Parallel Realization of Grid-Free Monte Carlo Algorithm for Boundary Value Problems

摘要

In many areas of the science there is a need to evaluate a functional of the solution of a given problem directly without computing the solution itself. The problem here is a linear functional of the solution of an elliptic boundary value problem to be estimated. Such kind of problems are similar to the air pollution problems in environmental sciences, where a rough estimate of the solution is acceptable. For practical computations it means that the relative error is about 5% - 10%. To solve this problem a grid-free Monte Carlo (MC) algorithm is used. The algorithm makes use of a Monte Carlo procedure called "Walk on the balls". Here we consider parallel realizations of the considered grid-free MC algorithm. Various numerical results are obtained by the implementation of the proposed parallel algorithms on several high performance machines: IBM +p690 Regata system and Sun Fire 15K server. One can see that the efficiency of the proposed parallel algorithm is close to 100%.