In the context of the construction of the European landscape of research, Inria and ULB (Université Libre de Bruxelles) signed in 2013 an agreement to foster joint research teams on topics of mutual interests. The team MEPHYSTO, a joint project of Inria, the Université Lille 1 and CNRS, and the Université Libre de Bruxelles, is the first such collaboration, in applied mathematics. It operates in two locations: Lille and Brussels. A. Gloria spends one day a week in Lille, G. Dujardin spends one day a week in Brussels, and S. Armstrong visits both Brussels and Lille on a monthly basis.

The main objective of the team is to develop mathematical and numerical tools to study in a quantitative way some specific physical models which display random and/or multiscale features. The emphasis is put on the interplay between analysis, probability, and numerics.

We focus our efforts on two prototypical examples: stochastic homogenization and the Schrödinger equations.

Whereas many models in physics involve randomness, they behave deterministically in suitable asymptotic regimes when stochastic effects average out. The qualitative and quantitative understanding of this deterministic behavior is the main challenge of this project.

From a mathematical point of view, our main fields of interest are stochastic homogenization of PDEs and random or deterministic one-dimensional nonlinear Schrödinger equations. These topics involve two challenges identified in the strategic plan of Inria “Objectif 2020": randomness and multiscale modeling.

From a physical point of view, the problems we shall consider find their origin in

the statistical physics of random polymer-chain networks;

light propagation in optical fibers.

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Stochastic homogenization
**

Homogenization is a theory which deals with oscillations in PDEs.
Let

for some suitable r. h. s.

The homogenized coefficients

In the periodic case, these correctors are well-behaved by standard PDE theory.
The convergence of *qualitative* ergodic theory ( , ). The following natural question, asked more than thirty years ago, is whether one can develop an associated *quantitative* ergodic theory.

One of our initial motivations to develop a quantitative stochastic homogenization theory
is the derivation of nonlinear elasticity from polymer physics, which is presented in the application section.
We plan to develop a complete quantitative theory of stochastic homogenization of elliptic equations.
In particular we aim at quantifying how well

**
Schrödinger equations
**

The linear Schrödinger equation, with an appropriate choice of geometry and boundary conditions, has been central to the description of all non-relativistic quantum mechanical systems for almost a century now. In addition, its nonlinear variant arises in the mean field description of Bose-Einstein condensates, where it is known as the Gross-Pitaevskii equation, but also in nonlinear classical optics, and in particular in fiber optics. The quantitative and qualitative description of its solutions (for both the evolution and the stationary equations), their time-asymptotic behavior, their stability or instability in terms of the parameters of the initial conditions and/or the potentials and boundary conditions continue to pose numerous physical and mathematical problems (see and for general references).

In view of our collaboration with the Lille laser physics laboratory PhLAM, we will focus more particularly on the one-dimensional nonlinear Schrödinger equation (NLS). Indeed, (NLS) drives the envelope of the propagation of a laser pulse in a Kerr medium, such as an optical fiber . Many phenomena on (NLS) (and variants thereof, with higher order derivatives, various types of initial conditions, external fields, etc.) are put in evidence by physical experiments at PhLAM, are not fully understood, and raise exciting questions from the numerical and analytical perspectives.

The same type of equation also describes Bose-Einstein condensates, for which questions related to Anderson localization are also of interest theoretically and experimentally at PhLAM.

Whereas numerical methods in nonlinear elasticity are well-developed and reliable, constitutive laws used for rubber in practice are phenomenological and generally not very precise. On the contrary, at the scale of the polymer-chain network, the physics of rubber is very precisely described by statistical physics. The main challenge in this field is to understand how to derive macroscopic constitutive laws for rubber-like materials from statistical physics.

At the continuum level, rubber is modelled by an energy

Our aim is to relate qualitatively and quantitatively the (precise but unpractical) statistical physics picture to explicit macroscopic constitutive laws that can be used for practical purposes.

In collaboration with R. Alicandro (Univ. Cassino, Italy) and M. Cicalese (Univ. Munich, Germany), A. Gloria analyzed in
the (asymptotic)

These preliminary results show that the variational model has the potential to explain qualitatively and quantitatively how rubber elasticity emerges from polymer physics. In order to go further and obtain more quantitative results and rigorously justify the model, we have to address several questions of analysis, modelling, scientific computing, inverse problems, and physics.

