arXiv subjects

Antoine Gloria

Publications and source records attributed to Antoine Gloria.

At least 55 records · Page 3Linked to original sources

arXiv2018

Loss of strong ellipticity through homogenization in 2D linear elasticity: A phase diagram

Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perform a more general analysis in the stationary, ergodic setting.

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arXiv2016

Isotropy prohibits the loss of strong ellipticity through homogenization in linear elasticity

Since the seminal contribution of Geymonat, M{ü}ller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved by homogenization in linear elasticity. This phenomenon is typically related to microscopic buckling of the composite material. The present contribution is concerned with the interplay between isotropy and strong ellipticity in the framework of periodic homogenization in linear elasticity. Mixtures of two isotropic phases may indeed lead to loss of strong ellipticity when arranged in a laminate manner. We show that if a matrix/inclusion type mixture of isotropic phases produces macroscopic isotropy, then strong ellipticity cannot be lost. R{é}sum{é}. Nous savons depuis l'article fondateur de Geymonat, M{ü}ller et Triantafyl-lidis qu'en{é}lasticit{é}en{é}lasticit{é} lin{é}aire l'homog{é}n{é}isation p{é}riodique ne conserve pas n{é}cessairement l'ellipticit{é} forte. Ce ph{é}nom{è}ne est li{é} au flambage microscopique des composites. Notre contribution consiste examiner le r{ô}le de l'isotropie dans ce type de pathologie. Le m{é}lange de deux phases isotropes peut en effet conduir{è} a cette perte si l'arrangement est celui d'un lamin{é}. Nous montrons qu'en revanche, si un arrangement de type ma-trice/inclusion produit un tenseur homog{é}n{é}is{é} isotrope, alors la forte ellipticit{é} est con-serv{é}e.

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arXiv2016

The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux $(\nablaϕ,a(\nabla ϕ+e))$ of the corrector $ϕ$, when spatially averaged over a scale $R\gg 1$ decay like the CLT scaling $R^{-\frac{d}{2}}$. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization.

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arXiv2016

Bounded correctors in almost periodic homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov. The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by the first author and Shen for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.

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arXiv2015

Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius-Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions.

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arXiv2015

Stochastic homogenization of nonconvex unbounded integral functionals with convex growth

We consider the well-travelled problem of homogenization of random integral functionals. When the integrand has standard growth conditions, the qualitative theory is well-understood. When it comes to unbounded functionals, that is, when the domain of the integrand is not the whole space and may depend on the space-variable, there is no satisfactory theory. In this contribution we develop a complete qualitative stochastic homogenization theory for nonconvex unbounded functionals with convex growth. We first prove that if the integrand is convex and has $p$-growth from below (with $p>d$, the dimension), then it admits homogenization regardless of growth conditions from above. This result, that crucially relies on the existence and sublinearity at infinity of correctors, is also new in the periodic case. In the case of nonconvex integrands, we prove that a similar homogenization result holds provided the nonconvex integrand admits a two-sided estimate by a convex integrand (the domain of which may depend on the space-variable) that itself admits homogenization. This result is of interest to the rigorous derivation of rubber elasticity from polymer physics, which involves the stochastic homogenization of such unbounded functionals.

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arXiv2015

Annealed estimates on the Green functions and uncertainty quantification

We prove optimal annealed decay estimates on the derivative and mixed second derivative of the elliptic Green functions on $\mathbb{R}^d$ for random stationary measurable coefficients that satisfy a certain logarithmic Sobolev inequality and for periodic coefficients, extending to the continuum setting results by Otto and the second author for discrete elliptic equations. As a main application we obtain optimal estimates on the fluctuations of solutions of linear elliptic PDEs with "noisy" diffusion coefficients, an uncertainty quantification result. As a direct corollary of the decay estimates we also prove that for these classes of coefficients the Hölder exponent of the celebrated De Giorgi-Nash-Moser theory can be taken arbitrarily close to 1 in the large (that is, away from the singularity).

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arXiv2015

A quantitative central limit theorem for the effective conductance on the discrete torus

We study a random conductance problem on a $d$-dimensional discrete torus of size $L > 0$. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance $A_L$ of the network is a random variable, depending on $L$, and the main result is a quantitative central limit theorem for this quantity as $L \to \infty$. In terms of scalings we prove that this nonlinear nonlocal function $A_L$ essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of $A_L$.

