Approximate homogenization of convex nonlinear elliptic PDEs

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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 H as an average of the operator against the optimal invariant measure, generalizing the linear case. Several nontrivial analytic formulas for 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.