Whereas the approximation of homogenized coefficients is an easy task in periodic homogenization, this is a highly nontrivial task for stochastic coefficients. This is in order to analyze numerical approximation methods of the homogenized coefficients that F. Otto (MPI for mathematics in the sciences, Leipzig, Germany) and A. Gloria obtained the first quantitative results in stochastic homogenization . The development of a complete stochastic homogenization theory seems to be ripe for the analysis and constitutes the second major objective of this section.

In order to develop a quantitative theory of stochastic homogenization, one needs to quantitatively understand the corrector equation ().
Provided

They also proved that the variance of spatial averages of the energy density *first optimal quantitative results* in stochastic homogenization.

As well known, the (non)linear Schrödinger equation

with coupling constants

where

If

If now

The kicked rotor is a unitary discrete time dynamics proposed in the seventies in the context of studies on quantum chaos, and used recently as a “quantum simulator” for the Anderson model. It is a quantum equivalent of the standard map and is obtained by integrating a time-dependent linear Schrödinger equation with a time-periodic, very singular (delta comb) potential. It continues to pose considerable mathematical challenges, in particular the so-called “quantum suppression of classical chaos” in the presence of a strong potential, which remains an open problem from the mathematical point of view. It can be rephrased as follows: show that the

This problem displays both numerical and analytical challenges, in particular because of the difficulty to obtain long time simulations of the system and because of the presence of instabilities due to the nonlinearity. Preliminary theoretical results motivate some conjectures on the behavior of these systems, that we plan to validate empirically in a first step. Indeed, reliable long-time simulations of the system should allow us to get more insight into the behavior of the exact solutions in the unstable cases. One of the main difficulties for the numerical simulation is the intrinsic instability of the system, which magnifies quite rapidly the numerical error due to machine precision. This requires the use of multiprecision techniques in order to handle reasonably long times, even for moderate nonlinearities, and of the transparent boundary conditions recently introduced by members of the former SIMPAF project-team.

The mechanics of heterogeneous materials aims at characterizing the macroscopic properties of heterogeneous materials using the properties of their constituents.

The homogenization theory is a natural tool for this task. In particular, for linear problems (linear conductivity or linear elasticity), the macroscopic properties are encoded into a single (conductivity or elasticity) homogenized tensor. The numerical approximation of this homogenized tensor is a typical objective of quantitative homogenization.

For nonlinear problems, such as rubber elasticity, the macroscopic properties are no longer characterized by a single tensor, but rather by a nonlinear energy density. Our aim is to relate qualitatively and quantitatively the (precise but unpractical) statistical physics picture to explicit macroscopic constitutive laws that can be used for practical purposes. This endeavor is relevant both in science and technology. The rigorous derivation of rubber elasticity from polymer-physics was indeed emphasized by John Ball as an important open problem of nonlinear elasticity in his survey on the field. Its solution could shed light on some aspects of polymer-physics. The associated ab initio derivation of constitutive laws (as an alternative to phenomenological laws) would also be of interest to computational mechanics and rubber industry.

For this application domain, we work in close collaboration with physicists (François Lequeux, ESPCI) and researchers from mechanics and computational mechanics (Patrick Le Tallec, Ecole polytechnique).

Solving numerically PDEs in highly heterogeneous media is a problem encountered in many situations, such as the transport of pollutants or the design of oil extraction strategies in geological undergrounds. When such problems are discretized by standard numerical methods the number of degrees of freedom may become prohibitive in practice, whence the need for other strategies.

Numerical solution methods inspired by asymptotic analysis are among the very few feasible alternatives, and started fifteen years ago with the contributions of Hou and Wu ,
Arbogast etc.
We refer to , , for a recent state of the art.
Numerical homogenization methods usually amount to looking for the solution of the problem () in the form

Relying on our quantitative insight in stochastic homogenization, a first task consists in addressing the three following prototypical academic examples: periodic, quasi-periodic, and stationary ergodic coefficients with short range dependence. The more ambitious challenge is to address more complex coefficients (of interest to practioners), and design adaptive and efficient algorithms for diffusion in heterogeneous media.

