Stochastic Homogenization Theory Research Articles

Convergence rates for linear elasticity systems on perforated domains

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $$L^{\frac{2d}{d-1-\tau }}$$ -error estimates $$O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )$$ for all $$\tau \in (0,1)$$ in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s inequalities (given by Lehrback and Vahakangas in J Funct Anal 271(2):330–364, 2016) and a suboptimal error estimate for the square function of the first-order approximating corrector (earliest investigated by Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036, 2012) under additional regularity assumption on coefficient). The new approach relied on the weighted quenched Calderon–Zygmund estimate (initially appeared in Gloria et al. work Milan J Math 88(1):99–170, 2020 for a quantitative stochastic homogenization theory). The second effort was $$L^2$$ -error estimates $$O\big (\varepsilon ^{\frac{5}{6}} \ln ^{\frac{2}{3}}(1/\varepsilon )\big )$$ for a Lipschitz domain, followed from a duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem and local-type Sobolev–Poincare inequalities on perforated domains. Throughout the paper, we do not impose any smoothness assumption on the coefficients.

Deep relaxation: partial differential equations for optimizing deep neural networks

Entropy-SGD is a first-order optimization method which has been used successfully to train deep neural networks. This algorithm, which was motivated by statistical physics, is now interpreted as gradient descent on a modified loss function. The modified, or relaxed, loss function is the solution of a viscous Hamilton–Jacobi partial differential equation (PDE). Experimental results on modern, high-dimensional neural networks demonstrate that the algorithm converges faster than the benchmark stochastic gradient descent (SGD). Well-established PDE regularity results allow us to analyze the geometry of the relaxed energy landscape, confirming empirical evidence. Stochastic homogenization theory allows us to better understand the convergence of the algorithm. A stochastic control interpretation is used to prove that a modified algorithm converges faster than SGD in expectation.

Implementation of the Multiscale Stochastic Finite Element Method on Elliptic PDE Problems

In this study, a multi-scale finite element method was proposed to solve two linear scale-coupling stochastic elliptic PDE problems, a tightly stretched wire and flow through porous media. At microscopic level, the main idea was to form coarse-scale equations with a prescribed analytic form that may differ from the underlying fine-scale equations. The relevant stochastic homogenization theory was proposed to model the effective global material coefficient matrix. At the macroscopic level, the Karhunen–Loeve decomposition was coupled with a Polynomial Chaos expansion in conjunction with a Galerkin projection to achieve an efficient implementation of the randomness into the solution procedure. Various stochastic methods were used to plug the microscopic cell to the global system. Strategy and relevant algorithms were developed to boost computational efficiency and to break the curse of dimension. The results of numerical examples were shown consistent with ones from literature. It indicates that the proposed numerical method can act as a paradigm for general stochastic partial differential equations involving multi-scale stochastic data. After some modification, the proposed numerical method could be extended to diverse scientific disciplines such as geophysics, material science, biological systems, chemical physics, oceanography, and astrophysics, etc.

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.

On effective attenuation in multiscale composite media

Experiments show that waves propagating through the earth’s crust experience frequency-dependent attenuation. Three regimes have been identified with specific attenuation properties: the low-, mid-, and high-frequency regimes with attenuation in general increasing with frequency. This paper shows how the observed behavior can be explained via theory for waves in random media. It considers multiple scattering of waves propagating in non-lossy one-dimensional random media with short- and/or long-range correlations. Using stochastic homogenization theory, it is possible to show that pulse propagation is described by effective fractional damping exponents. The damping exponents are related to the Hurst parameters of the random media which are characteristic parameters of the correlation properties of the fluctuations of the random media. This description is the link in between the random medium properties and the observed damping behavior. In particular, a simple binary medium is shown to reproduce well the experimental attenuation properties in the low-, mid-, and high-frequency regimes.

Random walk in random environment, corrector equation and homogenized coefficients: from theory to numerics, back and forth

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.

On a variant of random homogenization theory: convergence of the residual process and approximation of the homogenized coefficients

We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de Mathematiques Pures et Appliquees 2007]. 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. Serie I 2006]. 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.

Stochastic Σ-convergence and applications

Motivated by the fact that in nature almost all phenomena behave randomly in some scales and deterministically in some other scales, we build up a framework suitable to tackle both deterministic and stochastic homogenization problems simultaneously, and also separately. Our approach, the stochastic Σ-convergence, can be seen either as a multiscale stochastic approach since deterministic homogenization theory can be seen as a special case of stochastic homogenization theory (see Theorem 3), or as a conjunction of the stochastic and deterministic approaches, both taken globally, but also each separately. One of the main applications of our results is the homogenization of a model of rotating fluids.

