The KIPS Transactions:PartA (정보처리학회논문지A)

Korea Information Processing Society (한국정보처리학회)
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A Scalable Parallel Preconditioner on the CRAY-T3E for Large Nonsymmetric Spares Linear Systems

대형비대칭 이산행렬의 CRAY-T3E에서의 해법을 위한 확장가능한 병렬준비행렬

  • Ma, Sang-Baek (Dept. of Electronic computer Engineering, Hanyang University)
  • 마상백 (한양대학교 전자컴퓨터공학부)
  • Published : 2001.09.01
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Abstract

In this paper we propose a block-type parallel preconditioner for solving large sparse nonsymmetric linear systems, which we expect to be scalable. It is Multi-Color Block SOR preconditioner, combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. We compared it with four other preconditioners, which are ILU(0)-wavefront ordering, ILU(0)-Multi-Color ordering, SPAI(SParse Approximate Inverse), and SSOR preconditiner. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to $1025{\times}1024$. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications, The results show that Multi-Color Block SOR is scalabl and gives the best performances.

Keywords

References

  1. C. C. Jay Kuo and Tony Chan, 'Two-color Fourier analysis of iterative algorithms for elliptic problems with red/black ordering,' SIAM J. Sci Stat, Vol.11, No.4, 1990, pp.767-793 https://doi.org/10.1137/0911045
  2. Y. Saad, 'Krylov subspace methods on supercomputers,' SIAM J. Sci. Stat, Vol.10, 1989, pp.1200-1232 https://doi.org/10.1137/0910073
  3. M. Grote and T. Huckle, 'Parallel preconditioning with sparse approximate inverses,' J. Sci. Computing, Vol. 18, 1997, pp.838-853
  4. D. P. O'Leary, 'Ordering schmes for parallel processing of certain mesh problems,' SIAM J. Sci Stat, Vol. 5, 1984, pp. 620-632 https://doi.org/10.1137/0905044
  5. D. Boley, B. Buzbee and S. Parter, 'On block relaxation techniques,' MIRC Technical Summary Report 1860, Mathematics Research Center, University of Wisconsin, 1978
  6. R. Varga, Matrix Iterative Analysis, Prentice-Hall, New York, 1962
  7. Y. Saad and G. Golub, 'Highly parallel preconditioners for general sparse matrices,' in Recent Advances in Iterative Methods, IMA Volumes in Mathematics and its Applications, Vol.60, G.Golub, M.Luskin and A.Greenbaum eds, Springer-Verlag, Berlin, 1994, pp.165-199
  8. 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,' Math Comp., Vol.31, 1977, pp.148-162 https://doi.org/10.2307/2005786
  9. H. Elman, 'Iterative methods for large, sparse, nonsymmetric systems of linear equations,' Ph. D Thesis, Yale University, 1982
  10. Y. Saad and M. Schultz, 'GMRES : A Generalized minimal residual algorithm for solving nonsymmetric linear systems,' SIAM J.Sci. Stat, Vol.7, July, 1986
  11. H. Van der Vorst, 'BI-CGSTAB : A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,' SIAM. J. Sci. Statist. Comput, Vol.13, 1992, pp.631-644 https://doi.org/10.1137/0913035
  12. Y. Saad, 'SPARSKIT : A basic tool kit for sparse computations,' in Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996