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A global random walk on grid algorithm for second order elliptic equations

  • Karl K. Sabelfeld ORCID logo EMAIL logo and Dmitrii Smirnov

Abstract

We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

MSC 2010: 65C05; 65C40; 65Z05

Award Identifier / Grant number: 19-11-00019

Award Identifier / Grant number: 20-51-18009

Funding statement: This work of the GRWG algorithm development is supported by the Russian Science Foundation under Grant 19-11-00019, and the Russian Foundation for Basic Research under Grant 20-51-18009 in the part of random walk on grid process implementations.

Acknowledgements

Thanks also go to I. A. Shalimova for the results in the numerical example of Section 6.2 obtained by the random walk on ellipsoids algorithm.

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Received: 2021-02-09
Revised: 2021-06-30
Accepted: 2021-07-07
Published Online: 2021-08-08
Published in Print: 2021-09-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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