Our contribution to the analysis of models in laser physics is motivated by the LabEx CEMPI (Centre Européen pour les Mathématiques, la Physique et leurs Interactions, a large eight-year research and training project approved by the French government in February 2012 as a “Laboratoire d'Excellence" and an initiative of mathematicians and physicists of the Université Lille 1). For this application domain, we work in close collaboration with physicists, which ensures our direct impact on these scientific issues. We focus on two applications: optical fibers and cold atoms.

In collaboration with physicists from the PhLAM laboratory in Lille, we aim at developping new techniques for the numerical integration of a family of 1D Schrödinger-like equations modelling the propagation of laser pulses in optical fibers. The questions arising are challenging since physicists would like to have fairly fast and cheap methods for their problems, with correct qualitative and quantitative behaviors. Another point is that they are interested in methods and codes that are able to handle different physical situations, hence different terms in the NLS equation. To meet these requirements, we will have to use numerical time-integration techniques such as splitting methods or exponential Runge-Kutta methods, space discretization techniques such as finite differences and fast Fourier transforms, and absorbent boundary conditions. Our goal, together with the physicists is to be able to reproduce numerically the results of the experiments they make in actual optical fibers, and then to be able to tune parameters numerically to get more insight into the appearance of rogue waves beyond the dispersive blowup phenomenon.

Recall that the Schrödinger equation also describes Bose-Einstein condensates. A second experimental team at PhLAM projects to study questions related to Anderson localization in such condensates. In fact, they will realize the “kicked rotor” (see ), which provides a paradigm for Anderson localization, in a Bose-Einstein condensate. We plan to collaborate with them on the theoretical underpinnings of their findings, which pose many challenging questions.

The numerical method to approximate the constitutive laws for rubber elasticity derived from polymer physics (as used in , ) are implemented in the Inria software Modulef (joint work of M. Vidrascu, projetc-team REO, and A. Gloria).

It is based on

algorithms from stochastic geometry to generate suitable polymer networks;

Delaunay tessellation algorithms to deal with steric effects (courtesy of the Inria project-team GAMMA2);

the introduction of 1-dimensional finite elements for the polymer-chains in Modulef.

To solve the inverse problem for the reconstruction of an explicit constitutive law from in silico experiments in , we relied on the Covariance Matrix Adaptation Evolution Strategy developed in the project-team TAO.

The numerical methods proposed in for the approximation of homogenized coefficients were implemented in FreeFEM++, a user-friendly PDE-solver.

A. Gloria, S. Neukamm, and F. Otto published their recent contribution on quantitative homogenization in Inventiones Mathematicae.

As a plenary speaker of the World Congress of Computational Mechanics in Barcelone in July 2014, P. Le Tallec (Ecole polytechnique) presented our joint results , .

A. Gloria, S. Neukamm (Univ. Dresden), and F. Otto (MPI for mathematics in the sciences, Leipzig) developed in a general approach to quantify ergodicity in stochastic homogenization of discrete elliptic equations. Using a parabolic approach, they obtained optimal estimates on the time-decay of the so-called environment seen from the particle. This allowed them to prove optimal bounds on the corrector gradient and the corrector itself in any dimension (thus improving on ). They also obtained the first error analysis of the popular periodization method to approximate the homogenized coefficients.

In , A. Gloria and F. Otto extended their results , on discrete elliptic equations to the continuum setting. They treated in addition
the case of non-symmetric coefficients, and obtained optimal estimates in all dimensions by the elliptic approach (whereas , were suboptimal for

In , A. Gloria and D. Marahrens (MPI for mathematics in the sciences, Leipzig) extended the annealed results on the discrete Green function by D. Marahrens and F. Otto to the continuum setting. As a by-product of their result, they obtained new results in uncertainty quantification by estimating optimally the variance of the solution of an elliptic PDE whose coefficients are perturbed by some noise with short range of dependence.

