Approximate homogenization of convex nonlinear elliptic PDEs

Abstract

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

Approximate Homogenization of Fully Nonlinear Elliptic PDEs: Estimates and Numerical Results for Pucci Type Equations

“…For separable examples, the linear approximation is still effective. However, we also found nonseparable examples where the linear approximation is poor, which we will address in a companion paper [FO17] with a closer bound.…”

“…Remark 2.7. In a companion paper [FO17], we show that for convex operators in one dimension, Corollary 2.8. Consider the operator H a1,a2 given by (2).…”

“…we may have control of D 2 u Q for small values of Q. This is the case in one dimension in [FO17], where we obtain an analytical formula for u Q xx through (16), which gives |u Q xx | ≤ C|Q|. The main theorem is a formal result in the sense that it relies on the fact that u Q is a classical solution, which does not hold in general.…”

Discontinuous Galerkin and C0-IP finite element approximation of periodic Hamilton–Jacobi–Bellman–Isaacs problems with application to numerical homogenization

“…The numerical homogenization of HJB equations via a mixed finite element approximation of the approximate correctors has been proposed and analyzed in Gallistl, Sprekeler, Süli [25]. A finite difference approach for numerical effective Hamiltonians to HJB operators can be found in Camilli, Marchi [13], and some exact formulas and numerical simulations for effective Hamiltonians to certain types of HJB operators are available in Finlay, Oberman [21,22].…”

Mixed Finite Element Approximation of Periodic Hamilton--Jacobi--Bellman Problems With Application to Numerical Homogenization

“…For the case of second-order HJB equations, a finite difference scheme for the whole-space problem has been proposed in Camilli and Marchi [10]. In Finlay and Oberman [18,19], the effective Hamiltonian is computed exactly for HJB operators of certain types and numerical simulations have been conducted. It seems that finite element schemes for the numerical homogenization of the problem (1.3) have not yet been constructed.…”