A Parallel Quasi-Monte Carlo Method for Computing Extremal Eigenvalues

  • Michael Mascagni
  • Aneta Karaivanova

Abstract

The convergence of Monte Carlo methods for numerical integration can often be improved by replacing pseudorandom numbers (PRNs) with more uniformly distributed numbers known as quasirandom numbers (QRNs). In this paper the convergence of a Monte Carlo method for evaluating the extremal eigenvalues of a given matrix is studied when quasirandom sequences are used. An error bound is established and numerical experiments with large sparse matrices are performed using three different QRN sequences: Sobol', Halton and Faure. The results indicate: • An improvement in both the magnitude of the error and in the convergence rate that can be achieved when using QRNs in place of PRNs. • The high parallel efficiency established for Monte Carlo methods is preserved for quasi-Monte Carlo methods in this case. The execution time for computing an extremal eigenvalue of a large, sparse matrix on p processors is bounded by O(lN/p), where l is the length of the Markov chain in the stochastic process and N is the number of chains, both of which are independent of the matrix size.

Keywords

Assure Libr Terion 

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References

  1. 1.
    P. BRATLEY, B. L. FOX and H. NIEDERREITER, “Implementation and tests of low-discrepancy point sets,” ACM Trans. on Modeling and Comp. Simul., 2: 195–213, 1992Google Scholar
  2. 2.
    R. L. BURDEN and J. D. FAIRES, Numerical Analysis, Fifth Edition, Brooks/Cole Publishing Company, Pacific Grove, California, 1996.Google Scholar
  3. 3.
    R. E. CAFLISCH, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numerica, 7: 1–49, 1998.MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. DIMOV, A. KARAIVANOVA, “Parallel computations of eigenvalues based on a Monte Carlo approach,” Journal of Monte Carlo Methods and Applications, Vol.4, Num.1, pp.33–52, 1998.Google Scholar
  5. 5.
    H. FAURE, “Discrepance de suites associées à un système de numération (en dimension s),” Acta Arithmetica, XLI: 337–351, 1982.Google Scholar
  6. 6.
    G. H. GOLUB and C. F. VAN LOAN, Matrix computations, Second Edition, The Johns Hopkins University Press, Baltimore, 1996.MATHGoogle Scholar
  7. 7.
    J. H. HALTON, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math., 2: 84–90, 1960.MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. M. HAMMERSLEY and D. C. HANDSCOMB, Monte Carlo Methods, Methuen, London, 1964.MATHCrossRefGoogle Scholar
  9. 9.
    J. F. KOKSMA, “Een algemeene Stelling uit de théorie der gelijkmatige verdeeling modulo 1,” Mathematica B (Zutphen), 11: 7–11, 1942/43.MathSciNetGoogle Scholar
  10. 10.
    M. MASCAGNI, A. KARAIVANOVA and Y. LI, “A Quasi-Monte Carlo Method for Elliptic Partial Differential Equations,” Monte Carlo Methods and Applications, in the press, 2001.Google Scholar
  11. 11.
    M. MASCAGNI and A. KARAIVANOVA, “Are Quasirandom Numbers Good for Anything Besides Integration?” in Proceedings of Advances in Reactor Physics and Mathematics and Computation into the Next Millennium (PHYSOR2000), 2000.Google Scholar
  12. 12.
    M. MASCAGNI and A. SRINIVASAN, “Algorithm 806: SPRNG: A Scalable Library for Pseudorandom Number Generation,” ACM Transactions on Mathematical Software, 26: 436–461, 2000, and at the URL http://sprng.cs.fsu.edu..edu.CrossRefGoogle Scholar
  13. 13.
    B. MOSKOWITZ and R. E. CAFLISCH, “Smoothness and dimension reduction in quasi-Monte Carlo methods”, J. Math. Comput. Modeling, 23: 37–54, 1996.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    H. NIEDERREITER, “Low-discrepancy and low-dispersion sequences,” J. Number Theory, 30: 51–70, 1988.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    H. NIEDERREITER, Random number generation and quasi-Monte Carlo methods, SIAM: Philadelphia, 1992.MATHCrossRefGoogle Scholar
  16. 16.
    K. F. ROTH, “On irregularities of distribution,” Mathematika, 1: 73–79, 1954.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    I. M. SOBOL', “The distribution of points in a cube and approximate evaluation of integrals,” Zh. Vychisl. Mat. Mat Fiz., 7: 784–802, 1967.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Michael Mascagni
    • 2
  • Aneta Karaivanova
    • 2
    • 1
  1. 1.Bulgarian Academy of SciencesCentral Laboratory for Parallel ProcessingSofiaBulgaria
  2. 2.Department of Computer ScienceFlorida State UniversityTallahasseeUSA

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