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Approximate effective coefficients of random heterogeneous materials could be obtained by solving auxiliary problems with certain boundary condition. When the coefficients of the auxiliary problems have great jump caused by the random heterogeneous materials with high-contrast properties, Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) provide broad upper and lower bounds of effective coefficients, respectively. Two possible factors that result in inaccurate approximations of effective coefficients are discussed in this paper. Effect of large condition number of stiffness matrix caused by the high contrast on the numerical accuracy of approximate effective coefficients is analysed. Since DBC and NBC are not effective for the high-contrast materials, an alternative Robin boundary condition (RBC) is presented to provide much better approximations of effective coefficients. Convergence of the approximate effective coefficients under RBC is proved. Numerical examples indicate that proper adjusting factor introduced in RBC makes it more flexible than other boundary conditions. RBC is more suitable for the high-contrast materials and has potential to be an optimal boundary condition. [ABSTRACT FROM AUTHOR]

We study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations. [ABSTRACT FROM PUBLISHER]

In this paper, we focus on the Keller–Segel chemotaxis system in a random heterogeneous domain. We assume that the corresponding diffusion and chemotaxis coefficients are given by stationary ergodic random fields and apply stochastic two-scale convergence methods to derive the homogenized macroscopic equations. In establishing our results, we also derive a priori estimates for the Keller–Segel system that rely only on the boundedness of the coefficients; in particular, no differentiability assumption on the diffusion and chemotaxis coefficients for the chemotactic species is required. Finally, we prove the convergence of a periodization procedure for approximating the homogenized macroscopic coefficients. [ABSTRACT FROM AUTHOR]

We prove stochastic homogenization for a general class of coercive, nonconvex Hamilton–Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed. [ABSTRACT FROM AUTHOR]

Nondominated sorting is a combinatorial algorithm that sorts points in Euclidean space into layers according to a partial order. It was recently shown that nondominated sorting of random points has a Hamilton–Jacobi equation continuum limit. The original proof, given in Calder et al. (2014), relies on a continuum variational problem. In this paper, we give a new proof using a direct verification argument that completely avoids the variational interpretation. We believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle. [ABSTRACT FROM AUTHOR]

It is proved that for the 2-dimensional case with random shear flow of the G-equation model with strain term, the strain term reduces the front propagation. Also an improvement of the main result by Armstrong-Souganidis is provided. [ABSTRACT FROM AUTHOR]

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fieldsain the context of stochastic homogenization. The large-scale regularity ofa-harmonic functions is encoded by Liouville principles: The space ofa-harmonic functions that grow at most like a polynomial of degreekhas the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scaleCk,α-regularity theory, which in the present work is developed in the form of a correspondingCk,α-“excess decay” estimate: For a givena-harmonic functionuon a ballBR, its energy distance on some ballBrto the above space ofa-harmonic functions that grow at most like a polynomial of degreekhas the natural decay in the radiusrabove some minimal radiusr0. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realizationaof the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space ofa-harmonic functions that grow at most like a polynomial of degreek, establish the above excess decay, and then the corresponding Liouville principle. [ABSTRACT FROM AUTHOR]

The paper aims at analytically exhibiting for the first time the fundamental mechanisms underlying the fact that effective biological tissue electrical properties and their frequency dependence reflect the tissue composition and physiology. For doing so, a homogenization theory is derived to describe the effective admittivity of cell suspensions. A new formula is reported for dilute cases that gives the frequency-dependent effective admittivity with respect to the membrane polarization. Different microstructures are shown to be distinguishable via spectroscopic measurements of the overall admittivity using the spectral properties of the membrane polarization. The Debye relaxation times associated with the membrane polarization tensor are shown to be able to give the microscopic structure of the medium. A natural measure of the admittivity anisotropy is introduced and its dependence on the frequency of applied current is derived. A Maxwell–Wagner–Fricke formula is given for concentric circular cells, and the results can be extended to the random cases. A randomly deformed periodic medium is also considered and a new formula is derived for the overall admittivity of a dilute suspension of randomly deformed cells. [ABSTRACT FROM AUTHOR]

We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science , , , consists in selecting random realizations that best satisfy some statistical properties (such as the volume fraction of each phase in a composite material) usually only obtained asymptotically. We study the approach theoretically in some simplified settings (one-dimensional setting, perturbative setting in higher dimensions), and numerically demonstrate its efficiency in more general cases. [ABSTRACT FROM AUTHOR]

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension d ≥ 3. Our derivation is based on a Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman-Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential. [ABSTRACT FROM AUTHOR]

We prove quenched L p -type estimates for the gradient of a solution of a quasilinear elliptic equation with random coefficients. [ABSTRACT FROM AUTHOR]

