Analysis and comparison of Green’s function first-passage algorithms with “Walk on Spheres” algorithms

We analyze the optimization of the running times of Green’s function first-passage (GFFP) algorithms. The running times for these new first-passage (FP) algorithms [J. Chem. Phys. 106 (9) (1997) 3721; Phys. Fluids A 12 (7) (2000) 1699; J. Comput. Phys. 174 (2) (2001) 925; Monte Carlo Meth. Appl. 7 (3–4) (2001) 213], which use exact Green’s functions for the Laplacian to eliminate the absorption layer in the “Walk on Spheres” (WOS) method [Ann. Math. Stat. 27 (1956) 569; J. Heat Transfer 89 (1967) 121; J. Chem. Phys. 100 (5) (1994) 3821; J. Appl. Phys. 71 (6) (1992) 2727; J. Comput. Phys. 39 (1981) 396], are compared with those for WOS algorithms. It has been empirically observed that GFFP algorithms are more efficient than WOS algorithms when high accuracy is required [Phys. Fluids A 12 (7) (2000) 1699; J. Comput. Phys. 174 (2) (2001) 925; Monte Carlo Meth. Appl. 7 (3–4) (2001) 213]. Additionally, it has been observed that there is always an optimal distance from the surface of the absorbing boundary, δI, for a GFFP algorithm within which a FP surface can be permitted to intersect the boundary [Phys. Fluids A 12(7) (2000) 1699; J. Comput. Phys. 174 (2) (2001) 925; Monte Carlo Meth. Appl. 7 (3–4) (2001) 213]. In this paper, we will provide a rigorous complexity analysis consistent with these observations. This analysis is based on estimating the numbers of WOS and GFFP steps needed for absorption on the boundary, and the complexity and running times of each WOS and GFFP step. As an illustration, we analyze the running times for calculating the capacitance of the unit cube using both GFFP and WOS.