| Grégoire Allaire | Introduction to periodic homogenization |
| This short course is an introduction to the simplest of all homogenization methods, the so-called periodic one. The goal of homogenization is to deduce a simplified macroscopic model from a detailed microscopic one. The underlying hypothesis in the present setting is to assume that the medium or the equations coefficients are periodic and that the ration between this period and a charcateristic length is small. We shall discuss the methods of two-scale asymptotic expansions, of two-scale convergence, various applications in mechanics and physics, as well as a brief overview of the general non-periodic case (H-convergence theory). |
| Josselin Garnier | Stochastic homogenization |
| We will introduce some elements of probability theory, including stationary and ergodic processes, Brownian motion and diffusion processes, and the probabilistic interpretation of second-order PDEs. We will study the homogenization of a linear elliptic equation in a stationary random medium and the associated abstract cell problem. We will revisit the homogenization of elliptic and parabolic PDEs in periodic media with probabilistic techniques based on the representation of the solution of the PDE as the expectation of a functional of a diffusion process. Finally we will extend these techniques to random media. |
| Antoine Gloria | Numerical homogenization: Issues, methods, and convergence analysis |
| This course is dedicated to the homogenization of linear elliptic PDEs from the numerical point of view. The aim of the course is to introduce and analyze numerical methods to approximate the effective behavior of solutions to PDEs with highly-oscillating coefficients, as well as their fine scale oscillations around the effective behavior. Starting point will be standard numerical homogenization methods, whose convergence is ensured by H-convergence arguments and general corrector results. The so-called resonance error associated with these methods will be put in evidence in the periodic case, as well as its origin. This will allow us to treat separately each source of the error, and therefore efficiently reduce the resonance error. The analysis of this method will be presented in the periodic case and in the stationary stochastic case with finite correlation-length. This course is based on a series of works in collaboration with Felix Otto (Max Planck Institute) and Jean-Christophe Mourrat (EPFL). |
| Marc Briane | Hall effect in composites: positivity properties and meta-materials |
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In a homogeneous conductor a low magnetic field induces a transversal electric field both orthogonal to the current and to the magnetic field. The perturbed resistivity is measured at the first-order by the Hall coefficient and at the second-order by the magneto-resistance. In a two-dimensional composite the effective Hall coefficient preserves the bounds of the local Hall coefficient. The key ingredient for this result is the positivity of the determinant of the local matrix-valued electric fields. In dimension three the situation is radically different. Three meta-materials illustrate various pathologies which are inconsistent with the classical principles: reversal of the sign of the Hall coefficient, arbitrary large Hall coefficients, and an effective Hall field parallel to the magnetic field. On the other hand, again in dimension two the effective magneto-resistance is shown to satisfy a positivity property. Moreover, in the case of a strong magnetic field new bounds satisfied by the effective resistivity are derived in columnar isotropic composites.
These results were obtained in collaboration with G.W. Milton from the University of Utah. |
| François Castella | Almost periodic functions and strong confinement for the Schroedinger equation : some results. |
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We study the limiting behavior of a nonlinear Schroedinger equation
describing a 3 dimensional gas that is strongly confined along the vertical, z
direction. The confinement induces fast oscillations in time,
that need to be averaged out. Since the Hamiltonian in the z direction is merely assumed confining,
without any further specification, the associated
spectrum is discrete but arbitrary, and the fast oscillations induced
by the nonlinear equation entail countably many frequencies that are
arbitrarily distributed. For that reason,
averaging can not rely on small denominator estimates or like.
To overcome these difficulties, we prove that the fast oscillations are almost periodic in time, with values in a Sobolev-like space that we completely identify. We then exploit the existence of long time averages for almost periodic function to perform the necessary averaging procedure in our nonlinear problem. This is a joint work with N. Ben Abdallah (Toulouse, deceased) and F. Mehats (Rennes). |
| Claude Le Bris | Some recent variants of, and some novel numerical approaches for stochastic homogenization |
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The talk will overview some recent contributions on several theoretical aspects and numerical approaches in the modelling of random materials. In particular, some variants of the classical theory of stochastic homogenization introduced in [4] will be presented. The relation between stochastic homogenization problems
and other multiscale problems in materials science [5] will be emphasized. Several numerical approaches will be presented, for acceleration of convergence in the deterministic [6] and the stochastic [7,9] cases, as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model [1,2,3,8].
