Monte Carlo solution of the Neumann problem for the nonlinear Helmholtz equation

In this paper we will consider the Neumann boundary-value problem for the Helmholtz equation with a polynomial nonlinearity on the right-hand side. We will assume that a solution to our problem exists, and this permits us to construct an unbiased Monte Carlo estimator using the trajectories of certain branching processes. To do so we utilize Green’s formula and an elliptic mean-value theorem. This allows us to derive a special integral equation, which equates the value of the function u(x)u(x) at the point xx with its integral over the domain DD and on boundary of the domain ∂D=G∂D=G. The solution of the problem is then given in the form of a mathematical expectation over some particular random variables. According to this probabilistic representation, a branching stochastic process is constructed and an unbiased estimator of the solution of the nonlinear problem is formed by taking the expectation over this branching process. The unbiased estimator which we derive has a finite variance. In addition, the proposed branching process has a finite average number of branches, and hence is easily simulated. Finally, we provide numerical results based on numerical experiments carried out with these algorithms to validate our approach.