|12:00 to 12:45||Mohamed Mogtaba (Majmaah University KSA and AIMS, South Africa)||
Homogenization and Correctors for Linear Stochastic Equations in Periodically Perforated Domains with Non-linear Robin Conditioins
In this talk, we present homogenization and corrector results for stochastic linear parabolic equations in periodically perforated domains with non-linear Robin conditions on the holes. We use the periodic unfolding method and probabilistic compactness results due to Prokhorov and Skorokhod. The main challenges here are: First, one is not dealing with random variables only but one deals with some type of noise (White, colored or Levy) which makes the use of deterministic compactness results not enough. Hence, probabilistic compactness and tightness of probability measurer needed in order to obtain some types of strong convergences. Second, the variational formulation contains some boundary terms on the boundary of the holes (which is a varying surface)for this, we use the periodic unfolding operator to transform the boundary integrals into integrals on a fixed surface. Homogenization results presented in this talk are stochastic counterparts of some fundamental work given in [D. Cioranescu; P. Donato and R. Zaki, Asymptotic Analysis, vol. 53, no. 4, pp. 209-235, 2007]. We show that the sequence of solutions of the original problem converges in suitable topologies to the solution of a homogenized problem, which is a parabolic stochastic equation in fixed domain with Drichlet condition on the boundary. In contrast to the two scale convergence method, the corrector results obtained in this work are without any additional regularity assumptions on the solutions of the original problems.
|12:45 to 13:30||Igor Velcic (University of Zagreb, Croatia)||Stochastic homogenization of high contrast media|
|14:00 to 14:45||Gregory Panasenko (Université de Lyon / Université de Saint Etienne, France)||
Mitlicontinuum effective model for the wave equation in a high contrast laminated beam
The talk deals with the wave equation in a thin laminated beam with contrasting stiffness and density of layers. The problem contains two parameters: ε is a geometric small parameter, ratio of the diameter and its characteristic longitudinal size, ω is a physical great parameter, ratio of stiffness and densities of alternating layers. The asymptotic behavior of the solution depends on the combination of parameters ε2ω. If this value is small then the limit model is a standard homogenized one-dimensional wave equation. On the contrary if ε2ω is not small then the limit model is presented by so called multi-continuum model, i.e. multiple one dimensional wave equations, coupled or non-coupled and "co-existing" in every point. The proof of these results uses the milticomponent homogenization method proposed in the paper G.P.Panasenko Multicomponent homogenization of processes in strongly non-homogeneous structures, Mathematics USSRSbornik,1990, 181,1,134-142(in Russian); English transl. in Math. USSR Sbornik,1991, 69, 1, 143-153.
|14:45 to 15:30||Umberto De Maio (Università degli Studi di Napoli Federico II, Italy)||
Exact controllability for wave equation in a domain with very rough boundary
The aim of this talk is to present some results, about the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. The limit problem with Neumann condition on the oscillating boundary is different and hence we need to study the exact controllability of this new type of problem.
|16:00 to 16:45||Barbara Verfürth (Karlsruher Institut für Technologie, Germany)||
Numerical homogenization approaches for nonlinear problems
Many applications, such as geophysical flow problems, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge. In this talk, we discuss how to construct a problem-adapted multiscale basis in a linearized and localized fashion for nonlinear problems such as the quasilinear diffusion equation or the nonlinear Helmholtz equation. The corresponding generalized finite element method gives optimal error estimates up to linearization errors. In particular, neither higher regularity of the exact solution nor structural properties of the coefficients such as scale separation or periodicity need to be assumed. Numerical examples show very promising results illustrating the theoretical convergence rates.
|16:45 to 17:30||Rajesh Mahadevan (Universidad de Concepción, Chile)||
Homogenization of an elliptic equation in a domain with oscillating boundary with non-homogeneous non-linear boundary conditions
We study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. Homogenization under non-homogeneous Neumann condition or the Robin boundary condition on an oscillating boundary has the induced difficulty of taking limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello  and in Mel'nyk  by converting the surface term into a volume term by using auxiliary boundary value problems. In this work, we show the efficacy of using the unfolding operator  in order to handle such terms. This is a joint work with A.K. Nandakumaran and Ravi Prakash . Some problems of this nature have also been studied using an extension of the notion of two-scale convergence [1, 6]. For more extensive references we refer to the article .
|20:30 to 21:15||Konstantin Lurie (Worcester Polytechnic Institute, US)||
Mathematical Optimization and Dynamic Materials
Material optimization, after several millennia of its progress as an art, has begun to be transformed into science (as a part of applied mathematics) only about 60 years ago. The first stage of its development was concerned with statics, its mathematical embodiment being the elliptic equations with coefficients variable in space alone, and the material implementation being the assemblies of original constituents in space. This concept has received major development in hundreds, if not thousands, of publications, but appeared insufficient in dynamics, i.e., the situations arising in a real world that exists in space and time. The static material formations are not sufficient for the adequate analysis of dynamic problems because they are not adjusted to the temporal variations of the environment. Such adjustment become possible if the material properties are themselves variable in space and time; this requirement has produced a novel concept of dynamic materials (DM) that has been put forth some 20 years ago. Mathematically, this concept is related to hyperbolic equations, with coefficient controls variable in space and time. In the talk, the basics of the DM will be exposed, with their conceptual classification and examples illustrating the effective properties of laminates in 1D plus time. The role of homogenization in the analysis of wave propagation will be discussed along with special examples illustrating the material layouts in space-time demonstrating focusing of the wave routes. The focusing may be accompanied by formation of moving clots, or develop asymptotically approaching the limit cycles - the linear analogs
|21:15 to 22:00||Graeme Milton (University of Utah, USA)||
In homogenization one takes a sequence of inhomogeneous materials and examines convergence of the associated fields in an appropriate subsequence. However, in physical experiments one just has one material, so this begs the question: how much of the analysis in homogenization theory carries over to a single inhomogeneous body? Using the framework of the abstract theory of composites we see that the Dirichlet-Neumann map, governing the response of the body, plays the same role as effective tensors in composite and as a result shares, for example, the same analyticity properties as effective tensors. Moreover, this carries through to wave equations. Additionally, an extension of the theory of exact relations in composites that provides microstructure independent identities satisfied by certain classes of composites, reveals itself as exact identities satisfied by the infinite body Green's function (fundamental solution) in certain classes of inhomogenous media. For bodies containing inhomogeneous media it implies exact relations satisfied by the Dirichlet-Neumann map. Equivalently, it implies certain boundary field equalities that can be viewed as extensions of conservation laws. This is joint work with Maxence Cassier, Daniel Onofrei, and Aaron Welters.