References : [1] A. Anantharaman, C. Le Bris, Homogenization of a weakly randomly perturbed periodic material, C. R. Math. Acad. Sci., 348 (2010) 529-534. [2] A. Anantharaman, C. Le Bris, A numerical approach related to defect-type theories for some weakly random problems in homogenization, submitted to SIAM MMS, http://arxiv.org/abs/1005.3910 [3] A. Anantharaman, C. Le Bris, Elements of mathematical foundations for a numerical approach for weakly random homogenization problems, submitted to Communications in Computational Physics, http://arxiv.org/abs/1005.3922 [4] X. Blanc, C. Le Bris, P.-L. Lions, Stochastic homogenization and random lattices, J. Math. Pures Appl., 88, pp 34-63, 2007. [5] X. Blanc, C. Le Bris, P.-L. Lions, The energy of some microscopic stochastic lattices, Arch. Rat. Mech. Anal., 184, pp 303-339, 2007. [6] X. Blanc, C. Le Bris, Improving on homogenized coefficients in the periodic and quasi-periodic settings, Netw. Heterog. Media, Volume 5, Number 1, March 2010, pp 1-29. [7] R. Costaouec, C. Le Bris, F. Legoll, Variance reduction in stochastic homogenization: proof of concept, using antithetic variables, Bol. Soc. Esp. Mat. Apl., 50, pp 9-27, 2010. [8] R. Costaouec, C. Le Bris, F. Legoll, Numerical approximation of a class of problems in stochastic homogenization, C. R. Math. Acad. Sci., 348, Série 1, p 99-103, 2010. [9] C. Le Bris, Some numerical approaches for "weakly" random homogenization, Proceedings of ENUMATH 2009, Lect. Notes Comput. Sci. Eng., Springer, in press. |
| Antoine Lejay | Simulation of stochastic processes in discontinuous media : some solutions and open problems |
| Monte Carlo methods offer a simple tool to simulate particles evolving in a complex media, simply by simulating the dynamic of the particles in view of their local environment. In geophysics, for example, several effective coefficients and physical quantities (pressure, ...) may be computed this way. However, it is necessary to describe the dynamic of the particles, which is done usually thanks to the theory of stochastic differential equations. During this talk, we deal we the case of discontinuous coefficients, that is when the diffusivity abruptely change. We will then see some of possible methods. Yet we also discuss the theoretical and practical difficulties to bypass in order to develop numerical schemes. |
| George Papanicolaou | Numerical methods for multiscale problems |
| Homogenization is now into its fourth decade and it continues to be an attractive and challenging research area in analysis and probability, with many open problems. But the real challenge today is to design, analyze and implement computational methods that capture homogenization effects efficiently without explicit intervention with analytical or asymptotic methods. Is this possible? Can a computational method "recognize" adaptively the multiscale structure of, say, an elliptic boundary value problem and produce homogenization-compatible solutions without resolving the smallest scales? I will address this issue and after a brief overview I will discuss a projection based approach that is compatible with homogenization. This joint work with J. Nolen and O. Pironneau. |
| Etienne Pardoux | Homogenization of non-uniformly elliptic second order operators with periodic coefficients |
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The purpose of this talk is to present the results obtained in [1], [2] and [3]. The results in [1] extend the classical homogenization results for second order periodic operators to cases when the diffusion matrix can be degenerate on an open set with non-zero Lebesgue measure. The assumptions ensuring the ergodicity of the underlying diffusion process are of nonlocal type. A necessary and sufficient non-degeneracy condition for the matrix of the second order coefficients of the limit operator is also provided. The results in [2] and [3] extend the results in [1] to the cases of semilinear parabolic and elliptic PDEs. [1] M. Hairer, E. Pardoux, Homogenisation of periodic linear degenerate PDEs, J. Funct. Anal., 255, No. 9, 2462-2487, 2008. [2] E. Pardoux, R. Rhodes, A. B. Sow, Homogenization of periodic semilinear parabolic degenerate PDEs, Ann. I. H. P. AN, 26, 979-998, 2009. [3] E. Pardoux, A. B. Sow, Homogenization of a periodic degenerate semilinear elliptic PDE, Stoch. and Dynamics, 2011, in press. |