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arXiv 1104.1291math.PR
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离散椭圆型方程随机均匀化的最优方差估计

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Antoine Gloria, Felix Otto

备注：
Published 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)
发表信息：
Annals of Probability 2011, Vol. 39, No. 3, 779-856
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摘要：

我们考虑一个离散椭圆方程的$d$ -维格$\mathbb{Z}^d$随机系数$A$的最简单的类型：他们是同分布和独立的边缘。在大型w.r.t.晶格间距（即，单位），已知解算子表现得像具有常数确定性系数的（连续）椭圆方程的解算子。该对称"均质化“矩阵$A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$的特征在于对于任何方向$ξ\in\mathbb {R}^d$的$ξ\cdot A_{\mathrm {hom}}ξ=\langle(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)\rangle$，其中随机场$ϕ$（”校正器“）是$-\nabla^*\cdot A(ξ+\nablaϕ)=0$的唯一解，使得$ϕ(0)=0$、$\nablaϕ$是固定的，并且$\langle\nablaϕ\rangle=0$、$\langle\cdot\rangle$表示总体平均（或期望）。已知（"通过遍历性“），能量密度$\mathcal {E}=(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)$的上述系综平均（其是平稳随机场）可以通过系统平均来恢复。我们通过证明长度尺度$L$上的$\mathcal {E}$的空间平均的方差满足最优估计，即$\operatorname {var}[\sum \mathcal {E}η_L]\lesssim L^{-d}$，来量化这一点，其中平均函数[即，$\sumη_L=1$，$\operatorname {supp}(η_L)\subset\{|x|\le L\}$ ]必须在$|\nablaη_L|\lesssim L^{-1-d}$的意义上平滑。在两个空间维度（即，$d=2$），则存在对数校正。该估计是最佳的，因为它表明能量密度$\mathcal {E}$的平滑平均值在$L$中衰减，就好像$\mathcal {E}$将从边缘到边缘独立（对于$d>1$不是这样）。这一结果是有实际意义的，因为它允许估计的主要误差时，数值计算$a_{\mathrm {hom}}$。

英文摘要：

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 $ξ\cdot A_{\mathrm {hom}}ξ=\langle(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)\rangle$ for any direction $ξ\in\mathbb {R}^d$, where the random field $ϕ$ (the ``corrector'') is the unique solution of $-\nabla^*\cdot A(ξ+\nablaϕ)=0$ such that $ϕ(0)=0$, $\nablaϕ$ is stationary and $\langle\nablaϕ\rangle=0$, $\langle\cdot\rangle$ denoting the ensemble average (or expectation). It is known (``by ergodicity'') that the above ensemble average of the energy density $\mathcal {E}=(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)$, 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, $\operatorname {var}[\sum \mathcal {E}η_L]\lesssim L^{-d}$, where the averaging function [i.e., $\sumη_L=1$, $\operatorname {supp}(η_L)\subset\{|x|\le L\}$] has to be smooth in the sense that $|\nablaη_L|\lesssim L^{-1-d}$. 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 $a_{\mathrm {hom}}$.

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