Skip to main content
SpringerLink
Log in

A parallel implementation of the block preconditioned GCR method

  • Workshop: High Performance Numerical Computation and Applications
  • Conference paper
  • First Online:
High-Performance Computing and Networking (HPCN-Europe 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1593))

Included in the following conference series:

Abstract

The parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR. To precondition the GCR method a block Gauss-Jacobi method is used. Approximate solvers are used to obtain a solution of the diagonal blocks. Experiments on a cluster of HP workstations and on a Cray T3E are given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. O. Axelsson and G. Linskog. On the eigenvalue distribution of a class of preconditioning methods. Numerische Mathematik, 48:479–498, 1986.

    Article  MathSciNet  Google Scholar 

  2. Z. Bai, D. Hu, and L. Reichel. A Newton-basis GMRES implementation. IMA Journal of Numerical Analysis, 14:563–581, 1994.

    MathSciNet  Google Scholar 

  3. E. Brakkee, A. Segal, and C. G. M. Kassels. A parallel domain decomposition algorithm for the incompressible Navier-Stokes equations. Simulation Practice and Theory, 3:185–205, 1995.

    Article  Google Scholar 

  4. E. Brakkee, C. Vuik, and P. Wesseling. Domain decomposition for the incompressible Navier-Stokes equations: Solving subdomain problems accurately and in-accurately. International Journal for Numerical Methods in Fluids, 26:1217–1237, 1998.

    Article  MathSciNet  Google Scholar 

  5. Erik Brakkee. Domain Decomposition for the Incompressible Navier-Stokes Equations. PhD thesis, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands, April 1996.

    Google Scholar 

  6. E. de Sturler and H. A. van der Vorst. Reducing the effect of global communication in GMRES(m) and CG on parallel distributed memory computers. Applied Numerical Mathematics, 18:441–459, 1995.

    Article  Google Scholar 

  7. Stanley C. Eisenstat, Howard C. Elman, and Martin H. Schultz. Variational iterative methods for nonsymmetric systems of linear equations. SIAM Journal on Numerical Analysis, 20(2):345–357, April 1983.

    Article  MathSciNet  Google Scholar 

  8. J. Erhel. A parallel GMRES version for general sparse matrices. Electronic Transactions on Numerical Analysis (http://etna.mcs.kent.edu). 3:160–176, 1995.

    MathSciNet  Google Scholar 

  9. J. Frank and C. Vuik. Parallel implementation of a multiblock method with approximate subdomain solution. App. Num. Math., 1998. to appear.

    Google Scholar 

  10. Ivar Gustafsson. A class of first order factorization methods. BIT, 18:142–156, 1978.

    Article  MathSciNet  Google Scholar 

  11. Walter Hoffman. Iterative algorithms for Gram-Schmidt orthogonalization. Computing, 41:335–348, 1989.

    Article  MathSciNet  Google Scholar 

  12. J. van Kan. A second-order accurate pressure-correction scheme for viscous incompressible flow. SIAM Journal on Scientific and Statistical Computing, 7(3):870–891, 1986.

    Article  Google Scholar 

  13. G. Li. A block variant of the GMRES method on massively parallel processors. Parallel Computing 23, 23:1005–1019, 1997.

    Article  MathSciNet  Google Scholar 

  14. J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation, 31:148–162, 1977.

    Article  MathSciNet  Google Scholar 

  15. A. Segal, P. Wesseling, J. van Kan, C.W. Oosterlee, and K. Kassels. Invariant discretization of the incompressible Navier-Stokes equations in boundary-fitted coordinates. International Journal for Numerical Methods in Fluids, 15:411–426, 1992.

    Article  Google Scholar 

  16. R. B. Sidje. Alternatives for parallel Krylov subspace basis computation. Numerical Linear Algebra with Applications, 4(4):305–331, 1997.

    Article  MathSciNet  Google Scholar 

  17. Henk A. van der Vorst and C. Vuik. GMRESR: a family of nested GMRES methods. Numerical Linear Algebra with Applications, 1(4):369–386, 1994.

    Article  MathSciNet  Google Scholar 

  18. Homer F. Walker. Implementation of the GMRES method using Householder transformations. SIAM Journal on Scientific and Statistical Computing, 9(1):152–163, 1988.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    C. Vuik & J. Frank

  2. Center for Mathematics and Computer Science (CWI), P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands

    J. Frank

Authors
  1. C. Vuik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Frank
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Peter Sloot Marian Bubak Alfons Hoekstra Bob Hertzberger

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag

About this paper

Cite this paper

Vuik, C., Frank, J. (1999). A parallel implementation of the block preconditioned GCR method. In: Sloot, P., Bubak, M., Hoekstra, A., Hertzberger, B. (eds) High-Performance Computing and Networking. HPCN-Europe 1999. Lecture Notes in Computer Science, vol 1593. Springer, Berlin, Heidelberg . https://doi.org/10.1007/BFb0100666

Download citation

  • DOI: https://doi.org/10.1007/BFb0100666

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65821-4

  • Online ISBN: 978-3-540-48933-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation