  In this paper, we propose and analyze a new stochastic homogenization method
for diffusion equations with random and fast oscillatory coefficients. In the
proposed method, the homogenized solutions are sought through a two-stage
procedure. In the first stage, the original oscillatory diffusion equation is
approximated, for each fixed random sample w, by a spatially homogenized
diffusion equation with piecewise constant coefficients, resulting a random
diffusion equation. In the second stage, the resulted random diffusion equation
is approximated and computed by using an efficient multi-modes Monte Carlo
method which only requires to solve a diffusion equation with a constant
diffusion coefficient and a random right-hand side. The main advantage of the
proposed method is that it separates the computational difficulty caused by the
spatial fast oscillation of the solution and that caused by the randomness of
the solution, so they can be overcome separately using different strategies.
The convergence of the solution of the spatially homogenized equation (from the
first stage) to the solution of the original random diffusion equation is
established and the optimal rate of convergence is also obtained for the
proposed multi-modes Monte Carlo method. Numerical experiments on some
benchmark test problems for random composite materials are also presented to
gauge the efficiency and accuracy of the proposed two-stage stochastic
homogenization method.
