Advertisement

Iterative Monte Carlo algorithms for linear algebra problems

  • I. T. Dimov
  • A. N. Karaivanova
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1196)

Abstract

A common Monte Carlo approach for linear algebra problems is presented. The considered problems are inverting a matrix B, solving systems of linear algebraic equations of the form Bu=b and calculating eigenvalues of symmetric matrices. Several algorithms using the same Markov chains with different random variables are described.

The presented algorithms contain iterations with a resolvent matrix (used as iterative operator) of a given matrix. For inverting matrices and solving linear systems a mapping of the spectral parameter domain of convergence is established. This transformation leads to a new resolvent matrix and, respectively, to a new random variable constructed on the corresponding Markov chain, which allows the use of smaller number of iterations for reaching a given error. For calculating of eigenvalues an additional parameter in resolvent operator is involved to accelerate the algorithm convergence. The convergence of the iterative processes is proved and the convergence rate is compared with the rate of existing Monte Carlo algorithms for similar problems.

An error analysis is done.

Numerical experiments for Monte Carlo Almost Optimal (MAO) algorithms are performed on CRAY Y-MP C92A. It is shown that the accuracy and the algorithm complexity practically does not depend on the size of the matrix.

Key words

Monte Carlo method matrix computations convergence eigenvalues convergent iterative process 

MSC subject classification

65 C 05 65 U 05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Cu54]
    Curtiss, J.H.: Monte Carlo methods for the iteration of linear operators. J. Math Phys. 32, No 4 (1954) 209–232.Google Scholar
  2. [Cu56]
    Curtiss, J.H.: A Theoretical Comparison of the Efficiencies of two classical methods and a Monte Carlo method for Computing one component of the solution of a set of Linear Algebraic Equations. Proc. Symposium on Monte Carlo Methods, John Wiley and Sons (1956) 191–233.Google Scholar
  3. [Di91]
    Dimov, I.: Minimization of the Probable Error for Some Monte Carlo methods. Proc. Int. Conf. on Mathematical Modeling and Scientific Computation, Varna (1991).Google Scholar
  4. [DT93]
    Dimov, I., Tonev, O.: Random walk on distant mesh points Monte Carlo methods. Journal of Statistical Physics 70(5/6) (1993) 1333–1342.Google Scholar
  5. [DT93a]
    Dimov, I., Tonev, O.: Monte Carlo algorithms: performance analysis for some computer architectures. Journal of Computational and Applied Mathematics 48 (1993) 253–277.Google Scholar
  6. [DK96]
    Dimov, I., Karaivanova, A.: A Fast Monte Carlo Method for Matrix Computations, in Iterative Methods in Linear Algebra II, IMACS Series in Computational and Applied Mathematics (S. Margenov and P.S. Vassilevski eds.), (1996) 204–213.Google Scholar
  7. [Ka50]
    Kahn, H.: Random sampling (Monte Carlo) techniques in neutron attenuation problems. Nucleonics 6 No 5 (1950), 27–33; 6, No 6 (1950) 60–65.Google Scholar
  8. [KA64]
    Kantorovich, L.V., Akilov, S.P.: Functional analysis in normed spases. Pergamon Press, New York, 1964.Google Scholar
  9. [MAD94]
    Megson, G., Aleksandrov, V., Dimov, I.: Systolic Matrix Inversion Using a Monte Carlo Method. Journal of Parallel Algorithms and Applications 3, No 1 (1994) 311–330.Google Scholar
  10. [Mi87]
    Mikhailov, G.A.: Optimization of the ”weight” Monte Carlo methods. Nauka, Moscow, 1987.Google Scholar
  11. [So73]
    Sobol, I.M.: Monte Carlo numerical methods. Nauka, Moscow, 1973.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • I. T. Dimov
    • 1
  • A. N. Karaivanova
    • 1
  1. 1.Central Laboratory for Parallel ComputingBulgarian Academy of SciencesA SofiaBulgaria

Personalised recommendations