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arXiv2014

Reduction of the resonance error in numerical homogenisation II: correctors and extrapolation

This paper is the companion article of [Gloria, M3AS, 21 (2011), No. 3, pp 1601-1630]. One common drawback among numerical homogenization methods is the presence of the so-called resonance error, which roughly speaking is a function of the ratio $\frac{\varepsilon}ρ$, where $ρ$ is a typical macroscopic lengthscale and $\varepsilon$ is the typical size of the heterogeneities. In the present work, we make a systematic use of regularization and extrapolation to reduce this resonance error at the level of the approximation of homogenized coefficients and correctors for general non-necessarily symmetric stationary ergodic coefficients. We quantify this reduction for the class of periodic coefficients, for the Kozlov subclass of almost periodic coefficients, and for the subclass of random coefficients that satisfy a spectral gap estimate (e.g. Poisson random inclusions). We also report on a systematic numerical study in dimension 2, which demonstrates the efficiency of the method and the sharpness of the analysis. Last, we combine this approach to numerical homogenization methods, prove the asymptotic consistency in the case of locally stationary ergodic coefficients and give quantitative estimates in the case of periodic coefficients.

math.NA↗

arXiv2014

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for $d>2$. The main difference with respect to the first part of [Gloria-Otto, arXiv:1409.0801] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.

math.NA↗

arXiv2014

An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations

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 $L^2$-norm in probability of the \mbox{$H^1$-norm} in space of this error scales like $ε$, where $ε$ is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.

math.NA↗

arXiv2014

When are increment-stationary random point sets stationary?

In a recent work, Blanc, Le Bris, and Lions defined a notion of increment-stationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary. In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space. This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions $d>2$. In dimensions $d=1$ and $d=2$, we show that such sufficient conditions cannot exist.

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arXiv2014

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 $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the assumption that a spectral gap estimate holds in probability, we proved that there exists a stationary corrector field in dimensions $d>2$ 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.

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arXiv2013

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

This article is devoted to the analysis of a Monte Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time t>0, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions re-scaled by t of n independent random walks in n independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of PDE arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the re-scaled final position of the random walk in terms of t. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of n and t, and prove a large-deviation estimate, as well as a central limit theorem. Our estimates are optimal, up to a logarithmic correction in dimension 2.

math.PR↗

arXiv2012

Random parking, Euclidean functionals, and rubber elasticity

We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\mathbb{R}^d$ and of point sets that are (almost) subadditive in their first variable. Denoting by $ξ$ the random parking measure in $\mathbb{R}^d$, and by $ξ^R$ the random parking measure in the cube $Q_R=(-R,R)^d$, we show, under some natural assumptions on $S$, that there exists a constant $\bar{S}\in \mathbb{R}$ such that % $$ \lim_{R\to +\infty} \frac{S(Q_R,ξ)}{|Q_R|}\,=\,\lim_{R\to +\infty}\frac{S(Q_R,ξ^R)}{|Q_R|}\,=\,\bar{S} $$ % almost surely. If $ζ\mapsto S(Q_R,ζ)$ is the counting measure of $ζ$ in $Q_R$, then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.

math.PR↗

arXiv2012

An optimal error estimate in stochastic homogenization of discrete elliptic equations

This paper is the companion article to [Ann. Probab. 39 (2011) 779--856]. 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 $A_{\mathrm{hom}}=a_{\mathrm{hom}}\mathrm{Id}$ is characterized by $ξ\cdot A_{\mathrm{hom}}ξ=<(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)>$ for any direction $ξ\in\mathbb{R}^d$, where the random field $ϕ$ (the "corrector") is the unique solution of $-\nabla^*\cdot A(ξ+\nablaϕ)=0$ in $\mathbb{Z}^d$ such that $ϕ(0)=0$, $\nablaϕ$ is stationary and $<\nablaϕ>=0$, $<\cdot>$ denoting the ensemble average (or expectation).

math.PR↗

arXiv2011

An optimal variance estimate in stochastic homogenization of discrete elliptic equations

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 $ξ\cdot A_{\mathrm {hom}}ξ=\langle(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)\rangle$ for any direction $ξ\in\mathbb {R}^d$, where the random field $ϕ$ (the ``corrector'') is the unique solution of $-\nabla^*\cdot A(ξ+\nablaϕ)=0$ such that $ϕ(0)=0$, $\nablaϕ$ is stationary and $\langle\nablaϕ\rangle=0$, $\langle\cdot\rangle$ denoting the ensemble average (or expectation). It is known (``by ergodicity'') that the above ensemble average of the energy density $\mathcal {E}=(ξ+\nablaϕ)\cdot A(ξ+\nablaϕ)$, 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, $\operatorname {var}[\sum \mathcal {E}η_L]\lesssim L^{-d}$, where the averaging function [i.e., $\sumη_L=1$, $\operatorname {supp}(η_L)\subset\{|x|\le L\}$] has to be smooth in the sense that $|\nablaη_L|\lesssim L^{-1-d}$. 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 $a_{\mathrm {hom}}$.

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arXiv2010

Spectral measure and approximation of homogenized coefficients

This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

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