An optimal error estimate in stochastic homogenization of discrete elliptic equations

@article{Gloria2012AnOE,
title={An optimal error estimate in stochastic homogenization of discrete elliptic equations},
author={Antoine Gloria and Felix Otto},
journal={Annals of Applied Probability},
year={2012},
volume={22},
pages={1-28}
}
• Published 1 February 2012
• Mathematics
• Annals of Applied 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 % $$\xi\cdot A_{hom}\xi… 146 Citations Boundary Estimates in Elliptic Homogenization For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint • Mathematics Archive for Rational Mechanics and Analysis • 2021 We consider the stochastic heat equation \partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u, driven by a smooth space-time stationary Gaussian random field V(s,y), in dimensions d\geq Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations We introduce and analyze a numerical strategy to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. In particular, we consider the simplest case possible: Localized Bases for Finite-Dimensional Homogenization Approximations with Nonseparated Scales and High Contrast • Mathematics Multiscale Model. Simul. • 2011 Finite-dimensional approximations of solution spaces of divergence form operators with L^\infty-coefficients are constructed that can be generalized to vectorial equations (such as elasto-dynamics) and for high-contrast media, the accuracy is preserved. Normal approximation for the net flux through a random conductor We consider solutions of an elliptic partial differential equation in$${\mathbb R}^d$$Rd with a stationary, random conductivity coefficient. The boundary condition on a square domain of width L is Quantitative results on the corrector equation in stochastic homogenization • Mathematics • 2014 We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions d\ge 2. In previous works we studied the model problem of a discrete elliptic An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations • Mathematics • 2014 We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances • Mathematics Stochastics and Partial Differential Equations: Analysis and Computations • 2018 We study the random conductance model on the lattice$${\mathbb {Z}}^dZd, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random
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