Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

Abstract

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman-Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorohod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres (WOS) method, approximation of the boundary local time is obtained and the Feynman-Kac formula is calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

“…with a constant parameter q = κ/D ≥ 0 (see [44][45][46] for mathematical details and references). When the domain Ω is unbounded, one also needs to impose a regularity condition at infinity: G q (x, t|x 0 ) → 0 as |x| → ∞ (similar condition has to be imposed for the related boundary value problems (12,18,19), see below).…”

“…where we used the L 2 (∂Ω)-normalization of (0) as an eigenfunction of M 0 , and (v (0 In order to validate our analytical results and the quality of the numerical Laplace transform inversion, we undertake Monte Carlo simulations of reflected Brownian motion with diffusion coefficient D inside a disk and a ball of radius R. We employ a basic fixed time-step scheme, even though more advanced Monte Carlo techniques are available [46,[79][80][81][82]. We set R = 1 and D = 1 to fix units of length and time.…”

What is the fractional Laplacian? A comparative review with new results

“…The walk-on-spheres method is a type of Monte Carlo method for simulating solutions to the Dirichlet fractional Poisson problem with both zero and nonzero boundary conditions. It was originally proposed by Muller [99] in 1956 for solving Laplace equations with Dirichlet boundary conditions (see also [100]), and has been used for Neumann boundary conditions [101], and Robin boundary conditions [102] as well. This approach has recently been extended [22] to the following Riesz fractional Poisson problem:…”

A Path Integral Monte Carlo Method based on Feynman-Kac Formula for Electrical Impedance Tomography

“…In the context of complete electrode model, the second expectation is zero due to the zero Neumann boundary. One may find more details in [33].…”