An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Abstract

We consider a discrete elliptic equation on the $d$-dimensional lattice ${\mathbb{Z}}^d$ 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 $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $\xi \cdot A_{\mathrm{hom}}\xi =⟨(\xi +\nabla \varphi )\cdot A(\xi +\nabla \varphi )⟩$ for any direction $\xi \in {\mathbb{R}}^d$, where the random field $\varphi$ (the ``corrector'') is the unique solution of $-{\nabla }^{\ast }\cdot A(\xi +\nabla \varphi )=0$ such that $\varphi (0)=0$, $\nabla \varphi$ is stationary and $⟨\nabla \varphi ⟩=0$, $⟨\cdot ⟩$ denoting the ensemble average (or expectation). It is known (``by ergodicity'') that the above ensemble average of the energy density $\mathcal{E}=(\xi +\nabla \varphi )\cdot A(\xi +\nabla \varphi )$, 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, $\mathrm{var}[\sum \mathcal{E}{\eta }_L]\lesssim L^{-d}$, where the averaging function [i.e., $\sum {\eta }_L=1$, $\mathrm{supp}({\eta }_L)\subset \{|x|\le L\}$] has to be smooth in the sense that $|\nabla {\eta }_L|\lesssim L^{-1-d}$. 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 $a_{\mathrm{hom}}$.

arXiv commentPublished in at http://dx.doi.org/10.1214/10-AOP571 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (37)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Recent progress on the Random Conductance Model
• The additive structure of elliptic homogenization
• Lipschitz regularity for elliptic equations with random coefficients
• Disorder, entropy and harmonic functions
• Invariance principle for the random conductance model in a degenerate ergodic environment
• Correlation structure of the corrector in stochastic homogenization
• Mesoscopic higher regularity and subadditivity in elliptic homogenization
• Quantitative stochastic homogenization and regularity theory of parabolic equations
• Quantitative estimates in stochastic homogenization for correlated coefficient fields
• Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas
• Noise-stability and central limit theorems for effective resistance of random electric networks
• Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity
• The choice of representative volumes in the approximation of effective properties of random materials
• Bounded correctors in almost periodic homogenization
• A central limit theorem for the effective conductance: Linear boundary data and small ellipticity contrasts
• A diagram-free approach to the stochastic estimates in regularity structures
• Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters
• Optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form
• Some variance reduction methods for numerical stochastic homogenization
• A Fuzzy-Stochastic Multiscale Model for Fiber Composites: A one-dimensional study
• Propagation of seismic waves through a spatio-temporally fluctuating medium: Homogenization
• On the averaged Green's function of an elliptic equation with random coefficients
• Higher-order linearization and regularity in nonlinear homogenization
• Numerical Methods for Multilattices
• Computation of effective elastic moduli of rocks using hierarchical homogenization
• The random heat equation in dimensions three and higher: the homogenization viewpoint
• An informal introduction to quantitative stochastic homogenization
• Quantitative version of the Kipnis-Varadhan theorem and Monte Carlo approximation of homogenized coefficients
• Quantitative nonlinear homogenization: control of oscillations
• When are increment-stationary random point sets stationary?
• Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media
• Uniform estimate of an iterative method for elliptic problems with rapidly oscillating coefficients
• Quantitative stochastic homogenization of elliptic equations in nondivergence form
• Homogenization theory of random walks among deterministic conductances
• Coupling between Brownian motion and random walks on the infinite percolation cluster
• Surface lattice Green's functions for high-entropy alloys