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We study the effective behavior of random, heterogeneous, anisotropic, second-order phase transitions energies that arise in the study of pattern formations in physical–chemical systems. Specifically, we study the asymptotic behavior, as ε goes to zero, of random heterogeneous anisotropic functionals in which the second-order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first-order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals Γ -converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed. [ABSTRACT FROM AUTHOR]

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior a hom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨ · ⟩ and the corresponding linear elliptic operator - ∇ · a ∇ . In line with the theory of homogenization, the method proceeds by computing d = 3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨ · ⟩ . We make this point by investigating the bias (or systematic error), i.e., the difference between a hom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O (L - 1) . In case of a suitable periodization of ⟨ · ⟩ , we rigorously show that it is O (L - d) . In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨ · ⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. [ABSTRACT FROM AUTHOR]

We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure. [ABSTRACT FROM AUTHOR]

The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner. [ABSTRACT FROM AUTHOR]

Recent stochastic homogenization results for the Francfort–Marigo model of brittle fracture under anti-plane shear indicate the existence of a representative volume element. This homogenization result includes a cell formula which relies on Dirichlet boundary conditions. For other material classes, the boundary conditions do not effect the effective properties upon the infinite volume limit but may have a strong influence on the necessary size of the computational domain. We investigate the influence of the boundary conditions on the effective crack energy evaluated on microstructure cells of finite size. For periodic boundary conditions recent computational methods based on FFT-based solvers exploiting the minimum cut/maximum flow duality are available. In this work, we provide a different approach based on fast marching algorithms which enables a liberal choice of the boundary conditions in the 2D case. We conduct representative volume element studies for two-dimensional fiber reinforced composite structures with tough inclusions, comparing Dirichlet with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

We consider continuous-time random walks on a random locally finite subset of R d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group G = R d or G = Z d . We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on Z d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters. [ABSTRACT FROM AUTHOR]

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size ℓ ≫ 1 and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size L ≫ ℓ . More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of L ≫ ℓ ≫ 1 , using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and ℓ is optimal, where in this paper we focus on the case of d = 2 . This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor. [ABSTRACT FROM AUTHOR]

This paper presents the computational stochastic homogenization of a heterogeneous 3D-linear anisotropic elastic microstructure that cannot be described in terms of constituents at microscale, as live tissues. The random apparent elasticity field at mesoscale is then modeled in a class of non-Gaussian positive-definite tensor-valued homogeneous random fields. We present an extension of previous works consisting of a novel probabilistic model to take into account uncertainties in the spectral measure of the random apparent elasticity field. A probabilistic analysis of the random effective elasticity tensor at macroscale is performed as a function of the level of spectrum uncertainties, which allows for studying the scale separation and the representative volume element size in a robust probabilistic framework. [ABSTRACT FROM AUTHOR]

We study the large scale equilibrium behavior of Van der Waals–Cahn–Hilliard phase transitions in stationary ergodic media. Specifically, we are interested in free energy functionals of the following form F ω (u) = ∫ R d 1 2 φ ω (x , D u (x)) 2 + W (u (x)) d x , where W is a double-well potential and φ ω (x , ·) is a stationary ergodic Finsler metric. We show that, at large scales, the random energy F ω can be approximated by the anisotropic perimeter associated with a deterministic Finsler norm φ ~ . To find φ ~ , we build on existing work of Alberti, Bellettini, and Presutti, showing, in particular, that there is a natural sub-additive quantity in this context. [ABSTRACT FROM AUTHOR]

Stability under a variational convergence of nonlinear time delays reaction-diffusion equations is discussed. Problems considered cover various models of population dynamics or diseases in heterogeneous environments where delays terms may depend on the space variable. As a consequence a stochastic homogenization theorem is established and applied to vector disease and logistic models. The results illustrate the interplay between the growth rates and the time delays which are mixed in the homogenized model. [ABSTRACT FROM AUTHOR]

A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale C 1 , α -regularity estimate for caloric functions. [ABSTRACT FROM AUTHOR]

