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Discrete random walk on large spherical grids generated by spherical means for PDEs

Author

Listed:
  • Sabelfeld K. K.

    (Weierstrass Institute for Applied Analysis and Stochastics Mohrenstraße 39, D - 10117 Berlin, Germany, E-mail: sabelfel@wias-berlin.de)

  • Shalimova I. A.

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sci. Lavrentieva str., 6, 630090 Novosibirsk, Russia, E-mails: , lai@osmf.sscc.ru ias@osmf.sscc.ru)

  • Levykin A. I.

    (Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sci. Lavrentieva str., 6, 630090 Novosibirsk, Russia, E-mails: , lai@osmf.sscc.ru ias@osmf.sscc.ru)

Abstract

A new general stochastic-deterministic approach for a numerical solution of boundary value problems of potential and elasticity theories is suggested. It is based on the use of the Poisson-like integral formulae for overlapping spheres. An equivalent system of integral equations is derived and then approximated by a system of linear algebraic equations. We develop two classes of special Monte Carlo iterative methods for solving these systems of equations which are a kind of stochastic versions of the Chebyshev iteration method and successive overrelaxation method (SOR). In the case of classical potential theory this approach accelerates the convergence of the well known Random Walk on Spheres method (RWS). What is however much more important, this approach suggests a first construction of a fast convergent finite-variance Monte Carlo method for the system of Lamé equations.

Suggested Citation

  • Sabelfeld K. K. & Shalimova I. A. & Levykin A. I., 2004. "Discrete random walk on large spherical grids generated by spherical means for PDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 10(3-4), pages 559-574.
    • Handle: RePEc:bpj:mcmeap:v:10:y:2004:i:3-4:p:559-574:n:1037
    DOI: 10.1515/mcma.2004.10.3-4.559
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