grid problems (discrete PDEs, etc.). Grid partitioning and blockstructuring — with communication only along the subgrid or block boundaries — are the natural approches in this context. Any grid oriented method, in particularly any MG method can be efficiently parallelized using these approaches. In the SUPRENUM project, powerful software tools (e.g. a mapping library for the process-processor mapping and a communication library for the intergrid data exchanges) are developed that make it very easy to implement single grid and MG methods on local memory multiprocessor systems. Parallel MG programs have been run on the SUPRENUM simulator [16], the SUPRENUM pre-prototype [22] and some other local memory machines like the Intel iPSC and the CalTech hypercube.


Coarse Grid Local Memory Multigrid Method Grid Level Single Grid 
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]
    Bjørstad, P.E., Widlund, O.B.: Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23, 6, 1986.Google Scholar
  2. [2]
    Brandt, A.: Multigrid techniques: 1984 guide with applications to fluid dynamics. GMD-Studie No. 85, 1984.Google Scholar
  3. [3]
    Brandt, A.: Multigrid solvers on parallel computers. In “Elliptic Problem solvers (M. Schultz, ed.)”, Academic Press, New York, 1981.Google Scholar
  4. [4]
    Braess, D., Hackbusch, W., Trottenberg, U. (eds.): Advances in Multigrid methods Proceedings of the Conference Held in Oberwolfach, December, 8–13, 1984. Notes on Numerical Fluid Mechanics, Vol. 11, Vieweg, Braunschweig, 1984.Google Scholar
  5. [5]
    Dihn, Q.V., Glowinski, R., Periaux, J.: Solving elliptic problems by domain decomposition methods with applications. In “Elliptic Problem Solvers II (G. Birkhoff and A. Schoenstadt, eds.)”, Academic Press, New York, 1984, pp. 395–426.Google Scholar
  6. [6]
    Finnemann H., Volkert, J.: Parallel multigrid solvers for the neutron diffusion equation. Proceedings of the International Topical Meeting on Advances in Reactor Physics, Mathematics and Computation, Paris, April, 27–30, 1987.Google Scholar
  7. [7]
    Frederickson, P.O., McBryan, O.: Parallel superconvergent multigrid. Cornell Theory Center Technical Report CTC87TR12 7/87.Google Scholar
  8. [8]
    Gannon, D.B., Rosendale, J,R. van: Highly parallel multigrid solvers for elliptic PDEs: An experimental analysis. Report 82-36, ICASE, NASA Langley Research Center, Hampton, VA, 1978.Google Scholar
  9. [9]
    Giloi, W.K., Mühlenbein, H.: Rationale and concepts for the Suprenum supercomputer architecture. GMD, St. Augustin 1985.Google Scholar
  10. [10]
    Greenbaum, A.: A multigrid method for multiprocessors. Appl. Math. Comp. 19, pp. 75–88, 1986.Google Scholar
  11. [11]
    Grosch, C.E.: Performance analysis of Poisson solvers on array computers. Report TR 79-3, Old Domion University, Norfork, VA, 1979.Google Scholar
  12. [12]
    Grosch, C.E.: Poisson solvers on large array computer. Proceedings 1978 LANL Workshop on vector and parallel processors (B.L. Buzbee and J.F. Morrison, eds.), 1978.Google Scholar
  13. [13]
    Hackbusch, W., Trottenberg, U. (eds.): Multigrid methods. Proceedings of the Conference held at Köln-Porz, November 23–27, 1981. Lecture Notes in Mathematics Vol. 960, Springer, Berlin, 1982.Google Scholar
  14. [14]
    Hackbusch, W., Trottenberg, U. (eds.): Multigrid methods II. Proceedings of the 2nd Conference on Multigrid Methods, Cologne, Oct. 1–4, 1985. Lecture Notes in Mathematics Vol. 1228, Springer, Berlin, 1986.Google Scholar
  15. [15]
    Hempel, R., Schüller, A.: Vereinheitlichung und Portabilität paralleler Anwendersoftware durch Verwendung einer Kommunikationsbibliothek. Arbeitspapiere der GMD, Nr. 234, GMD, St. Augustin, 1986.Google Scholar
  16. [16]
    Limburger, F., Scheidler, Ch., Tietz, Ch., Wessels, A.: Benutzeranleitung des SUPRENUM-Simulationssystems SUSI. GMD, St. Augustin, 1986.Google Scholar
  17. [17]
    Linden, J., Stüben, K.: Multigrid methods: An overview with emphasis on grid generation processes. Arbeitspapiere der GMD Nr. 207, GMD, St. Augustin, 1986.Google Scholar
  18. [18]
    McBryan, O.: Numerical computation on massively parallel hypercubes., to appear.Google Scholar
  19. [19]
    McBryan, O., Van de Velde, E.: The multigrid method on parallel processors. In [14].Google Scholar
  20. [20]
    McCormick, S.F. (ed.): Proceedings of the 2nd International Multigrid Conference, April 1985, Copper Mountain. Appl. Math. Comp. Vol. 19, North Holland, 1986.Google Scholar
  21. [21]
    Niestegge, A., Stüben, K.: A parallel multigrid method for the Stokes problem. GMD-Arbeitspapier, GMD, St. Augustin, to appear.Google Scholar
  22. [22]
    Peinze, K., Thole, C.A., Thomas, B., Werner, K.H.: The SUPRENUM prototyping programme. Suprenum-Report 5, SUPRENUM GmbH, Bonn, 1987.Google Scholar
  23. [23]
    Rice, J.: Parallel methods for PDEs. Report CSD-TR-587, Purdue Univercity, West Lafayette, Indiana, 1986.Google Scholar
  24. [24]
    Solchenbach, K.: Parallel multigrid methods: Efficient coarse grid techniques. Suprenum-Report, SUPRENUM GmbH, Bonn, to appear.Google Scholar
  25. [25]
    Stüben, K., Trottenberg, U.: Multigrid methods: Fundamental algorithms, model problem analysis and applications. In [13]Google Scholar
  26. [26]
    Thole, C.A.: Experiments with multigrid methods on the CalTech-hypercube. GMD-Studie Nr. 103, GMD, St. Augustin, 1985.Google Scholar
  27. [27]
    Thole, C.A., Trottenberg, U.: A short note on standard parallel multigrid algorithms for 3D-problems. Suprenum-Report 3, SUPRENUM GmbH, Bonn, 1987.Google Scholar
  28. [28]
    Wagner, B., Leicher, S., Schmidt, W.: Applications of a multigrid finite volume method with Runge-Kutta time integration for solving the Euler and Navier-Stokes equations. In GMD-Studie 110 (U. Trottenberg, W. Hackbusch, eds.), GMD, St. Augustin, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Karl Solchenbach
    • 1
    • 2
  • Clemens-August Thole
    • 1
    • 2
  • Ulrich Trottenberg
    • 1
    • 2
  1. 1.Suprenum GmbHBonn
  2. 2.Gesellschaft für Mathematik und DatenverarbeitungSt. Augustin

Personalised recommendations