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Journal of Engineering Mathematics

, Volume 93, Issue 1, pp 21–39 | Cite as

Partitioning strategies for the block Cimmino algorithm

  • L. A. Drummond
  • Iain S. Duff
  • Ronan Guivarch
  • Daniel Ruiz
  • Mohamed Zenadi
Article

Abstract

In the context of the block Cimmino algorithm, we study preprocessing strategies to obtain block partitionings that can be applied to general linear systems of equations \(\mathbf{A}\mathbf{x}= \mathbf{b}\). We study strategies that transform the matrix \(\mathbf{A}\mathbf{A}^\mathrm{{T}}\) into a matrix with a block tridiagonal structure. This provides a partitioning of the linear system for row projection methods because block Cimmino is essentially equivalent to block Jacobi on the normal equations, and the resulting partition will yield a two-block partition of the original matrix. Therefore, the resulting block partitioning should improve the rate of convergence of block row projection methods such as block Cimmino. We discuss a method for obtaining a partitioning using a dropping strategy that gives more blocks at the cost of relaxing the two-block partitioning. We then use a hypergraph partitioning that works directly on the matrix \(\mathbf{A}\) to reduce directly the connections between blocks. We give numerical results showing the performance of these techniques both in their effect on the convergence of the block Cimmino algorithm and in their ability to exploit parallelism.

Keywords

Cuthill McKee Hypergraph partitioning Iterative methods Sparse matrices Unsymmetric matrices 

Notes

Acknowledgments

This work was partially funded by the ANR-BARESAFE project, ANR-11-MONU-004, Programme Modèles Numériques 2011, supported by the French National Agency for Research. The authors were granted access to the HPC resources of CALMIP under allocation 2013-P0989. We thank the four referees for their detailed comments on the first version of the manuscript. The research of I.S. Duff was supported in part by the EPSRC Grant EP/I013067/1.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • L. A. Drummond
    • 1
  • Iain S. Duff
    • 2
    • 3
  • Ronan Guivarch
    • 4
  • Daniel Ruiz
    • 4
  • Mohamed Zenadi
    • 4
  1. 1.Computational Research Division, Lawrence Berkeley National LaboratoryBerkeleyUSA
  2. 2.Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique (CERFACS)Toulouse CedexFrance
  3. 3.R18, Rutherford Appleton LaboratoryOxonEngland
  4. 4.ENSEEIHT-IRITToulouse Cedex 7France

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