Monte Carlo methods for solving boundary value problems of the second and third kinds

In this paper we consider two ways of obtaining the estimates of the solutions of boundary value problems of the second and third kinds for the Helmholtz multidimensional equation Delta u - cu = -g. In the framework of both approaches we obtain the global estimates of the solutions and the estimates of the solutions at a single point. We prove that the variance is finite. In the first case the estimates of the solutions are based on the 'walk on spheres' process with reflection from the boundary. We consider different ways of termination of the trajectory both in the interior of the domain and in a neighbourhood of the boundary. In the second case we study the algorithms of the Monte Carlo method for solving the systems of algebraic equations, which correspond to the standard difference approximation of the boundary value problems studied. Finally, we obtain the global estimates of the solution of the problem and solve the problem of minimizing the computational cost.