We study the average Green’s function of stochastic, uniformly elliptic operators of divergence form on ZdZd. When the randomness is independent and has small variance, we prove regularity of the Fourier transform of the self-energy. The proof relies on the Schur complement formula and the analysis of singular integral operators combined with a Steinhaus system. [ABSTRACT FROM AUTHOR]

The results of this paper build upon those first obtained by Sznitman and Zeitouni (Invent Math 164(3), 455-567, 2006). We establish, for spacial dimensions $$d\ge 3$$ , the existence of a unique invariant measure for isotropic diffusions in random environment on $$\mathbb {R}^d$$ which are small perturbations of Brownian motion. Furthermore, we establish a general homogenization result for initial data which are locally measurable with respect to the coefficients. [ABSTRACT FROM AUTHOR]

We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We prove some convergence results with respect to stochastic two-scale convergence, which are related to classical $$\Gamma $$ -convergence results. The main result is a general $$\liminf $$ -estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rate-independent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandtl-Reuss plasticity, Tresca friction on a macroscopic surface and Tresca friction on microscopic fissures. [ABSTRACT FROM AUTHOR]

We study the qualitative homogenization of second-order Hamilton-Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion. [ABSTRACT FROM AUTHOR]

We consider a random, uniformly elliptic coefficient field $$a(x)$$ on the $$d$$ -dimensional integer lattice $$\mathbb {Z}^d$$ . We are interested in the spatial decay of the quenched elliptic Green function $$G(a;x,y)$$ . Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble $$\langle \cdot \rangle $$ . We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $$\langle |\nabla _x G(x,y)|^p\rangle $$ and $$\langle |\nabla _x\nabla _y G(x,y)|^p\rangle $$ , have the same decay rates in $$|x-y|\gg 1$$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358-390, ), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $$G$$ , that is, $$\langle |\nabla _x G(x,y)|^2\rangle $$ and $$\langle |\nabla _x\nabla _y G(x,y)|\rangle $$ . As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast. [ABSTRACT FROM AUTHOR]

We consider-in a uniformly strictly convex potential regime-two versions of random gradient models with disorder. In model (A) the interface feels a bulk term of random fields while in model (B) the disorder enters through the potential acting on the gradients. We assume a general distribution on the disorder with uniformly-bounded finite second moments. It is well known that for gradient models without disorder there are no Gibbs measures in infinite volume in dimension $$d = 2$$ , while there are shift-invariant gradient Gibbs measures describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn (Commun Math Phys 185:1-36, ). Van Enter and Külske proved in (Ann Appl Probab 18(1):109-119, ) that adding a disorder term as in model (A) prohibits the existence of such gradient Gibbs measures for general interaction potentials in $$d = 2$$ . In Cotar and Külske (Ann Appl Probab 22(5):1650-1692, ) we proved the existence of shift-covariant random gradient Gibbs measures for model (A) when $$d\ge 3$$ , the disorder is i.i.d and has mean zero, and for model (B) when $$d\ge 1$$ and the disorder has a stationary distribution. In the present paper, we prove existence and uniqueness of shift-covariant random gradient Gibbs measures with a given expected tilt $$u\in {\mathbb R}^{d}$$ and with the corresponding annealed measure being ergodic: for model (A) when $$d\ge 3$$ and the disordered random fields are i.i.d. and symmetrically-distributed, and for model (B) when $$d\ge 1$$ and for any stationary disorder-dependence structure. We also compute for both models for any gradient Gibbs measure constructed as in Cotar and Külske (Ann Appl Probab 22(5):1650-1692, ), when the disorder is i.i.d. and its distribution satisfies a Poincaré inequality assumption, the optimal decay of covariances with respect to the averaged-over-the-disorder gradient Gibbs measure. [ABSTRACT FROM AUTHOR]

