Monte Carlo algorithms for numerical estimation of the elliptic BVP solution and its gradient

We propose numerical algorithms for estimating the solution of the boundary value problems for elliptic equation and the solution gradient. The problem of error reduction is considered for deterministic as well as statistical computational errors. We also propose external extrapolation for the Euler scheme in order to reduce the deterministic error. A variant of antithetic variates method is suggested for reducing the variance of the weight Monte Carlo estimate for the solution gradient. Numerical results are presented for Dirichlet BVP