  We approximate the homogenization of fully nonlinear, convex, uniformly
elliptic Partial Differential Equations in the periodic setting, using a
variational formula for the optimal invariant measure, which may be derived via
Legendre-Fenchel duality. The variational formula expresses $\bar H$ as an
average of the operator against the optimal invariant measure, generalizing the
linear case. Several nontrivial analytic formulas for $\bar H$ are obtained.
These formulas are compared to numerical simulations, using both PDE and
variational methods. We also perform a numerical study of convergence rates for
homogenization in the periodic and random setting and compare these to
theoretical results.
