We consider a discrete elliptic equation with random coefficients $A$, which (to fix ideas) are identically distributed and independent from grid point to grid point $x\in\mathbb{Z}^d$. On scales large w.\ r.\ t.\ the grid size (i.\ e.\ unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ``homogenized'' coefficients $A_{hom}$ are characterized by % $$ \xi\cdot A_{hom}\xi\;=\;\langle\left((\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)\right)(0)\rangle, \quad\xi\in\mathbb{R}^d, $$ % where the random field $\phi$ is the unique stationary solution of the ``corrector problem'' % $$ -\nabla\cdot A(\xi+\nabla\phi)\;=\;0 $$ % and $\langle\cdot\rangle$ denotes the ensemble average. \medskip It is known (``by ergodicity'') that the above ensemble average of the energy density $e=(\xi+\nabla\phi)\cdot A(\xi+\nabla\phi)$, 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 $e$ on length scales $L$ is estimated as follows: % $$ {\rm var}\left[\sum_{x\in\mathbb{Z}^d}\eta_L(x)\,e(x)\right] \;\lesssim\;L^{-d}, $$ % where the averaging function (i.\ e.\ $\sum_{x\in\mathbb{Z}^d}\eta_L(x)=1$, ${\rm supp}\eta_L\subset[-L,L]^d$) has to be smooth in the sense that $|\nabla\eta_L|\lesssim L^{-1}$. In two space dimensions (i.\ e.\ $d=2$), there is a logarithmic correction. \medskip In other words, smooth averages of the energy density $e$ behave like as if $e$ would be independent from grid point to grid point (which it is not for $d>1$). This result is of practical significance, since it allows to estimate the error when numerically computing $A_{hom}$.

/pdf/an-optimal-error-estimate-in-stochastic-homogenization-of-538k7nm1wg.pdf

An optimal error estimate in stochastic homogenization of discrete elliptic equations

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice $$\mathbb Z^d$$
 with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions $$d>2$$
 ) and prove new optimal estimates for regularized versions of the corrector (in dimensions $$d\ge 2$$
 ). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.

/pdf/quantification-of-ergodicity-in-stochastic-homogenization-4nnnad9u4v.pdf

Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics

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 $A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$ is characterized by $\xi\cdot A_{\mathrm {hom}}\xi=\langle(\xi+\nabla\phi )\cdot A(\xi+\nabla\phi)\rangle$ for any direction $\xi\in\mathbb {R}^d$, where the random field $\phi$ (the ``corrector'') is the unique solution of $-\nabla^*\cdot A(\xi+\nabla\phi)=0$ such that $\phi(0)=0$, $\nabla\phi$ is stationary and $\langle\nabla\phi\rangle=0$, $\langle\cdot\rangle$ 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., $\sum\eta_L=1$, $\operatorname {supp}(\eta_L)\subset\{|x|\le L\}$] 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 $a_{\mathrm {hom}}$.

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

We study the effective large-scale behavior of discrete elliptic equations on the lattice $\mathbb Z^d$ with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w.~r.~t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i.~e. for the ''random environment as seen from a random walker''). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension $d>2$) and optimal estimates for regularized versions of the corrector (in dimensions $d\geq 2$). We also give a self-contained proof for a new estimate on the gradient of the parabolic, variable-coefficient Green's function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative methods, we develop an error analysis that gives optimal bounds in terms of scaling in the size of the representative volume element --- even for large ellipticity ratios.

/pdf/quantification-of-ergodicity-in-stochastic-homogenization-4fodd60pia.pdf

Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics --- long version

http://www.cc.ac.cn/05research_report/0518.pdf

Numerical methods for multiscale elliptic problems

The author considers a natural class of difference equations whose coefficients have micro-inhomogeneities. A general compactness theorem is established, asserting that the solutions of these difference equations can converge only to solutions of differential equations as the lattice is refined. Difference equations with random micro-inhomogeneous coefficients are studied separately; their averaging properties are determined.Bibliography: 12 titles.

S M Kozlov

Papers

Averaging of difference schemes