An optimal error estimate in stochastic homogenization of discrete elliptic equations

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

An Optimal Quantitative Two-scale Expansion in Stochastic Homogenization of Discrete Elliptic Equations

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion...

Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations

We prove that the effective nonlinearities (ergodic constants) obtained in the stochastic homogenization of Hamilton-Jacobi, “viscous” Hamilton-Jacobi and nonlinear uniformly elliptic pde are approximated by the analogous quantities of appropriate “periodizations” of the equations. We also obtain an error estimate, when there is a rate of convergence for the stochastic homogenization.

Numerical Approximation of Effective Coefficients in Stochastic Homogenization of Discrete Elliptic Equations

We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: An elliptic equation on the d-dimensional lattice Z with independent and identically distributed conductivities on the associated edges. Recent results by Otto and the author quantify the error mad...

An Optimal Variance Estimate in Stochastic Homogenization of Discrete Elliptic Equations by Antoine Gloria and Felix Otto

We consider a discrete elliptic equation on the d-dimensional lattice Zd with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. Thi...