Monte Carlo methods for solving elliptic equations with boundary conditions containing the normal derivative

The development of new statistical modeling methods to use algorithm of walk on spheres after the trajectory reaches the reflecting boundary, is discussed. It is based on a relation for the mean value value of a function at a point on the boundary. It makes possible to improve the performance of the stochastic computational algorithm. A mean-value relation for the solution is constructed at a point x on the boundary of the domain. The value of the solution at a boundary point is estimated using the developed relations. The right-hand side of the relations are regarded as an integral operator that transform functions defined on the entire space into functions with domain τ. The kernel of the operator may be alternating, and the convergence of the von Neumann series after replacing the kernel by its modulus cannot be ensured. The approach applies to the exterior von Neumann problem virtually without changes.