The Annals of Probability

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Antoine Gloria and Felix Otto

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Abstract

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. This symmetric “homogenized” matrix Ahom=ahom Id is characterized by ξAhomξ=〈(ξ+∇ϕ)⋅A(ξ+∇ϕ)〉 for any direction ξ∈ℝd, where the random field ϕ (the “corrector”) is the unique solution of −∇A(ξ+∇ϕ)=0 such that ϕ(0)=0, ∇ϕ is stationary and 〈∇ϕ〉=0, 〈⋅〉 denoting the ensemble average (or expectation).

It is known (“by ergodicity”) that the above ensemble average of the energy density $\mathcal {E}=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of $\mathcal {E}$ on length scales L satisfies the optimal estimate, that is, $\operatorname{var}[\sum \mathcal {E}\eta_{L}]\lesssim L^{-d}$, where the averaging function [i.e., ∑ηL=1, supp(ηL)⊂{|x|≤L}] has to be smooth in the sense that |∇ηL|≲L−1−d. In two space dimensions (i.e., d=2), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density $\mathcal {E}$ decay in L as if $\mathcal {E}$ would be independent from edge to edge (which it is not for d>1).

This result is of practical significance, since it allows to estimate the dominant error when numerically computing ahom.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 779-856.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281725

Digital Object Identifier
doi:10.1214/10-AOP571

Mathematical Reviews number (MathSciNet)
MR2789576

Zentralblatt MATH identifier
1215.35025

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 39A70: Difference operators [See also 47B39] 60H25: Random operators and equations [See also 47B80] 60F99: None of the above, but in this section

Keywords
Stochastic homogenization variance estimate difference operator

Citation

Gloria, Antoine; Otto, Felix. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011), no. 3, 779--856. doi:10.1214/10-AOP571. https://projecteuclid.org/euclid.aop/1300281725


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