Stochastic Two-scale Convergence Research Articles

Stochastic two-scale convergence and Young measures

<p style='text-indent:20px;'>In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence.</p>

Homogenization of immiscible compressible two–phase flow in random porous media

The paper deals with stochastic homogenization of a system modeling immiscible compressible two-phase, such as water and gas, flow in random porous media. The problem is written in terms of the phase formulation, i.e. the saturation of one phase and the pressure of the second phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear degenerate parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion-convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. We consider the behavior of compressible two-phase flow in heterogeneous reservoirs with permeability and porosity being realizations of given statistically homogeneous random fields. We derive the effective (macroscopic) problem and prove the convergence of solutions. Our approach relies on stochastic two-scale convergence techniques, the realization-wise notion of stochastic two-scale convergence being used. Also, we exploit various a priori estimates as well as monotonicity and compactness arguments. To our best knowledge, this is the first stochastic homogenization result in the case of compressible two-phase flow in random porous media.

Stochastic homogenization of the Landau–Lifshitz–Gilbert equation

Following the ideas of Zhikov and Piatnitski (Izv Math 70(1):19–67, 2006), and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations: the equation of harmonic maps into the sphere and the Landau–Lifschitz–Gilbert equation. These equations have strong nonlinear features, and in general their solutions are not unique.

Homogenization of biomechanical models of plant tissues with randomly distributed cells

In this paper homogenization of a mathematical model for biomechanics of a plant tissue with randomly distributed cells is considered. Mechanical properties of a plant tissue are modelled by a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells and cell walls, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and the Stokes equations for fluid flow inside the cells. The nonlinear coupling between the mechanics and chemistry is given by the dependence of elastic properties of plant tissue on densities of chemical substances as well as by the dependence of chemical reactions on mechanical stresses present in a tissue. Using techniques of stochastic homogenization we derive rigorously macroscopic model for plant tissue biomechanics with random distribution of cells. Strong stochastic two-scale convergence is shown to pass to the limit in the non-linear reaction terms. Appropriate meaning of the boundary terms is introduced to define the macroscopic equations with flux boundary conditions and transmission conditions on the microscopic scale.

Stochastic homogenization of \Lambda -convex gradient flows

<p style='text-indent:20px;'>In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a <inline-formula><tex-math id="M2">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace operator with <inline-formula><tex-math id="M4">\begin{document}$ p\in (1, \infty) $\end{document}</tex-math></inline-formula>. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (<inline-formula><tex-math id="M5">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-)convex functionals.

Stochastic homogenization of a scalar viscoelastic model exhibiting stress–strain hysteresis

Motivated by rate-independent stress–strain hysteresis observed in filled rubber, this article considers a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using a variant of stochastic two-scale convergence as introduced by Bourgeatet al., we obtain the homogenized limit of the evolution, and demonstrate that under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. These results are illustrated by means of numerical simulations in a particular one-dimensional instance of the model.

Stochastic homogenization of rate-dependent models of monotone type in plasticity

In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Γ-convergence theory.

Convergences of the squareroot approximation scheme to the Fokker–Planck operator

We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.

Stochastic Unfolding and Homogenization of Spring Network Models

The aim of our work is to provide a simple homogenization and discrete-to-continuum procedure for energy driven problems involving stochastic rapidly-oscillating coefficients. Our intention is to extend the periodic unfolding method to the stochastic setting. Specifically, we recast the notion of stochastic two-scale convergence in the mean by introducing an appropriate stochastic unfolding operator. This operator admits similar properties as the periodic unfolding operator, leading to an uncomplicated method for stochastic homogenization. Secondly, we analyze the discrete-to-continuum (resp. stochastic homogenization) limit for a rate-independent system describing a network of linear elasto-plastic springs with random coefficients.

Stochastic homogenization of the bending plate model

We use the notion of stochastic two-scale convergence introduced in [32] to solve the problem of stochastic homogenization of the elastic plate in the bending regime.

Stochastic homogenization of rate-independent systems and applications

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.

Stochastic homogenization of the Keller–Segel chemotaxis system

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.

Stochastic Two-Scale Convergence and Iterated Function System

We will use the random self-similar structure to generalize the concept of stochastic two-scale convergence extending some ideas elaborated in the deterministic case by J. Kolumban.

Acoustic propagation in a random saturated medium: the biphasic case

Homogenized acoustic properties of a non-periodic porous linear elastic solid filled with a viscous Newtonian fluid are derived. A small parameter of the problem is the ratio between typical size of the microstructural inhomogeneity and the macroscopic length scale. We consider the ratio of elasticity coefficients to viscosity coefficients to be of order which leads to the biphasic macroscopic behaviour. To pass to the limit in the governing equations, we employ stochastic two-scale convergence in the mean. Periodic approximation is then used and elimination of corrector terms is carried out in detail for the periodic geometry. The effective equations for the solid displacement, fluid displacement and fluid pressure are obtained and compared to the classical Biot system.

Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities

The heat transfer problem in a polycrystal with nonlinear jump conditions on the grain boundaries will be homogenized using the method of stochastic two-scale convergence developed by Zhikov and Pyatnitskii [V.V. Zhikov and A.L. Pyatnitskii, Homogenization of random singular structures and random measures, Izv. Math. 70(1) (2006), pp. 19–67] and recently extended by the author [M. Heida, An extension of stochastic two-scale convergence and application, Asympt. Anal. (2010) (in press)]. It will be shown that for monotone Lipschitz jump conditions differentiable in 0, the nonlinearity vanishes in the limit. Additionally, existing Poincaré inequalities will be extended to more general geometric settings with the only restriction of local C 1-interfaces with finite intensity. In particular, the result can now be applied to the Poisson–Voronoi tessellation.

Stochastic two-scale convergence of an integral functional

Stochastic two-scale convergence of an integral functional

An extension of the stochastic two-scale convergence method and application

The stochastic two-scale convergence method, recently developed in an article by Zhikov and Piatnitskii (Izv. Math. 70(1) (2006), 19-67) is extended to arbitrary probability spaces and is now based on the theory of stochastic geometry instead of random measures. It will be shown that former results on stochastic and periodic two-scale convergence fit into the new approach in a natural way. These results will be applied to functions with jumps on (n − 1)-dimensional manifolds, in particular to a homogenization problem of heat transfer through a composite material or polycrystal.

Stochastic homogenization and macroscopic modelling of composites and flow through porous media

The aim of this contribution is mainly twofold. First, the stochastic two-scale convergence in the mean developed by Bourgeat et al. [13] is used to derive the macroscopic models of: (i) diffusion in random porous medium, (ii) nonstationary flow of Stokesian fluid through random linear elastic porous medium. Second, the multi-scale convergence method developed by Allaire and Briane [7] for the case of several microperiodic scales is extended to random distribution of heterogeneities characterized by separated scales (stochastic reiterated homogenization). .

On diffusion of a single‐phase, slightly compressible fluid through a randomly fissured medium

AbstractIn this paper, the Douglas–Peszyńska–Showalter model of diffusion through a partially fissured medium is given a stochastic formulation using the framework for problems in random media as set forth by Jikov, Kozlov and Oleinik. The concept of stochastic two‐scale convergence in the mean is then used to homogenize the randomized micromodels which result. As a consequence of this homogenization procedure, exact stochastic generalizations of results obtained by Clark and Showalter on diffusion through periodically fissured media are derived. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Modèle de double porosité aléatoire

We consider a weakly compressible single phase fluid flow through a randomly fissured porous medium. Under standard assumptions of the stochastic homogenization theory we obtain the stochastic version of the periodic case. The basic tool is the stochastic two-scale convergence. The source-like terms are given by the coupling with a newly introduced stochastic auxiliary problem.