epsilon-shell error analysis for "Walk On Spheres" algorithms
The "Walk On Spheres" (WOS) algorithm and its relatives have long been used to solve a wide variety of boundary value problems [Ann. Math. Stat. 27 (1956) 569; J. Heat Transfer 89 (1967) 121; J. Chem. Phys. 100 (1994) 3821; J. Appl. Phys. 71 (1992) 2727]. All WOS algorithms that require the construction of random walks that terminate, employ an c-shell to ensure their termination in a finite number of steps. To remove the error related to this e-shell, Green's function first-passage (GFFP) algorithms have been proposed [J. Chem. Phys. 106 (1997) 3721] and used in several applications [Phys. Fluids A 12 (2000) 1699; Monte Carlo Meth. Appl. 7 (2001) 213; The simulation-tabulation method for classical diffusion Monte Carlo, J. Comput. Phys. submitted]. One way to think of the GFFP algorithm is as an epsilon = 0 extension of WOS. Thus, an important open question in the use of GFFP is to understand the tradeoff made in the efficiency of GFFP versus the c-dependent error in WOS. In this paper, we present empirical evidence and analytic analysis of the c-shell error in some simple boundary value problems for the Laplace and Poisson equations and show that the error associated with the epsilon-shell is O(epsilon), for small epsilon. This fact supports the conclusion that GFFP is preferable to WOS in cases where both are applicable. (C) 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.
error laplace poisson walk on spheres (wos)