In this work, stochastic homogenization analysis of heterogeneous materials is addressed in the context of elasticity under finite deformations. The randomness of the morphology and of the material properties of the constituents as well as the correlation among these random properties are fully accounted for, and random effective quantities such as tangent tensor, first Piola-Kirchhoff stress, and strain energy along with their numerical characteristics are tackled under different boundary conditions by a multiscale finite element strategy combined with the Montecarlo method. The size of the representative volume element (RVE) with randomly distributed particles for different particle volume fractions is first identified by a numerical convergence scheme. Then, different types of displacement-controlled boundary conditions are applied to the RVE while fully considering the uncertainty in the microstructure. The influence of different random cases including correlation on the random effective quantities is finally analyzed. [ABSTRACT FROM AUTHOR]

In this paper we describe efficient methods of generation of representative volume elements (RVEs) suitable for producing the samples for analysis of effective properties of composite materials via and for stochastic homogenization. We are interested in composites reinforced by a mixture of spherical and cylindrical inclusions. For these geometries we give explicit conditions of intersection in a convenient form for verification. Based on those conditions we present two methods to generate RVEs: one is based on the random sequential adsorption scheme, the other one on the time driven molecular dynamics. We test the efficiency of these methods and show that the first one is extremely powerful for low volume fraction of inclusions, while the second one allows us to construct denser configurations. All the algorithms are given explicitly so they can be implemented directly. [ABSTRACT FROM AUTHOR]

We estimate the variance of the value function for a random optimal control problem. The value function is the solution w of a Hamilton-Jacobi equation with random Hamiltonian H( p, x, ω)= K( p)− V( x/ ϵ, ω) in dimension d≥2. It is known that homogenization occurs as ϵ→0, but little is known about the statistical fluctuations of w. Our main result shows that the variance of the solution w is bounded by O( ϵ/|log ϵ|). The proof relies on a modified Poincaré inequality of Talagrand. [ABSTRACT FROM AUTHOR]

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 d ≥ 2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem. [ABSTRACT FROM AUTHOR]

In this paper, a multiscale problem arising in material science is considered. The problem involves a random coefficient which is assumed to be a perturbation of a deterministic coefficient, in a sense made precisely in the body of the text. The homogenized limit is then computed by using a perturbation approach. This computation requires repeatedly solving a corrector-like equation for various configurations of the material. For this purpose, the reduced basis approach is employed and adapted to the specific context. The authors perform numerical tests that demonstrate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

We consider first–order corrections to the classical theory by Lifshitz, Slyozov and Wagner (LSW) for systems with a finite number of particles. Numerical simulations in V. E Fradkov etal. [ Phys. Rev. E 53:3925–3932 (1996)] show a cross–over in the scaling of the correction term from φ1/3 to φ1/2 (φ is the volume fraction of particles), when the system size becomes larger than the screening length. We rigorously derive this cross–over for the rate of change of the energy, starting from the monopole approximation. The proof exploits the fact that the rate of change of energy has a variational characterization. [ABSTRACT FROM AUTHOR]

We establish some ergodic theorems with the view to obtaining a convergence result of sequences of random Radon measures. We also give an application in stochastic homogenization of nonconvex integral functionals. [ABSTRACT FROM AUTHOR]

This paper provides a rigorous convergence rate and complexity analysis for a recently introduced framework, called PDE acceleration, for solving problems in the calculus of variations and explores applications to obstacle problems. PDE acceleration grew out of a variational interpretation of momentum methods, such as Nesterov's accelerated gradient method and Polyak's heavy ball method, that views acceleration methods as equations of motion for a generalized Lagrangian action. Its application to convex variational problems yields equations of motion in the form of a damped nonlinear wave equation rather than nonlinear diffusion arising from gradient descent. These accelerated PDEs can be efficiently solved with simple explicit finite difference schemes where acceleration is realized by an improvement in the CFL condition from d t ∼ d x 2 for diffusion equations to d t ∼ d x for wave equations. In this paper, we prove a linear convergence rate for PDE acceleration for strongly convex problems, provide a complexity analysis of the discrete scheme, and show how to optimally select the damping parameter for linear problems. We then apply PDE acceleration to solve minimal surface obstacle problems, including double obstacles with forcing, and stochastic homogenization problems with obstacles, obtaining state-of-the-art computational results. [ABSTRACT FROM AUTHOR]