Solving Poisson's equation in high dimensions by a hybrid Monte-Carlo finite difference method

We introduce and implement a hybrid Monte-Carlo finite difference method for approximating the solution of Poisson's equation. This method solves smaller problems multiple times to collectively solve a larger main problem, when the solution of the main problem is unattainable by known regular direct and iterative methods. The method thereby resolves features that a single smaller problem may not. This hybrid Monte-Carlo finite difference method achieves second order accuracy on generic problems, and on problems with sharp features.