Abstract : A rich variety of models in mechanics are heterogenous and multi-scale in nature, and the derivation of „averaged“ or „effective“ behaviours on large-scales are well-known to be challenging and of particular interest. The complexity of the micro-structure often requires stochastic modeling and advanced methods, combining tools from PDE and probability, to understand and compute such effective properties. The purpose of this mini-symposium is to offer an overview of recent developments on the theory of stochastic homogenization and its applications in several areas of applied mathematics, ranging from fluids mechanics, wave propagation, nonlinear elasticity and statistical mechanics.
Organizer(s) : Nicolas Clozeau, Laure Giovangigli, Lihan Wang
[04699] Recent advances in quantitative stochastic homogenisation of nonlinear models
Format : Talk at Waseda University
Author(s) :
Nicolas Clozeau (Institute of science and technology Austria)
Antoine Gloria (Sorbonne Université)
Mathias Schäffner (Uni Halle)
Julian Fischer (Institute of science and technology Austria)
Antonio Agresti (Institute of science and technology Austria)
Abstract : I will present recent advances on the quantitative homogenisation of stochastic nonlinear models. First, I will discuss
the case of convex variational models described by its Euler-Lagrange equation, taking the form of a nonlinear elliptic equation in divergence form with monotone and random coefficients. I will present the quantitative homogenisation theory in current development with Antoine Gloria and Mathias Schäffner, aiming at describing the oscillations and fluctuations of solutions at the microscopic scale as well as the large-scale regularity theory of the random nonlinear operator. Second, I will discuss the case of non-convex variational models of Griffith type in fracture mechanics and a quantitative result concerning the convergence of the cell formula recently obtained in collaboration with Julian Fischer and Antonio Agresti.
[04559] Quantitative Homogenization for Nondivergence Form Equations
Format : Talk at Waseda University
Author(s) :
Jessica Lin (McGill University)
Abstract : In this talk, I will first give an overview of stochastic homogenization for nondivergence form equations (from the PDE perspective) and quenched invariance principles for nonreversible diffusion processes (from the probability perspective). I will then present various quantitative stochastic homogenization results and discuss challenges specific to the homogenization of nondivergence form equations. This talk is based on joint work with Scott Armstrong (NYU) and Benjamin Fehrman (Oxford).
[04584] Quantitative homogenization of elliptic system with periodic and high contrast coefficients
Format : Talk at Waseda University
Author(s) :
Wenjia Jing (Tsinghua University)
Xin Fu (Tsinghua University)
Abstract : We present several results about the quantitative estimates of the homogenization of elliptic systems in high contrast periodic media. The periodically distributed high contrast parts have physical parameters that are either extremely large or extremely small compared to those in the background. We develop a method that is somewhat unified and can treat both types of high contrast limits. We obtain quantitative convergence rates with proper correctors, uniform Lipschitz regularity for the solutions of the heterogeneous equations and, as an application, a quantitative description of the spectral convergence for the double-porosity problem. We also discuss possible extensions of the method to some other systems, e.g., linear elasticity, with richer high contrast structures.
[05476] On the lower spectrum of heterogeneous acoustic operators
Format : Online Talk on Zoom
Author(s) :
Mitia Duerinckx (Université Libre de Bruxelles)
Abstract : In this talk, we describe a quantitative link between homogenization and Anderson localization for heterogeneous acoustic operators: we draw consequences on the spatial spreading of eigenstates in the lower spectrum (if any) from the long-time homogenization of the wave equation, through dispersive estimates. This yields an alternative proof (avoiding Floquet theory) that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic coefficients, and it further provides nontrivial quantitative lower bounds on the spatial spreading of potential eigenstates in case of quasiperiodic and random coefficients. This is based on joint work with Antoine Gloria.
[03179] Boundary effects in radiative transfer of acoustic waves in a randomly-fluctuating medium delimited by boundaries
Format : Talk at Waseda University
Author(s) :
Adel Messaoudi (Aix-Marseille université)
Régis Cottereau (CNRS)
Christophe Gomez (Aix-Marseille université)
Abstract : This presentation discusses the derivation of radiative transfer equations for acoustic waves propagating in a randomly-fluctuating half-space and slab in the weak-scattering regime, and the study of boundary effects. These radiative transfer equations allow to model the transport of wave energy density, taking into account the scattering by random heterogeneities. The approach builds on an asymptotic analysis of the Wigner transform of the wave solution and the method of images.
[05197] Bloch analysis extended to weakly disordered periodic media
Format : Online Talk on Zoom
Author(s) :
Régis Cottereau (CNRS)
Yilun Li (CentraleSupélec)
Bing Tie (CNRS)
Abstract : The dispersion properties of periodic metamaterials can be tailored in order to obtain desirable effects, for instance band gaps over chosen frequency ranges. However, these patterns are sometimes completely destroyed by the small (random) defects introduced by the manufacturing processes, and the induced loss of periodicity of the metamaterials. This contribution explores the extension of the classical Bloch-Floquet theory to problems that are weakly non-periodic, using asymptotic analysis and a random mapping of the properties to a periodic reference.
[04385] Gamma-convergence and stochastic homogenisation for phase-transition models
Format : Talk at Waseda University
Author(s) :
Roberta Marziani (TU Dortmund)
Abstract : In this talk we discuss the gamma-convergence of general phase-transition functionals of Modica-Mortola type whose integrands depend both on the space variable and on the regularization parameter (which represents the characteristic length scale of phase transition). In particular we show that the limit is a surface functional whose integrand is characterized by the limit of a suitable cell formula. We then extend our analysis to the case of stochastic homogenization and prove a gamma-convergence result for stationary random integrands.
[05141] Anomalous diffusion of a passive tracer advected by the curl of the GFF in 2D
Format : Talk at Waseda University
Author(s) :
Peter Morfe (Max Planck Institute, Leipzig)
Georgiana Chatzigeorgiou (Max Planck Institute, Leipzig)
Lihan Wang (Max Planck Institute, Leipzig)
Felix Otto (Max Planck Institute, Leipzig)
Abstract : I will discuss the long-time asymptotics of the displacement of a passive tracer in a (time-independent) turbulent flow, where the velocity field equals the curl of the GFF, in two dimensions. Physicists long ago predicted that the mean-squared displacement scales like time with a logarithmic correction. In our contribution, we prove that this is indeed the case via a novel iterative argument that exploits fundamental ideas from the theory of stochastic homogenization.
[04734] Variance reduction methods in random homogenization by using surrogate models
Format : Online Talk on Zoom
Author(s) :
Frederic Legoll (Ecole des Ponts ParisTech and Inria)
Sebastien Brisard (Ecole des Ponts ParisTech)
Michael Bertin (Ecole des Ponts ParisTech and Inria)
Abstract : We consider the homogenization of elliptic PDEs with random coefficients. The associated corrector problem is set on the entire space, and is thus practically intractable. A standard approximation consists in restricting this problem to a large but bounded domain. The obtained effective coefficients are random. It is thus natural to consider several realizations.
To improve the accuracy on the expectation of the effective coefficients, we introduce a variance reduction approach based on a surrogate model.
