We consider a discrete elliptic equation on the $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. spacing (i.e., unity), solution operator is known behave like (continuous) constant deterministic coefficients. This symmetric ``homogenized'' matrix $A_{\mathrm {hom}}=a_{\mathrm {hom}}\operatorname {Id}$ characterized by $ξ\cdot A_{\mathrm...

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. We consider a discrete elliptic equation on $d$-dimensional lattice $\mathbb{Z}^d$ with random coefficients $A$ of simplest type: They are identically distributed and independent from edge edge. On scales large w.r.t. spacing (i.e., unity), solution operator known behave like (continuous) constant deterministic coefficients. symmetric "homogenized" matrix $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ characterized by...

We derive optimal estimates in stochastic homogenization of linear elliptic equations divergence form dimensions d \geq 2 . In previous works we studied the model problem a discrete equation on \mathbb Z^d Under assumption that spectral gap estimate holds probability, proved there exists stationary corrector field > and energy density behaves as if it had finite range correlation terms variance spatial averages – latter decays at rate central limit theorem. this article extend these...

This paper is concerned with the approximation of effective coefficients in homogenization linear elliptic equations. One common drawback among numerical methods presence so-called resonance error, which roughly speaking a function ratio ε/η, where η typical macroscopic length scale and ε size heterogeneities. In present work, we propose an alternative for computation homogenized (or more generally modified cell-problem), first brick design methods. We show that this approach drastically...

A number of methods have been proposed in recent years to perform the numerical homogenization (possibly nonlinear) elliptic operators. These are usually defined at discrete level. Most them compute a operator, close, sense be made precise, homogenized operator for problem. The purpose present work is clarify construction this convex case by interpreting method continuous level and extend it quasiconvex setting. discretization new may performed several ways, recovering generalizing variety...

We establish an optimal, linear rate of convergence for the stochastic homogenization discrete elliptic equations. consider model problem independent and identically distributed coefficients on a discretized unit torus. show that difference between solution to random torus first two terms two-scale asymptotic expansion has same scaling as in periodic case. In particular $L^2$-norm probability \mbox{$H^1$-norm} space this error scales like $\epsilon$, where $\epsilon$ is discretization...

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of finite range dependence. show the gradient and flux $(\nablaϕ,a(\nabla ϕ+e))$ corrector $ϕ$, when spatially averaged over scale $R\gg 1$ decay like CLT scaling $R^{-\frac{d}{2}}$. establish this optimal rate on level sub-Gaussian bounds in terms stochastic integrability, also suboptimal Gaussian integrability. The proof unravels exploits self-averaging property associated...

This paper is about the homogenization of linear elliptic operators in divergence form with stationary random coefficients that have only slowly decaying correlations.It deduces optimal estimates error from growth (extended) corrector.In line heuristics, there are transitions at dimension d = 2, and for a correlation-decay exponent β 2; we capture correct power logarithms coming these two sources criticality.The decay correlations sharply encoded terms multiscale logarithmic Sobolev...

This work is devoted to the homogenization of elliptic equations in high-contrast media so-called `double-porosity' resonant regime, for which we solve two open problems literature random setting. First, prove qualitative under very weak conditions, that cover case inclusions are not uniformly bounded or separated. Second, stronger assumptions, provide sharp error estimates two-scale expansion.

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their analysis—which has been made possible by recent contributions on quantitative theory two the authors, Neukamm Otto. makes connection between our theoretical results computations. We give a complete picture found literature, compare them terms known (or expected) convergence rates empirically study them. Two types are presented:...

We establish quantitative results on the periodic approximation of corrector equation for stochastic homogenization linear elliptic equations in divergence form, when diffusion coefficients satisfy a spectral gap estimate probability, and d> 2. This work is based [5], which complete continuum version [6, 7] (with addition optimal d = 2). The main difference with respect to first part [5] that we avoid here use Green’s functions more directly rely De Giorgi-Nash-Moser theory.

Abstract We study a random conductance problem on d ‐dimensional discrete torus of size L > 0. The conductances are independent, identically distributed variables uniformly bounded from above and below by positive constants. effective A the network is variable, depending , that converges almost surely to homogenized hom . Our main result quantitative central limit theorem for this quantity as → ∞. In particular, we prove there exists some σ 0 such...

Consider an ergodic stationary random field A on the ambient space ℝ d . In order to establish concentration properties for nonlinear functions Z(A), it is standard appeal functional inequalities like Poincaré or logarithmic Sobolev in probability space. These are however only known hold a restricted class of laws (product measures, Gaussian measures with integrable covariance, more general Gibbs nicely behaved Hamiltonians). this contribution, we introduce variants these inequalities, which...

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: An equation on d-dimensional lattice with independent identically distributed conductivities associated edges. Recent results by Otto author quantify error made approximating homogenized coefficient averaged energy regularized corrector (with parameter T) some box finite size L. this article,...

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