In their recent work , A. Gloria, S. Neukamm, and F. Otto developed a regularity theory for random elliptic operators inspired by the contributions of Avellaneda and Lin in the periodic setting and of our visitor S. Armstrong with C. Smart . This allowed them to consider coefficients with arbritarily slow decaying correlations in the form of a family of correlated Gaussian fields.

In , A. Gloria and J. Nolen (Duke Univ.) proved a quantitative central limit theorem for the effective conductance on the discrete torus. In particular, they quantified the Wasserstein distance between a normal random variable and the CLT-like rescaling of the difference between the approximation of the effective conductance by periodization and the effective conductance. Their estimate is sharp and shows that the Wasserstein distance goes to zero (up to logarithmic factors) as if the energy density of the corrector was iid (which it is not). This completes and settles the analysis started in on the approximation of homogenized coefficients by periodization by characterizing the limiting law in addition to the scaling.

In , A. Gloria, P. Le Tallec (Mechanics department, Ecole polytechnique), and M. Vidrascu (Project-team REO, Inria) numerically investigated the nonlinear elasticity model obtained in by discrete stochastic homogenization, and compared it to the standard measurements by Treloar on natural rubber. The implementation was realized in the Modulef software. These results are in rather good agreement, which shows that the approach seems to be promising.

In , M. de Buhan (CNRS, Univ. Paris Descartes), A. Gloria, P. Le Tallec and M. Vidrascu proposed a numerical method to produce analytical approximations (that can be used in practical nonlinear elasticity softwares) of the numerical approximations obtained in of the discrete-to-continuum energy density derived theoretically in . This numerical method is based on the parametrization of the set of polyconvex Ogden laws and on the combination of a least square method and a genetic algorithm (cf. CMA-ES).

Inspired by the quantitative analysis of and , Z. Habibi (former SIMPAF post-doctoral fellow) and A. Gloria introduced in a general method to reduce the so-called resonance error in numerical homogenization, both at the levels of the approximation of the homogenized coefficients and of the correctors. This method significantly extends . The method relies on the introduction of a massive term in the corrector equation and of a systematic use of Richardson extrapolation. In the three academic examples of heterogeneous coefficients (periodic, quasiperiodic, and Poisson random inclusions), the method yields optimal theoretical and empirical convergence rates, and outperforms most of the other existing methods.

S. De Bièvre, S. Rota Nodari (CEMPI postdoc 2013-2015) and F. Genoud (CEMPI visitor, September 2013) have explained the geometry underlying the so-called energy-momentum method for proving orbital stability in infinite dimensional Hamiltonian systems. Applications include the orbital stability of solitons of the NLS and Manakov equations. This work is to appear as a chapter (120p) in the first volume of the CEMPI Lecture Notes in Mathematics, cf. .

S. De Bièvre and his PhD student E. Soret rigorously proved the growth rate of the energy in a Markovian model for stochastic acceleration of a particle in a random medium, cf. .

G. Dujardin introduced an energy preserving method for Hamiltonian dynamics perturbed by a multiplicative noise, cf. .

The MEPHYSTO team is currently hosting former members of the project-team SIMPAF who focus on numerical methods for dissipative systems:

a posteriori estimates .

T. Gallouët also made contributions in optimal transport , .

The team (C. Chainais and A. Gloria) has had its third bilateral contract with ANDRA (French nuclear waste storage agency) from December 2012 to June 2014. The post-doctoral position of T. Gallouët was funded by this contract.

This collaboration concerned mathematical and numerical issues on a corrosion model, and in particular the identification of steady-states and the design of asymptotic-preserving schemes for a free interface problem.

G. Dujardin and I. Lacroix are members of the ANR BECASIM project (http://

Title: Simulation numérique avancée pour les condensats de Bose-Einstein.

Type: Modèles Numériques - 2012

ANR reference: ANR-12-MONU-0007

Coordinator: Ionut DANAILA, Université de Rouen.

Duration: January 2013 - December 2016.

Partners: Université Lille 1, UPMC, Ecole des Ponts ParisTech, Inria-Nancy Grand-Est, Université Montpellier 2.

Title: Centre Européen pour les Mathématiques, la Physique et leurs interactions

Coordinator: Stephan De Bièvre.

Duration: January 2012 - December 2019.