This article is about the quantitative homogenization theory of linear elliptic equations in divergence form with random coefficients. We derive gradient estimates on the homogenization error, i. e. on the difference between the actual solution and the two-scale expansion of the homogenized solution, both in terms of strong norms (oscillation) and weak norms (fluctuation). These estimates are optimal in terms of scaling in the ratio between the microscopic and the macroscopic scale. The purpose of this article is to highlight the usage of the recently introduced annealed Calderón-Zygmund (CZ) estimates in obtaining the above, previously known, error estimates. Moreover, the article provides a novel proof of these annealed CZ estimate that completely avoids quenched regularity theory, but rather relies on functional analysis. It is based on the observation that even on the level of operator norms, the Helmholtz projection is close to the one for the homogenized coefficient (for which annealed CZ estimates are easily obtained). In this article, we strive for simple proofs, and thus restrict ourselves to ensembles of coefficient fields that are local transformations of Gaussian random fields with integrable correlations and Hölder continuous realizations. As in earlier work, we use the natural objects from the general theory of homogenization, like the (potential and flux) correctors and the homogenization commutator. Both oscillation and fluctuation estimates rely on a sensitivity calculus, i.e. on estimating how sensitively the quantity of interest does depend on an infinitesimal change in the coefficient field, which is fed into the Spectral Gap inequality. In this article, the annealed CZ estimate is the only form in which elliptic regularity theory enters. [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]

In our previous papers we have described efficient and reliable methods of generation of representative volume elements (RVE) perfectly suitable for analysis of composite materials via stochastic homogenization. In this paper we profit from these methods to analyze the influence of the morphology on the effective mechanical properties of the samples. More precisely, we study the dependence of main mechanical characteristics of a composite medium on various parameters of the mixture of inclusions composed of spheres and cylinders. On top of that we introduce various imperfections to inclusions and observe the evolution of effective properties related to that. The main computational approach used throughout the work is the FFT-based homogenization technique, validated however by comparison with the direct finite elements method. We give details on the features of the method and the validation campaign as well. [ABSTRACT FROM AUTHOR]

We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behavior of a particle undergoing Brownian motion on the surface, viewed as a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor D. In one dimension, the effective diffusion coefficient is given by 1/Z2, where Z is the average line element of the surface. In two-dimensions, D will not have a closed-form expression in general. However, we are able to derive variational bounds for the effective diffusion tensor. Moreover, in the special case when D is isotropic, we show that D = 1/Z I, where Z is the average area element of the random surface. We also describe a numerical scheme for approximating the effective diffusion tensor and illustrate this scheme with three examples. [ABSTRACT FROM AUTHOR]

In this article, we study the application of multilevel Monte Carlo (MLMC) approaches to numerical random homogenization. Our objective is to compute the expectation of some functionals of the homogenized coefficients, or of the homogenized solutions. This is accomplished within MLMC by considering different sizes of representative volumes (RVEs). Many inexpensive computations with the smallest RVE size are combined with fewer expensive computations performed on larger RVEs. Likewise, when it comes to homogenized solutions, different levels of coarse-grid meshes are used to solve the homogenized equation. We show that, by carefully selecting the number of realizations at each level, we can achieve a speed-up in the computations in comparison to a standard Monte Carlo method. Numerical results are presented for both one-dimensional and two-dimensional test-cases that illustrate the efficiency of the approach. [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]

We study the homogenization of elliptic systems of equations in divergence form where the coefficients are compositions of periodic functions with a random diffeomorphism with stationary gradient. This is done in the spirit of scalar stochastic homogenization by Blanc, Le Bris and Lions. An application of the abstract result is given for Maxwell's equations in random dissipative bianisotropic media. [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]

We consider a variance reduction approach for the stochastic homogenization of divergence form linear elliptic problems. Although the exact homogenized coefficients are deterministic, their practical approximations are random. We intro duce a control variate technique to reduce the variance of the computed approximations of the homogenized coefficients. Our approach is based on a surrogate model inspired by a defect-type theory, where a perfect periodic material is perturbed by rare defects. This model has been introduced in [A. Anantharaman and C. Le Bris, C. R. Math. Acad. Sci. Paris, 348 (2010), pp. 529-534] in the context of weakly random models. In this work, we address the fully random case and show that the perturbative approaches proposed in [A. Anantharaman and C. Le Bris, C. R. Math. Acad. Sci. Paris, 348 (2010), pp. 529-534], [A. Anantharaman and C. Le Bris, Multiscale Model. Simul., 9 (2011), pp. 513-544] can be turned into an efficient control variable. We theoretically demonstrate the efficiency of our approach in simple cases. We next provide illustrating numerical results and compare our approach with other variance reduction strategies. We also show how to use the reduced basis approach proposed in [C. Le Bris and F. Thomines, Chin. Ann. Math. Ser. B, 33 (2012), pp. 657-672] so that the cost of building the surrogate model remains limited. [ABSTRACT FROM AUTHOR]

This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis--which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of the authors, and by Neukamm and Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates and empirically study them. Two types of methods are presented: methods based on the corrector equation and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures and calls for new theoretical developments. [ABSTRACT FROM AUTHOR]

We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian. [ABSTRACT FROM AUTHOR]

