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Monday, 15 February 2021
Time Speaker Title Resources
12:35 to 12:50 A. K. Nandakumaran & P. Donato Welcome Remarks (Organizers)
12:50 to 13:00 Rajesh Gopakumar Welcome Remarks (ICTS, Center Director)
13:00 to 14:00 Olivier Guibé (University of Rouen Normandie, France) Renormalized Solutions for Elliptic Equations with L1 Data
14:10 to 15:10 François Murat (Sorbonne University and CNRS, France) An example of boundary homogenization: the homogenization of the Neumann's brush problem
15:20 to 16:20 Daniel Peterseim (Institut of Mathematics, University of Ausburg, Ausburg) Numerical Homogenization by Localized Orthogonal Decomposition

This series of lectures provides an introduction to the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. We discuss its mathematical derivation and error analysis, present aspects of the implementation of the LOD method in MATLAB and reveal how the method relates to classical homogenization and domain decomposition.

16:30 to 17:30 Andrey Piatnitski (The Arctic University of Norway, Norway and IITP RAS, Russia) Stochastic homogenization

The course will focus on several classical and more modern results in the files of homogenization of random operators.  In the first part we will discuss homogenization of elliptic differential operators in divergence form with stationary coefficients. We will consider classical approaches based on correctors, asymptotic expansions, and div-curl lemma.

Then we will introduce stochastic two-scale convergence and study the properties of this convergence. The application of stochastic two-scale convergence method to homogenization problems in random media will be illustrated with a number of examples such as homogenization in random perforated domain.

18:00 to 20:00 -- Informal Meeting
Tuesday, 16 February 2021
Time Speaker Title Resources
13:00 to 14:00 Olivier Guibé (University of Rouen Normandie, France) Renormalized Solutions for Elliptic Equations with L1 Data - 2
14:10 to 15:10 François Murat (Sorbonne University and CNRS, France) An example of boundary homogenization: the homogenization of the Neumann's brush problem - 2
15:20 to 16:20 Daniel Peterseim (Institut of Mathematics, University of Ausburg, Ausburg) Numerical Homogenization by Localized Orthogonal Decomposition - 2

This series of lectures provides an introduction to the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. We discuss its mathematical derivation and error analysis, present aspects of the implementation of the LOD method in MATLAB and reveal how the method relates to classical homogenization and domain decomposition.

16:30 to 17:30 Andrey Piatnitski (The Arctic University of Norway, Norway and IITP RAS, Russia) Stochastic homogenization - 2

The course will focus on several classical and more modern results in the files of homogenization of random operators.  In the first part we will discuss homogenization of elliptic differential operators in divergence form with stationary coefficients. We will consider classical approaches based on correctors, asymptotic expansions, and div-curl lemma.

Then we will introduce stochastic two-scale convergence and study the properties of this convergence. The application of stochastic two-scale convergence method to homogenization problems in random media will be illustrated with a number of examples such as homogenization in random perforated domain.

18:00 to 20:00 -- Brief presentation by Participants
Wednesday, 17 February 2021
Time Speaker Title Resources
13:00 to 14:00 Olivier Guibé (University of Rouen Normandie, France) Renormalized Solutions for Elliptic Equations with L1 Data - 3
14:10 to 15:10 François Murat (Sorbonne University and CNRS, France) An example of boundary homogenization: the homogenization of the Neumann's brush problem - 3
15:20 to 16:20 Daniel Peterseim (Institut of Mathematics, University of Ausburg, Ausburg) Numerical Homogenization by Localized Orthogonal Decomposition - 3

This series of lectures provides an introduction to the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. We discuss its mathematical derivation and error analysis, present aspects of the implementation of the LOD method in MATLAB and reveal how the method relates to classical homogenization and domain decomposition.

16:30 to 17:30 Andrey Piatnitski (The Arctic University of Norway, Norway and IITP RAS, Russia) Stochastic homogenization - 3

The course will focus on several classical and more modern results in the files of homogenization of random operators.  In the first part we will discuss homogenization of elliptic differential operators in divergence form with stationary coefficients. We will consider classical approaches based on correctors, asymptotic expansions, and div-curl lemma.

Then we will introduce stochastic two-scale convergence and study the properties of this convergence. The application of stochastic two-scale convergence method to homogenization problems in random media will be illustrated with a number of examples such as homogenization in random perforated domain.

18:00 to 20:00 -- Brief presentation by Participants
Thursday, 18 February 2021
Time Speaker Title Resources
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.
The present result is published in G.P.Panasenko, Multicontinuum wave propagation in a laminated beam with contrasting stiffness and density of layers, Journal of Mathematical Science, 232,  4, 2018, 503-515, https://doi.org/10.1007/s10958-018-3889-7,  translated from Problemy Matematicheskogo Analiza 93, 2018, pp. 89-99 https://doi.org/1072-3374/18/2286-0601, and was supported by the grant number 19-11-00033 of  Russian Science  Foundation executed by National Research University "Moscow Power Engineering Institute”.

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 [3] and in Mel'nyk [4] 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 [2] in order to handle such terms. This is a joint work with A.K. Nandakumaran and Ravi Prakash [5]. 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 [5].

