A global root-finding method for high dimensional problems

A method to solve the problem f(x) = 0 efficiently on any n-dimensional domain Ω under very broad hypoteses is proposed. The position of the root of f , assumed unique, is found by computing the center of mass of an Ω-shaped object having a singular mass density. It is shown that although the mass of the object is infinite, the position of its center of mass can be computed exactly and corresponds to the solution of the problem. The exact analytical result is implemented numerically by means of an adaptive Monte Carlo sampling technique which provides an exponential rate of convergence. The method can be extended to functions with multiple roots, providing an efficient automated root finding algorithm.