We are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign. The problem can be reformulated as the homogenization of a Hamilton–Jacobi equation with a positively homogeneous of degree one non-coercive Hamiltonian. The periodic setting was studied earlier by Cardaliaguet, Lions and Souganidis (2009) [16] . Here we concentrate in the random media and show that the solutions of the oscillatory Hamilton–Jacobi equation converge in L ∞ -weak ⋆ to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective Hamiltonian(s), determined by the properties of the random media. [ABSTRACT FROM AUTHOR]

We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspectives. Under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a logarithmic rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed logarithmic rate. A general methodology to study the stochastic homogenization and rates of convergence for stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Souganidis, and Wang [1] , and Caffarelli and Souganidis [2] . We extend their approach to fully nonlinear uniformly parabolic equations, developing a number of new arguments to handle the parabolic structure of the problem. [ABSTRACT FROM AUTHOR]

We consider a nonlinear convex stochastic homogenization problem, in a stationary setting. In practice, the deterministic homogenized energy density is approximated by a random apparent energy density, obtained by solving the corrector problem on a truncated domain. We show that the technique of antithetic variables can be used to reduce the variance of the computed quantities, and thereby decrease the computational cost at equal accuracy. This leads to an efficient approach for approximating expectations of the apparent homogenized energy density and of related quantities. The efficiency of the approach is numerically illustrated on several test cases. Some elements of analysis are also provided. [ABSTRACT FROM AUTHOR]

In [G. Bouchitté and D. Felbacq, C. R. Math. Acad. Sci. Paris 339 (2004) 377-382; D. Felbacq and G. Bouchitté, Phys. Rev. Lett. 94 (2005) 183902; D. Felbacq and G. Bouchitté, New J. Phys. 7 (2005) 159], a theory for artificial magnetism in two-dimensional photonic crystals has been developed for large wavelength using homogenization techniques. In this paper we pursue this approach within a rigorous stochastic framework: dielectric parallel nanorods are randomly disposed, each of them having, up to a large scaling factor, a random permittivity ε(ω) whose law is represented by a density on a window Δh = [a-, a+] × [0, h] of the complex plane. We give precise conditions on the initial probability law (permittivity, radius and position of the rods) under which the homogenization process can be performed leading to a deterministic dispersion law for the effective permeability with possibly negative real part. Subsequently a limit analysis h → 0, accounting a density law of ε which concentrates on the real axis, reveals singular behavior due to the presence of resonances in the microstructure. [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 propose a procedure for the computation of natural frequencies for structures made of metal foam. Because the heterogeneity of the foam geometry has an influence on macroscopic properties, the irregular geometry of the foam has to be taken into account. This is done by adapting a Laguerre tessellation to statistical descriptors of the geometry obtained from CT image analysis. As the length scale of the representative volume element is nearly of the same order as the length scale of the structures under consideration, classical homogenization techniques for the computation of effective properties cannot be applied. Therefore, we introduce a stochastic homogenization method based on empirical marginal distribution functions and correlation functions of apparent properties. This information allows us to define random fields for elastic properties and the mass density on the macroscopic level. Statistical properties of the natural frequencies can then be inferred. [ABSTRACT FROM AUTHOR]

The article discusses a study presenting the first quantitative homogenization results for Hamilton-Jacobi equations in the stochastic setting. It assumes that the Hamiltonian H = H (p,y,w) satisfies a finite range dependence hypothesis in its spatial dependence. Its arguments rely on adaptations of some of the probabilistic techniques based on Azuma's inequality and the martingale method of bounded differences and do not yield sharp error estimates or convergence rates for homogenization.

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl. 88 (2007) 34–63.]. The equation under consideration is a standard linear elliptic equation in divergence form, where the highly oscillatory coefficient is the composition of a periodic matrix with a stochastic diffeomorphism. The homogenized limit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We first establish, in the one-dimensional case, a convergence result (with an explicit rate) on the residual process, defined as the difference between the solution to the highly oscillatory problem and the solution to the homogenized problem. We next return to the multidimensional situation. As often in random homogenization, the homogenized matrix is defined from a so-called corrector function, which is the solution to a problem set on the entire space. We describe and prove the almost sure convergence of an approximation strategy based on truncated versions of the corrector problem. [ABSTRACT FROM AUTHOR]

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 L2-norm in probability of the H1-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. [ABSTRACT FROM AUTHOR]

Effective boundary conditions for the flow of a viscous fluid across randomly leaky permeable membranes are studied. Threshold leak conditions of subgradient type, introduced by Fujita [Res. Inst. Math. Sci. Kokyuroku 888 (1994), 199-216, J. Comp. Appl. Math. 149(1) (2002), 56-79], are considered on the randomly distributed solid part of the membrane. The effective conditions are of subgradient type with an effective yield limit, in the case of a densely distributed solid part, or of Navier slip type, in the critical case; in the dilute case the tangential slip cancels, whereas the normal velocity and stress are continuous. We thus extend our results [Complex Variables and Elliptic Equations 57(2-4) (2012), 437-453], from the periodic permeable membrane, to a randomly permeable membrane model. [ABSTRACT FROM AUTHOR]