Solving the Multidimensional Difference Biharmonic Equation by the Monte Carlo Method

Abstract

We construct and justify new weighted Monte Carlo methods for estimation of a solution to the Dirichlet problem for the multidimensional difference biharmonic equation by modeling a “random walk by a grid.” Vector versions of our algorithms extend to the difference metaharmonic equations, with the shape of unbiasedness conditions of estimators preserved together with boundedness of their variances. In this connection, we construct a simple algorithm for estimation of the first eigenvalue of the multidimensional difference Laplace operator. Moreover, we construct special algorithms of a “random walk by a grid” which under certain conditions allow us to estimate solutions of the Dirichlet problem for the biharmonic equation with a weak nonlinearity as well as solutions to problems with mixed boundary conditions, the Neumann condition inclusively.