Partners: Laboratoire Paul Painlevé and Laser physics department (PhLAM), Université Lille 1.

The “Laboratoire d'Excellence” Centre Européen pour les Mathématiques, la Physique et leurs interactions (CEMPI), a project of the Laboratoire de Mathématiques Paul Painlevé and the Laboratoire de Physique des Lasers, Atomes et Molécules (PhLAM), was created in the context of the "Programme d'Investissements d'Avenir" in February 2012.

The association Painlevé-PhLAM creates in Lille a research unit for fundamental and applied research and for training and technological development that covers a wide spectrum of knowledge stretching from pure and applied mathematics to experimental and applied physics.

One of the three focus areas of CEMPI research is the interface between mathematics and physics. This focus area encompasses three themes. The first is concerned with key problems of a mathematical, physical and technological nature coming from the study of complex behaviour in cold atoms physics and non-linear optics, in particular fibre optics. The two other themes deal with fields of mathematics such as algebraic geometry, modular forms, operator algebras, harmonic analysis and quantum groups that have promising interactions with several branches of theoretical physics.

Type: FP7

Instrument: ERC Starting Grant

Duration: February 2014 - January 2019

Coordinator: Antoine Gloria

Partner: Département de mathématique, Université Libre de Bruxelles (Belgium)

Inria contact: Antoine Gloria

Abstract: Quantitative methods in stochastic homogenization

Max Planck Institute for Mathematics in the Sciences (Germany).

Development of a quantitative theory of stochastic homogenization.

The activity around quantitative stochastic homogenization was developed in collaboration with F. Otto, director of the Max Planck Institute for Mathematics in the Sciences, Leipzig (Germany).

Daniel Marahrens (MPIMS, Leipzig): one week in March (A. Gloria), annealed estimates on Green's functions.

Felix Otto (MPIMS, Leipzig): one week in April (A. Gloria), quantitative stochastic homogenization.

Gilbert Reinisch (physicist at University of Reykjavik): from May 12th 2014 to May 28th 2014 (G. Dujardin and M. Gazeau), numerical simulations of several differential systems modelling the evolution of quantum dots. This visit was cofounded by Inria and the LabEx CEMPI. This work is a follow up of the paper .

Marco Cicalese (Univ. Munich): one week in May (A. Gloria), derivation of nonlinear elasticity from polymer-physics.

Jean-Christophe Mourrat (ENS Lyon): 10 days in May (A. Gloria), quantitative stochastic homogenization.

Stefan Neukamm (Weierstrass Institute, Berlin, now at Univ. Dresden): 10 days in May (A. Gloria), quantitative stochastic homogenization.

Ansgar Jüngel (Univ. Vienna): one week in July (C. Chainais and I. Lacroix-Violet), discrete functional inequalities for asymptotic preserving schemes.

S. De Bièvre is the scientific coordinator of CEMPI

D. Bonheure and A. Gloria organize a PDE and analysis seminar at Brussels (http://

G. Dujardin and I. Lacroix are in charge of the PDE and numerical analysis seminar of the Paul Painlevé mathematics department in Lille (http://

G. Francfort (Paris 13), A. Gloria, and M. Kruzik organized a workshop "Relaxation, homogenization and dimensional reduction in hyperelasticity" at Université Paris-Nord, March 25-27, 2014 (http://

D. Bonheure is associate editor at the Bulletin of the Belgian Mathematical Society - Simon Stevin (http://

C. Chainais and A. Gloria are associate editors at the North-Western European Journal of Mathematics (http://

The members of the team reviewed numerous papers for numerous international journals.

The members of the team are mainly academics. We teach at the licence and master levels at the Université Lille 1 and ULB.

PhD in progress: P.-L. Colin, Theoretical and numerical study of some corrosion models, since October 2012, advised by C. Chainais and I. Lacroix-Violet.

PhD in progress: M. Duerinckx, Problems in stochastic homogenization, since October 2014, advised by A. Gloria and S. Serfaty (UPMC).

PhD in progress: E. Soret, Stochastic acceleration and thermalization, since October 2011, advised by S. De Bièvre and T. Simon (Lille 1).

C. Calgaro is the organizer of the action "Mathématiques itinérantes" (http://