Monte Carlo Solution of Semi-linear Helmholtz Boundary Value Problem

In these work we will study a probabilistic representation of the solution of the Helmholtz boundary problem for the non-linear problem \(\begin{aligned} -\Delta u(x) + cu(x) = g \cdot f(u),\quad x \in D,\quad u{|_\Gamma } = \psi \end{aligned}\) where, f(u) in our case could be hyperbolic functions sh(u) or ch(u). Under the assumption of the existence of a solution, an unbiased estimator is constructed on the trajectories of the proposed branching process “walk on spheres”. To do this, using Green’s formula, a special integral equation is written that connects the value of the function with its integrals over a ball and a sphere of maximum radius centered at a point and entirely contained in the region under consideration. A probabilistic representation of the solution of the problem in the form of the mathematical expectation of some random variable is obtained. In accordance with the probabilistic representation, a branching process of walk on spheres is constructed and an unbiased estimator of the solution of the problem with finite variance is constructed on its trajectories.