Antoine Gloria

Quantitative results on the corrector equation in stochastic homogenization

We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions

An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies

d\ge 2

AN OPTIMAL QUANTITATIVE TWO-SCALE EXPANSION IN STOCHASTIC HOMOGENIZATION OF DISCRETE ELLIPTIC EQUATIONS

. In previous works we studied the model problem of a discrete elliptic equation on

Reduction of the resonance error. Part 1: Approximation of homogenized coefficients

\mathbb{Z}^d

An analytical framework for numerical homogenization - Part II: windowing and oversampling

. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions

Bounded Correctors in Almost Periodic Homogenization

d>2

Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas

and that the energy density of that corrector behaves as if it had finite range of correlation in terms of the variance of spatial averages - the latter decays at the rate of the central limit theorem. In this article we extend these results, and several other estimates, to the case of a continuum linear elliptic equation whose (not necessarily symmetric) coefficient field satisfies a continuum version of the spectral gap estimate. In particular, our results cover the example of Poisson random inclusions.

Variational description of bulk energies for bounded and unbounded spin systems

A number of methods have been proposed in the recent years to perform the numerical homogenization of (possibly nonlinear) elliptic operators. These methods are usually defined at the discrete level. Most of them compute a numerical operator, close, in a sense to be made precise, to the homogenized elliptic operator for the problem. The purpose of the present work is to clarify the construction of this operator in the convex case by interpreting the method at the continuous level and to extend it to the nonconvex setting. The discretization of this new operator may be performed in several ways, recovering a variety of methods, such as the multiscale finite element method (MsFEM) or the heterogeneous multiscale method (HMM). In addition to the above, we introduce an original and general numerical corrector in the convex case.

Quantitative version of the Kipnis–Varadhan theorem and Monte Carlo approximation of homogenized coefficients

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 coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the

A partitioned Newton method for the interaction of a fluid and a 3D shell structure

L^2