Estimation of derivatives of solutions to boundary value problems by Monte Carlo methods

Monte Carlo methods offer the possibility to estimate derivatives of a solution to a boundary value problem even without estimating the solution to the problem making them attractive in various practical applications. In this paper, Monte Carlo methods are applied to estimate derivatives of a solution to the third boundary value problem of the diffusion equation with a constant complex parameter with respect to the parameter as well as with respect to a spatial variable. Moreover, the estimation of the spatial derivative of non-centric Green's function for the diffusion operator in a ball is used. All the derivatives estimators are obtained by the method of “walking on spheres” with boundary reflection. Based on the derivatives estimators of the solution to the diffusion equation problem, derivatives of a solution of the heat equation boundary value problem with respect to the time variable, the spatial variable, and a constant real parameter are obtained.