Bus coupled systems for multigrid algorithms

  • O. Kolp
  • H. Mierendorff
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1228)


Speedup and efficiency of some simple parallel multigrid algorithms for a class of bus coupled systems are investigated. We consider some basic multigrid methods (V-cycle, W-cycle) with regular grid generation and without local refinements. Our bus coupled systems consist of many independent processors each with its own local memory. A typical example for our abstract bus concept is a ring bus. The investigation of such systems is restricted to hierarchical orthogonal systems. Simple orthogonal bus systems, tree structures and mixed types are included in our general model. It can be shown that all systems are of identical suitability if the tasks are sufficiently large. The smaller however the degree of parallelism of an algorithm is, the clearer are the differences in the performance of the various systems. We can classify the most powerful systems and systems with lower performance but better technical properties. Complexity investigations enabled us to evaluate the different systems. These investigations are complemented by simulations based on the different parallel algorithms.

In general, the order of the speedup depends only on a few parameters, such as the dimension of the problem, the cycle type and the dimension of the system. The constant factors in our asymptotical expressions for the speedup depend on many parameters, especially on those of the processors and buses. We investigate these relations by simulation of some typical examples. The simulation also clarifies under which circumstances the asymptotical rules are useful for the description of system behavior.


Fine Grid Multigrid Method Computational Work Local Refinement Multigrid Algorithm 
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.

11. References

  1. [Br81]
    A. Brandt: Multigrid Solvers on Parallel Computers; in M. H. Schultz (ed.): Elliptic Problem Solvers; Academic Press, 1981.Google Scholar
  2. [CS85]
    T. F. Chan and Y. Saad: Multigrid Algorithms on the Hypercube Multiprocessor; Yale University, report no. YALEU/DCS/RR-368.Google Scholar
  3. [CS83]
    T. F. Chan and R. Schreiber: Parallel Networks for Multigrid Algorithms; Yale University, report no. YALEU/DCS/RR-262.Google Scholar
  4. [GM85]
    W. K. Giloi and H. Mühlenbein: Rationale and Concepts for the SUPRENUM Super-computer Architecture; unpublished manuscript.Google Scholar
  5. [GR84]
    D. B. Gannon and J. van Rosendale: On the Impact of Communication Complexity on the Design of Parallel Numerical Algorithms; IEEE Trans. on Comp., vol. C-33, no. 12, dec. 1984.Google Scholar
  6. [KM85]
    O. Kolp and H. Mierendorff: Efficient Multigrid Algorithms for Locally Constrained Parallel Systems; Proc. of the 2nd Copper Mountain Conf. on Multigrid Methods; march 31–april 3, 1985, Copper Mountain, Colorado; submitted to AMC.Google Scholar
  7. [ST82]
    K. Stüben and U. Trottenberg: Multigrid Methods: Fundamental Algorithms, Model Problem Analysis and Applications; in Hackbusch and Trottenberg (eds.): Multigrid Methods, Proc. of the Conf. in Köln-Porz, nov. 23–27, 1981; Lecture Notes in Mathematics, Springer, Berlin 1982.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • O. Kolp
    • 1
  • H. Mierendorff
    • 1
  1. 1.Gesellschaft fuer Mathematik und Datenverarbeitung mbH Schloss BirlinghovenSankt Augustin 1Germany

Personalised recommendations