  In this article, we study the application of Multi-Level Monte Carlo (MLMC)
approaches to numerical random homogenization. Our objective is to compute the
expectation of some functionals of the homogenized coefficients, or of the
homogenized solutions. This is accomplished within MLMC by considering
different levels of representative volumes (RVE), and, when it comes to
homogenized solutions, different levels of coarse-grid meshes. Many inexpensive
computations with the smallest RVE size and the largest coarse mesh are
combined with fewer expensive computations performed on larger RVEs and smaller
coarse meshes. We show that, by carefully selecting the number of realizations
at each level, we can achieve a speed-up in the computations in comparison to a
standard Monte Carlo method. Numerical results are presented both for
one-dimensional and two-dimensional test-cases.