References:
[1] G. Allaire, A. Damlamian, and U. Hornung, Two-scale convergence on periodic surfaces and applications, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, World Scientifc Pub., Singapore (1996), 15{25.
[2] A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete and continuous dynamical systems 23 (2009), no. 1 & 2, 197{219.
[3] A. Gaudiello, Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary, Ricerche Mat. 43 (1994), no. 2, 239{292.
[4] T. A. Mel'nyk, Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3 : 2 : 1, Math. Methods Appl. Sci. 31 (2008), no. 9, 1005{1027.
[5] R. Mahadevan, A. K. Nandakumaran and Ravi Prakash, Homogenization of an elliptic equation in a domain with oscillating boundary with non-homogeneous non-linear boundary conditions, Appl. Math. Optimization 82 (2020), 245{278.
[6] M. Neuss-Radu, Some extensions of two-scale convergence, C. R. Acad. Sci. Paris Ser. I Math 322 (1996), no. 9, 899{904.

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
of shocks.

21:15 to 22:00 Graeme Milton (University of Utah, USA) Beyond Homogenization

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.

Friday, 19 February 2021
Time Speaker Title Resources
12:00 to 12:45 Harsha Hutridurga (IIT Bombay, India) Elastic cloaking theory

In this talk, we will discuss some results on the cloaking of displacement _elds in linear elasticity. We shall demonstrate that, in the spirit of transformation media theory, the transformed governing equations in Cosserat and Willis frameworks are equivalent to certain high contrast small defect problems for the usual Navier equations. We will discuss near- cloaking for elasticity systems via a regularised transform and we shall also illustrate it via numerical experiments. We also study transmission problems when the Lame parameters in the inclusion tend to extreme values. We discuss both the soft and the hard inclusion limits. We will also propose an approximate isotropic cloak for a symmetrized Cosserat cloak.

12:45 to 13:30 Taras Mel’nyk (National Taras Shevchenko University of Kyiv, Ukraine) Asymptotic analysis of spectral problems in thick junctions with the branched fractal structure

The asymptotic behavior of the eigenvalues and eigenfunctions of a spectral problem in a thick junction with the branched fractal structure and the perturbed Robin boundary conditions on the boundaries of the branches is studied. The Hausdorff convergence of the spectrum is proved, the leading terms of asymptotics are constructed and the corresponding asymptotic estimates are justified both for the eigenvalues and eigenfunctions.

14:00 to 14:45 Rheadel Fulgencio (University of Rouen Normandie, France and University of the Philippines, Philippines) Homogenization of a Quasilinear Elliptic Problem in a Two-Component Domain with L1 data
14:45 to 15:30 Agnes Lamacz-Keymling (Universität Duisburg-Essen, Germany) High-order homogenization in optimal control by the Bloch wave method

In this talk we examine a linear-quadratic optimal control problem in which the cost functional and the elliptic state equation involve a highly oscillatory periodic coefficient $A^\varepsilon$. The small parameter $\varepsilon>0$ denotes the periodicity length. We propose a high-order effective control problem with constant coefficients and prove a corrector result which allows to approximate the original optimal solution with error $O(\varepsilon^M)$, where $M\in\mathbb{N}$ is as large as one likes. Our analysis relies on a Bloch wave expansion of the optimal solution and is performed in two steps. In the first step, we expand the lowest Bloch eigenvalue in a Taylor series to obtain a high-order effective optimal control problem. In the second step, the original and the effective problem are rewritten in terms of the Bloch and the Fourier transform, respectively. This allows for a direct comparison of the optimal solutions via the corresponding variational inequalities.

16:00 to 16:45 Carmen Perugia (University of Sannio, Italy) Homogenization results for a coupled system of reaction-diffusion equations

The macroscopic behavior of the solution of a coupled system of partial differential equations arising in the modeling of reaction-diffusion processes in periodic porous media is analyzed. This mathematical model can be used for studying several metabolic processes taking place in living cells, in which biochemical species can diffuse in the cytosol and react both in the cytosol and also on the organellar membranes. The coupling of the concentrations of the biochemical species is realized via various properly scaled nonlinear reaction terms. These nonlinearities, which model, at the microscopic scale, various volume or surface reaction processes, give rise in the macroscopic model to different effects, such as the appearance of additional source or sink terms or of a non-standard diffusion matrix.

16:45 to 17:30 Ravi Prakash (Universidad de Concepción, Chile) Unfolding Operator in locally periodic domains

Among several methods developed in the last five decades to study the study the asymptotic behavior of partial differential equations, the periodic unfolding is the latest. The significance of this operator is known for the study of asymptotic behavior of optimal control/controllability problems and also to characterize optimal control, etc. The periodic unfolding was developed by Cioranescu et. al. and was modified, later, to study problems in periodic rectangular oscillating boundary domains (pillar-type domains) by Blanchard et. Arrieta et al. have study the homogenization problems posed on locally periodic oscillating domains with amplitude of order ε. Here, the oscillating part shrinks down to the boundary of non-oscillating part of the domain as ε tends to zero. In this talk, we will analyze general unfolding operators for a wider class of locally periodic oscillating boundary domains. Moreover, we will also analyze the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain.