Walk-on-Hemispheres first-passage algorithm.

Due to the isomorphism between an electrostatic problem and the corresponding Brownian diffusion one, the induced charge density on a conducting surface by a charge is isomorphic to the first-passage probability of the diffusion initiated at the location of the charge. Based on the isomorphism, many diffusion algorithms such as "Walk-on-Spheres" (WOS), "Walk-on-Planes" and so on have been developed. Among them, for fast diffusion simulations WOS algorithm is generally applied with an [Formula: see text]-layer, which is used for diffusion convergence on the boundary but induces another error from the [Formula: see text]-layer in addition to the intrinsic Monte Carlo error.