Stochastic Diffeomorphisms and Homogenization of Multiple Integrals

In a recent work, Blanc, Le Bris and Lions have introduced the notion of stochastic diffeomorphism together with a variant of stochastic homogenization theory for linear and monotone elliptic operators. Their proofs rely on the ergodic theorem and on the analysis of the associated corrector equation. In the present article, we provide another proof of their results using the formalism of integral functionals. We also extend the analysis to cover the case of quasiconvex integrands.

Effective fractional acoustic wave equations in one-dimensional random multiscale media

This paper considers multiple scattering of waves propagating in a non-lossy one-dimensional random medium with short- or long-range correlations. Using stochastic homogenization theory it is possible to show that pulse propagation is described by an effective deterministic fractional wave equation, which corresponds to an effective medium with a frequency-dependent attenuation that obeys a power law with an exponent between 0 and 2. The exponent is related to the Hurst parameter of the medium, which is a characteristic parameter of the correlation properties of the fluctuations of the random medium. Moreover the frequency-dependent attenuation is associated with a special frequency-dependent phase, which ensures that causality and Kramers-Kronig relations are satisfied. In the time domain the effective wave equation has the form of a linear integro-differential equation with a fractional derivative.

Scaling up through domain decomposition

In this article, we discuss domain decomposition parallel iterative solvers for highly heterogeneous problems of flow and transport in porous media. We are particularly interested in highly unstructured coefficient variation where standard periodic or stochastic homogenization theory is not applicable. When the smallest scale at which the coefficient varies is very small, it is often necessary to scale up the equation to a coarser grid to make the problem computationally feasible. Standard upscaling or multiscale techniques require the solution of local problems in each coarse element, leading to a computational complexity that is at least linear in the global number N of unknowns on the subgrid. Moreover, except for the periodic and the isotropic random case, a theoretical analysis of the accuracy of the upscaled solution is not yet available. Multilevel iterative methods for the original problem on the subgrid, such as multigrid or domain decomposition, lead to similar computational complexity (i.e. 𝒪(N)) and are therefore a viable alternative. However, previously no theory was available guaranteeing the robustness of these methods to large coefficient variation. We review a sequence of recent papers where simple variants of domain decomposition methods, such as overlapping Schwarz and one-level FETI, are proposed that are robust to strong coefficient variation. Moreover, we also extend the theoretical results, for the first time, to the dual-primal variant of FETI.

Stochastic homogenization and random lattices

We present some variants of stochastic homogenization theory for scalar elliptic equations of the form − div [ A ( x ε , ω ) ∇ u ( x , ω ) ] = f . These variants basically consist in defining stochastic coefficients A ( x ε , ω ) from stochastic deformations (using random diffeomorphisms) of the periodic setting, as announced in [X. Blanc, C. Le Bris, P.-L. Lions, Une variante de la théorie de l'homogénéisation stochastique des opérateurs elliptiques (A variant of stochastic homogenization theory for elliptic operators), C. R. Acad. Sci. Sér. I 343 (2006) 717–727]. The settings we define are not covered by the existing theories. We also clarify the relation between this type of questions and our construction, performed in [X. Blanc, C. Le Bris, P.-L. Lions, A definition of the ground state energy for systems composed of infinitely many particles, Commun. Partial Differential Equations 28 (1–2) (2003) 439–475; X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal. 184 (2) (2007) 303–339], of the energy of, both deterministic and stochastic, microscopic infinite sets of points in interaction.

Distributed algorithms in an ergodic Markovian environment

AbstractWe provide a probabilistic analysis of the d‐dimensional banker algorithm when transition probabilities may depend on time and space. The transition probabilities evolve, as time goes by, along the trajectory of an ergodic Markovian environment, whereas the spatial parameter just acts on long runs. Our model complements the one considered by Guillotin‐Plantard and Schott (Random Struct Algorithm 21 (2002) 3–4, 371–396) where transitions are governed by a dynamical system, and appears as a new (small) step towards more general time and space dependent protocols. Our analysis relies on well‐known techniques from stochastic homogenization theory and investigates the asymptotic behavior of the rescaled algorithm as the total amount of resources available for allocation tends to infinity. In the two‐dimensional setting, we manage to exhibit three different possible regimes for the deadlock time of the limit system. To the best of our knowledge, the way we distinguish these regimes is completely new. We interpret our results in terms of stabilization of the algorithm.© 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007

Remarks on the quasistatic problem of viscoelasticity: Existence, uniqueness and homogenization

This article is devoted to the problem (1.1) of quasistatic viscoelasticity. It turns out that (1.1) can be rewritten as an abstract initial value problem of the form (1.2). In Sect. 2 we consider the general abstract initial value problem (1.3). We prove existence and uniqueness of solutions, and stability with respect to the data. In Sects. 3 and 4 we apply our abstract results to the viscoelastic problem (1.1). In Sect. 3 we prove existence and uniqueness of solutions to the n-dimensional problem. In Sect. 4 we develop a stochastic homogenization theory for the 1-dimensional problem. Finally, we close our discussion with some remarks on the homogenization of the n-dimensional problem.

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.