Section: New Results

Interactions of Macro- and Microscopic scales

Homogenization methods

We have obtained three types of results regarding the homogenization theory and its applications. The first series of results is related to nonlinear elasticity. In [44] , A. Gloria has proved the convergence of a discrete model for rubber towards a nonlinear elasticity theory in collaboration with R. Alicandro and M. Cicalese. This analysis has motivated the study of a specific random point set to model the stochastic network of polymer chains, namely the random parking measure, and results have been obtained by A. Gloria and M. Penrose (University of Bath) in [42] . The numerical simulation of the model with the random parking measure has been addressed by A. Gloria, P. La Tallec and M. Vidrascu (project team REO) in [21] , and the comparisons with mechanical experiments are promising, A related inverse problem is currently under investigation by M. de Buhan, A. Gloria, P. Le Tallec, and M. Vidrascu.

A second type of results concerns a quantitative theory of stochastic homogenization of discrete linear elliptic equations. A breakthrough has been obtained by A. Gloria and F. Otto (MPI Leipzig) in [63] and [24] , who gave the first optimal variance estimate of the energy density of the corrector field for stochastic discrete elliptic equations. The proof makes extensive use of a spectral gap estimate and of deep elliptic regularity theory, bringing in fact the probabilistic arguments to a minimum. This analysis has enabled A. Gloria to propose efficient numerical homogenization methods, both in the discrete and continuum settings [62] , [20] , see the review article [33] . In [23] , A. Gloria and J.-C. Mourrat has pushed the approach forward and introduced new approximation formulas for the homogenized coefficient. In [22] they have considered a more probabilistic approach and given a complete error analysis of a Monte-Carlo approximation of the homogenized coefficients in the discrete case. Work in progress concerns the generalization of the results on discrete elliptic equations to the continuum case.

The third direction of research concerns the periodic homogenization of a coupled elliptic/parabolic system arising in the modelling of nuclear waste storage. This work is in collaboration with the French agency ANDRA. A. Gloria, T. Goudon, and S. Krell have made a complete theoretical analysis of the problem, derived effective equations, and devised an efficient method to solve the effective problem numerically, based on the reduced basis approach, see [41] . This subject has been pushed forward by Z. Habibi in collaboration with ANDRA.

Statistical physics : molecular dynamics

In [28] , the analysis of constrained molecular dynamics is proposed, with associated numerical schemes.

In [29] , the pobabilistic derivation of the chemotaxis equation from the individual motion of bacteriae have been carried out. In [30] , a numerical method with asymptotic variance reduction have been proposed.