Green’s function Monte Carlo algorithms for elliptic problems

In many large-scale problems, one is interested to obtain directly an approximate value of a functional of the solution. Here, we consider a special class of grid-free Monte Carlo algorithms for direct computing of linear functionals of the solution of an elliptic boundary-value problem. Such kind of problems appear in environmental sciences, computational physics and financial mathematics. To create the algorithms, we use the Green’s function analysis and define the conditions under which the integral transformation kernel of the integral representation for the boundary-value problem under consideration is non-negative. This analysis is done for a possible set of densities, and it is used to generate three different grid-free Monte Carlo algorithms based on different choices of the density of the radius of the balls used in Monte Carlo simulation. Only one of the generated algorithms was known before. We shall call it Sipin’s algorithm. It was proposed and studied by Sipin. The aim of this work is to study the two new algorithms proposed here and based on two other (than in Sipin’s algorithm) possible choices of the densities. The algorithms are described and analyzed. The performed numerical tests show that the efficiency of one of the new algorithms, which is based on a constant density is higher than the efficiency of Sipin’s algorithm.