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Parallel Monte Carlo Algorithms for Sparse SLAE Using MPI

  • V. Alexandrov
  • A. Karaivanova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1697)

Abstract

The problem of solving sparse Systems of Linear Algebraic Equations (SLAE) by parallel Monte Carlo numerical methods is considered. The almost optimal Monte Carlo algorithms are presented. In case when a copy of the non-zero matrix elements is sent to each processor the execution time for solving SLAE by Monte Carlo on p processors is bounded by O(nNdT/p) where N is the number of chains, T is the length of the chain in the stochastic process, which are independent of matrix size n, and d is the average number of non-zero elements in the row. Finding a component of the solution vector requires O(NdT/p) time on p processors, which is independent of the matrix size n.

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References

  1. 1.
    Alexandrov, V., Lakka, S.: Comparison of three Parallel Monte Carlo Methods for Matrix Inversion, Proc. of EUROPAR96, Lyon, France, Vol II (1996), 72–80Google Scholar
  2. 2.
    Alexandrov, V., Megson, G.M.: Solving Sytem of Linear algebraic Equations by Monte Carlo Method on Regular Arrays, Proc. of PARCELLA96, 16–20 September, Berlin, Germany, (1996) 137–146Google Scholar
  3. 3.
    Alexandrov, V., Rau-Chaplin, A., Dehne, F., Taft, K.: Efficient Coarse Grain Monte Carlo Algorithms for Matrix Computations using PVM, LNCS 1497, Springer, August, (1998) 323–330Google Scholar
  4. 4.
    Bertsekas, D.P., Tsitsiklis: Parallel and Distributed Computation, Prentice Hall, (1989)Google Scholar
  5. 5.
    Dehne, F., Fabri, A., Rau-Chaplin, A.: Scalable parallel geometric algorithms for multicomputers, Proc. 7th ACM Symp. on Computational Geometry, (1993)Google Scholar
  6. 6.
    Dimov, I., Alexandrov, V., Karaivanova, A.: Implementation of Monte Carlo Algorithms for Eigenvalue Problem using MPI, LNCS 1497, Springer, August (1998) 346–353Google Scholar
  7. 7.
    Dimov, I., Alexandrov, V.: A New Highly Convergent Monte Carlo Method for Matrix Computations, Mathematics and Computers in Simulation, Vol. 47, No 2–5, North-Holland, August (1998) 165–182CrossRefMathSciNetGoogle Scholar
  8. 8.
    Golub, G.H., Ch., F., Van Loon: Matrix Computations, The Johns Hopkins Univ. Press, Baltimore and London, (1996)zbMATHGoogle Scholar
  9. 9.
    Halton, J.H.: Sequential Monte Carlo Techniques for the Solution of Linear Systems, TR 92-033, University of North Carolina at Chapel Hill, Department of Computer Science, (1992)Google Scholar
  10. 10.
    Megson, G.M., Aleksandrov, V., Dimov, I.: Systolic Matrix Inversion Using Monte Carlo Method, J. Parallel Algorithms and Applications, Vol.3, (1994) 311–330zbMATHGoogle Scholar
  11. 11.
    Sobol’, I.M.: Monte Carlo numerical methods. Moscow, Nauka, (1973) (Russian) (English version Univ. of Chicago Press 1984).Google Scholar
  12. 12.
    Westlake J.R.: A Handbook of Numerical Matrix Inversion and Solution of Linear Equations, John Wiley and Sons, New York, (1968)zbMATHGoogle Scholar
  13. 13.
    Gallivan, K., Hansen, P.C., Ostromsky, Tz., Zlatev, Z.: A Locally Optimized Reordering Algorithm and its Application to a Parallel Sparse Linear System Solver, Computing V.54, (1995) 39–67zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • V. Alexandrov
    • 1
  • A. Karaivanova
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Central Laboratory for Parallel ProcessingBulgarian Academy of SciencesSofiaBulgaria

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