I will discuss a model of interacting particles in continuous space which is reversible with respect to Poisson point measures with constant density. Similar discrete models are known to ”homogenize”, in the sense that the evolution of the particle density can be approximated by the solution to a partial differential equation over large scales. The goal of the talk is to present some results that make this approximation quantitative.
Based on joint works with Arianna Giunti, Chenlin Gu and Maximilian Nitzschner.[-]

The simulation of random heterogeneous materials is often very expensive. For instance, in a homogenization setting, the homogenized coefficient is defined from the so-called corrector function, that solves a partial differential equation set on the entire space. This is in contrast with the periodic case, where he corrector function solves an equation set on a single periodic cell. As a consequence, in the stochastic setting, the numerical approximation of the corrector function (and therefore of the homogenized coefficient) is a challenging computational task.
In practice, the corrector problem is solved on a truncated domain, and the exact homogenized coefficient is recovered only in the limit of infinitely large domains. As a consequence of this truncation, the approximated homogenized coefficient turns out to be stochastic, even though the exact homogenized coefficient is deterministic. One then has to resort to Monte-Carlo methods, in order to compute the expectation of the (approximated, apparent) homogenized coefficient within a good accuracy. Variance reduction questions thus naturally come into play, in order to increase the accuracy (e.g. reduce the size of the confidence interval) for a fixed computational cost. In this talk, we will present some variance reduction approaches to address this question.[-]

Traditionally homogenization asks whether average behavior can be discerned from Hamilton-Jacobi equations that are subject to high-frequency fluctuations in spatial variables. A similar question can be asked for the associated Hamiltonian ODEs. When the Hamiltonian function is convex in momentum variable, these two questions turn out to be equivalent. This equivalence breaks down for general Hamiltonian functions. In this talk I will give a dynamical system formulation for homogenization and address some result concerning weak and strong homogenization phenomena.[-]

Lamination of two materials with tensors $L_{1}$ and $L_{2}$ generates an effective tensor $L_{*}$. For certain fractional linear transformations $W(L)$, dependent on the property under consideration (conduction, elasticity, thermoelasticity, piezoelectricity, poroelasticity, etc.) and on the direction of lamination, $W\left(L_{*}\right)$ is just a weighted average of $W\left(L_{1}\right)$ and $W\left(L_{2}\right)$, weighted by the volume fractions occupied by the two materials, and this gives $L_{*}$ in terms of $L_{1}$ and $L_{2}$. Given an original set of materials one may laminate them together iteratively on larger and larger widely separated length scales, at each stage possibly laminating together two materials both of which are already laminates. Ultimately, one gets a hierarchical laminate with the lamination process represented by a tree with the leaves corresponding to the original set of materials, and with the volume fractions and direction of lamination specified at each vertex. It is amazing to see the range of effective tensors $L_{*}$ one can obtain, or effective tensor functions $L_{*}\left(L_{1}, L_{2}\right)$ one can obtain if, say, there are just two original materials. These functions $L_{*}\left(L_{1}, L_{2}\right)$ are closely related to Herglotz-Nevanlinna-Stieltjes functions. Here we will survey many results, some old, some surprising, on what effective tensors, and effective tensor functions, can be obtained lamination. In some cases the effective tensor or effective tensor function of any microgeometry can be mimicked by one of a hierarchical laminate. The question of what can be achieved is closely tied to the classic problem of rank-1 convexification and whether and when it equals quasiconvexification.[-]