An optimal variance estimate in stochastic homogenization of discrete elliptic equations

Details An optimal variance estimate in stochastic homogenization of discrete elliptic equations An optimal variance estimate in stochastic homogenization of discrete elliptic equations

2011-04-07

arxiv.org/abs/1104.1291v1

Annals of Probability 2011, Vol. 39, No. 3, 779-856

Mathematics

Probability

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

Scientific paper

10.1214/10-AOP571

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 $a_{\mathrm {hom}}$.

Affiliated with

Also associated with

No associations

Rating

An optimal variance estimate in stochastic homogenization of discrete elliptic equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with An optimal variance estimate in stochastic homogenization of discrete elliptic equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and An optimal variance estimate in stochastic homogenization of discrete elliptic equations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-717892