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

@article{Gloria2011AnOV,
title={An optimal variance estimate in stochastic homogenization of discrete elliptic equations},
author={Antoine Gloria and Felix Otto},
journal={Annals of Probability},
year={2011},
volume={39},
pages={779-856}
}
• Published 7 April 2011
• Mathematics
• Annals of Probability
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 % $$\xi\cdot A_{hom}\xi… 217 Citations An optimal error estimate in stochastic homogenization of discrete elliptic equations • Mathematics • 2012 We consider a discrete elliptic equation with random coefficients A, which (to fix ideas) are identically distributed and independent from grid point to grid point x\in\mathbb{Z}^d. On scales Homogenization of iterated singular integrals with applications to random quasiconformal maps. • Mathematics • 2020 We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (F_j)_{j Quantitative Homogenization and Convergence of Moving Averages It is shown that local averages can result in faster convergence: for example, if a(x) = a(1-x), then for x \in (\varepsilon, 1-\varpsilon) , then subtracting an explicitly given linear function (depending on a(\cdot),f results in the same bound. The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint • Mathematics Archive for Rational Mechanics and Analysis • 2021 We consider the stochastic heat equation \partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u, driven by a smooth space-time stationary Gaussian random field V(s,y), in dimensions d\geq Boundary Estimates in Elliptic Homogenization For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: Lp Estimates • Mathematics, Computer Science • 2015 The Lp convergence results for solutions of the above system and for the non oscillating operator A_\varepsilon(x) = A(x)} Aε(x)=A(x), with the following convergence rate for all. Annealed estimates on the Green function • Mathematics • 2013 We consider a random, uniformly elliptic coefficient field$$a(x)$$a(x) on the$$d$$d-dimensional integer lattice$$\mathbb {Z}^d$$Zd. We are interested in the spatial decay of the quenched elliptic Normal approximation for a random elliptic equation We consider solutions of an elliptic partial differential equation in$$\mathbb{R }^d$$Rd with a stationary, random conductivity coefficient that is also periodic with period$$LL. Boundary
Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations
We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible:
Optimal Artificial Boundary Condition for Random Elliptic Media
• Computer Science, Mathematics
Found. Comput. Math.
• 2021
An error estimate is rigorously established that the order of the error estimate in both $L$ and $\ell$ is optimal, and this amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain.