The MATHERIALS project-team has been created jointly by the École des Ponts ParisTech (ENPC) and Inria in 2015. It is the follow-up and an extension of the former project-team MICMAC originally created in October 2002. It is hosted by the CERMICS laboratory (Centre d'Enseignement et de Recherches en Mathématiques et Calcul Scientifique) at École des Ponts. The permanent research scientists of the project-team have positions at CERMICS and at two other laboratories of École des Ponts: Institut Navier and Laboratoire Saint-Venant. The scientific focus of the project-team is to analyze and improve the numerical schemes used in the simulation of computational chemistry at the microscopic level and to create simulations coupling this microscopic scale with meso- or macroscopic scales (possibly using parallel algorithms). Over the years, the project-team has accumulated an increasingly solid expertise on such topics, which are traditionally not well known by the community in applied mathematics and scientific computing. One of the major achievements of the project-team is to have created a corpus of literature, authoring books and research monographs on the subject , , , , that other scientists may consult in order to enter the field.

Our group, originally only involved in electronic structure computations, continues to focus on many numerical issues in quantum chemistry, but now expands its expertise to cover several related problems at larger scales, such as molecular dynamics problems and multiscale problems. The mathematical derivation of continuum energies from quantum chemistry models is one instance of a long-term theoretical endeavour.

Quantum Chemistry aims at understanding the properties of matter through
the modelling of its behavior at a subatomic scale, where matter is
described as an assembly of nuclei and electrons.
At this scale, the equation that rules the interactions between these
constitutive elements is the Schrödinger equation. It can be
considered (except in few special cases notably those involving
relativistic phenomena or nuclear reactions)
as a universal model for at least three reasons. First it contains all
the physical
information of the system under consideration so that any of the
properties of this system can in theory be deduced from the
Schrödinger
equation associated to it. Second, the Schrödinger equation does not
involve any
empirical parameters, except some fundamental constants of Physics (the
Planck constant, the mass and charge of the electron, ...); it
can thus be written for any kind of molecular system provided its
chemical
composition, in terms of natures of nuclei and number of electrons,
is known. Third, this model enjoys remarkable predictive
capabilities, as confirmed by comparisons with a large amount of
experimental data of various types.
On the other hand, using this high quality model requires working with
space and time scales which are both very
tiny: the typical size of the electronic cloud of an isolated atom is
the Angström (*all* macroscopic properties can be
simply upscaled from the consideration of the short time behavior of a
tiny sample of matter. Many of them derive from ensemble or bulk
effects, that are far from being easy to understand and to model.
Striking examples are found in solid state materials or biological
systems. Cleavage, the ability of minerals to naturally split along
crystal surfaces (e.g. mica yields to thin flakes), is an ensemble
effect. Protein folding is
also an ensemble effect that originates from the presence of the
surrounding medium; it is responsible for peculiar properties
(e.g. unexpected acidity of some reactive site enhanced by special
interactions) upon which vital processes are based.
However, it is undoubtedly true that *many* macroscopic phenomena originate from
elementary processes which take place at the atomic scale. Let us
mention for instance the fact that
the elastic constants of a perfect crystal or the color of a chemical
compound (which is related to the wavelengths
absorbed or emitted during optic transitions between electronic
levels) can be evaluated by atomic scale calculations. In the same
fashion, the lubricative properties of graphite are essentially due to a
phenomenon which can be entirely modeled at the atomic scale.
It is therefore reasonable to simulate the behavior of matter at the
atomic scale in order to understand what is going on at the
macroscopic one.
The journey is however a long one. Starting from the basic
principles of Quantum Mechanics to model the matter at the subatomic
scale,
one finally uses statistical mechanics to reach the macroscopic
scale. It is often necessary to rely on intermediate steps to deal with
phenomena which take place on various *mesoscales*.
It may then be possible to couple one description of the system with some
others within the so-called *multiscale* models.
The sequel indicates how this journey can be completed
focusing on the first smallest scales (the subatomic one), rather than on the
larger ones.
It has already been mentioned that at the subatomic scale,
the behavior of nuclei and electrons is governed by the Schrödinger
equation, either in its time-dependent form
or in its time-independent form. Let us only mention at this point that

both equations involve the quantum Hamiltonian of the
molecular system under consideration; from a mathematical viewpoint,
it is a self-adjoint
operator on some Hilbert space; *both* the Hilbert
space and the Hamiltonian operator depend on the nature of the system;

also present into these equations is
the wavefunction of the system; it completely
describes its state; its

The time-dependent equation is a first-order linear evolution
equation, whereas the time-independent equation is a linear eigenvalue
equation.
For the reader more familiar with numerical analysis
than with quantum mechanics, the linear nature of the problems stated
above may look auspicious. What makes the
numerical simulation of these equations
extremely difficult is essentially the huge size of the Hilbert
space: indeed, this space is roughly some
symmetry-constrained subspace of *nonlinear* partial differential equations,
each
of these equations being posed on

As the size of the systems one wants to study increases, more efficient
numerical techniques need to be resorted to. In computational chemistry,
the typical scaling law for the complexity of computations with respect
to the size of the system under study is

how can one improve the nonlinear iterations that are the basis of any
*ab initio* models for computational chemistry?

how can one more efficiently solve the inner loop which most often consists in the solution procedure for the linear problem (with frozen nonlinearity)?

how can one design a sufficiently small variational space, whose dimension is kept limited while the size of the system increases?

An alternative strategy to reduce the complexity of *ab initio*
computations is to try to couple different models at different
scales. Such a mixed strategy can be either a sequential one or a
parallel one, in the sense that

in the former, the results of the model at the lower scale are simply used to evaluate some parameters that are inserted in the model for the larger scale: one example is the parameterized classical molecular dynamics, which makes use of force fields that are fitted to calculations at the quantum level;

while in the latter, the model at the lower scale is concurrently coupled to the model at the larger scale: an instance of such a strategy is the so called QM/MM coupling (standing for Quantum Mechanics/Molecular Mechanics coupling) where some part of the system (typically the reactive site of a protein) is modeled with quantum models, that therefore accounts for the change in the electronic structure and for the modification of chemical bonds, while the rest of the system (typically the inert part of a protein) is coarse grained and more crudely modeled by classical mechanics.

The coupling of different scales can even go up to the macroscopic scale, with methods that couple a microscopic representation of matter, or at least a mesoscopic one, with the equations of continuum mechanics at the macroscopic level.

The orders of magnitude used in the microscopic representation of
matter are far from the orders of magnitude of the macroscopic
quantities we are used to: The number of particles under
consideration in a macroscopic sample of material is of the order of
the Avogadro number

To give some insight into such a large number of particles contained in
a macroscopic sample, it is helpful to
compute the number of moles of water on earth. Recall that one mole of water
corresponds to 18 mL, so that a standard glass of water contains roughly
10 moles, and a typical bathtub contains *i.e.*

For practical numerical computations
of matter at the microscopic level, following the dynamics of every atom would
require simulating

Describing the macroscopic behavior of matter knowing its microscopic
description
therefore seems out of reach. Statistical physics allows us to bridge the gap
between microscopic and macroscopic descriptions of matter, at least on a
conceptual
level. The question is whether the estimated quantities for a system of

Despite its intrinsic limitations on spatial and timescales, molecular simulation has been used and developed over the past 50 years, and its number of users keeps increasing. As we understand it, it has two major aims nowadays.

First, it can be
used as a *numerical microscope*, which allows us to perform
“computer” experiments.
This was the initial motivation for simulations at the microscopic level:
physical theories were tested on computers.
This use of molecular simulation is particularly clear in its historic
development, which was triggered and sustained by the physics of simple
liquids. Indeed, there was no good analytical theory for these systems,
and the observation of computer trajectories was very helpful to guide the
physicists'
intuition about what was happening in the system, for instance the mechanisms
leading to molecular diffusion. In particular,
the pioneering works on Monte-Carlo methods by Metropolis *et al.*, and the first
molecular dynamics
simulation of Alder and Wainwright were performed because of such motivations.
Today, understanding the behavior of matter at the
microscopic level can still be difficult from an experimental viewpoint
(because of the high resolution required, both in time and in
space), or because we simply do not know what to look for!
Numerical simulations are then a valuable tool to test some
ideas or obtain some data to process and analyze in order
to help assessing experimental setups. This is
particularly true for current nanoscale systems.

Another major aim of molecular simulation, maybe even more important than the
previous one,
is to compute macroscopic
quantities or thermodynamic properties,
typically through averages of some functionals of the system.
In this case, molecular simulation is a
way to obtain *quantitative* information on a system,
instead of resorting to approximate theories, constructed for simplified models,
and giving only qualitative answers.
Sometimes, these properties are accessible through experiments,
but in some cases only numerical computations are possible
since experiments may be unfeasible or too costly
(for instance, when high pressure or large temperature regimes are considered,
or when studying materials not yet synthesized).
More generally, molecular simulation is a tool to explore the links between
the microscopic and macroscopic properties of a material, allowing
one to address modelling questions such as “Which microscopic ingredients are
necessary
(and which are not) to observe a given macroscopic behavior?”

Over the years, the project-team has developed an increasing expertise on how to couple models written at the atomistic scale with more macroscopic models, and, more generally, an expertise in multiscale modelling for materials science.

The following observation motivates the idea of coupling atomistic and
continuum representation of materials. In many situations of interest
(crack propagation, presence of defects in the atomistic lattice, ...),
using a model based on continuum mechanics is difficult. Indeed, such a
model is based on a macroscopic constitutive law, the derivation of
which requires a deep qualitative and quantitative understanding of the
physical and mechanical properties of the solid under consideration.
For many solids, reaching such an understanding is a challenge, as loads
they are subjected to become larger and more diverse, and as
experimental observations helping designing such models are not always
possible (think of materials used in the nuclear industry).
Using an atomistic model in the whole domain is not possible either, due
to its prohibitive computational cost. Recall indeed that a
macroscopic sample of matter contains a number of atoms on the order of
*only a small
part* of the solid. So, a natural idea is to try to take advantage of
both models, the continuum mechanics one and the atomistic one, and to
couple them, in a domain decomposition spirit. In most of the domain,
the deformation is expected to be smooth, and reliable continuum
mechanics models are then available. In the rest of the
domain, the expected deformation is singular, so that one needs an atomistic
model to describe it properly, the cost of which remains however limited
as this region is small.

From a mathematical viewpoint, the question is to couple a discrete model with a model described by PDEs. This raises many questions, both from the theoretical and numerical viewpoints:

first, one needs to derive, from an atomistic model, continuum mechanics models, under some regularity assumptions that encode the fact that the situation is smooth enough for such a macroscopic model to provide a good description of the materials;

second, couple these two models, e.g. in a domain decomposition spirit, with the specificity that models in both domains are written in a different language, that there is no natural way to write boundary conditions coupling these two models, and that one would like the decomposition to be self-adaptive.

More generally, the presence of numerous length scales in material
science problems represents a challenge for numerical simulation,
especially when some *randomness* is assumed on the
materials. It can take various forms, and includes defects in
crystals, thermal fluctuations, and impurities or heterogeneities in
continuous media. Standard methods available in the literature to
handle such problems often lead to very costly computations. Our
goal is to develop numerical methods that are more
affordable. Because we cannot embrace all difficulties at once, we
focus on a simple case, where the fine scale and the coarse-scale
models can be written similarly, in the form of a simple elliptic
partial differential equation in divergence form. The fine scale
model includes heterogeneities at a small scale, a situation which
is formalized by the fact that the coefficients in the fine scale
model vary on a small length scale. After homogenization, this model
yields an effective, macroscopic model, which includes no small
scale. In many cases, a sound theoretical groundwork exists for such
homogenization results. The difficulty stems from the fact that the models
generally lead to prohibitively costly computations. For such a
case, simple from the theoretical viewpoint, our aim is to focus on
different practical computational approaches to speed-up the
computations. One possibility, among others, is to look for specific
random materials, relevant from the practical viewpoint, and for
which a dedicated approach can be proposed, that is less expensive
than the general approach.

Claude Le Bris was selected to deliver the Coxeter lectures at the Fields Institute in Toronto and the Aziz lectures at the University of Maryland.

Florent Hédin received the “Best student/postdoc oral
presentation” award at the 7^{th} Workshop on Parallel-in-Time
methods, Roscoff, France, May.

Keywords: Molecular simulation - Quantum chemistry - Statistical physics - C++ - OpenMP

Functional Description: Molecular simulation software written in C++

Contact: Gabriel Stoltz

In electronic structure calculation as in most of our scientific endeavors, we pursue a twofold goal: placing the models on a sound mathematical grounding by an appropriate mathematical analysis, and improving the numerical approaches by a dedicated numerical analysis.

The members of the team have continued their systematic study of the properties of materials in the reduced Hartree-Fock approximation, a model striking a good balance between mathematical tractability and the ability to reproduce qualitatively complex effects.

E. Cancès and G. Stoltz have studied with L. Cao models for certain extended defects in materials . These extended defects typically correspond to taking out a slab of finite width in the three-dimensional homogeneous electron gas. The work is performed in the framework of the reduced Hartree-Fock model with either Yukawa or Coulomb interactions, using techniques previously developed to study local perturbations of the free-electron gas. It is shown that the model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. These minimizers are unique for Yukawa interactions, and are characterized by a self-consistent equation. Numerical simulations show evidence of Friedel oscillations in the total electronic density.

A. Levitt has examined the phenomenon of screening in materials. In he has studied the effect of adding a small charge to a periodic system modeled by the reduced Hartree-Fock at finite temperature. He has showed that the reaction potential created by the rearrangement of the electrons counteracts exactly the free charge, so that the effective interaction in such systems is short-range. The proof proceeds by studying the properties of the linear response operator, which also sheds some light on the charge-sloshing instability seen in numerical methods to solve the self-consistent equations.

E. Cancès has pursued his long-term collaboration with Y. Maday
(Sorbonne Université) on the numerical analysis of linear and nonlinear
eigenvalue problems. Together with G. Dusson (Warwick, United Kingdom),
B. Stamm (Aachen, Germany), and M. Vohralík (Inria SERENA), they have
designed *a posteriori* error estimates for conforming numerical
approximations of the Laplace eigenvalue problem with homogeneous
Dirichlet boundary conditions. In , they prove *a priori* error estimates for
the perturbation-based post-processing of the plane-wave approximation
of Schrödinger equations introduced and tested numerically in previous
works. They consider a Schrödinger operator

Implicit solvation models aim at computing the properties of a molecule in solution (most chemical reactions take place in the liquid phase) by replacing all the solvent molecules but the few ones strongly interacting with the solute, by an effective continuous medium accounting for long-range electrostatics. E. Cancès, Y. Maday (Sorbonne Université), and B. Stamm (Aachen, Germany) have introduced a few years ago a very efficient domain decomposition method for the simulation of large molecules in the framework of the so-called COSMO implicit solvation models. In collaboration with F. Lipparini and B. Mennucci (Chemistry, Pisa, Italy) and J.-P. Piquemal (Sorbonne Université), they have implemented this algorithm in widely used computational software products (Gaussian and Tinker). Together with L. Lagardère (Sorbonne Université) and G. Scalmani (Gaussian Inc., USA), they illustrate in the domain decomposition COSMO (ddCOSMO) implementation and how to couple it with an existing classical or quantum mechanical (QM) codes. They review in detail what input needs to be provided to ddCOSMO and how to assemble it, describe how the ddCOSMO equations are solved and how to process the results in order to assemble the required quantities, such as Fock matrix contributions for the QM case, or forces for the classical one. Throughout the paper, they make explicit references to the ddCOSMO module, which is an open source, Fortran 90 implementation of ddCOSMO that can be downloaded and distributed under the LGPL license.

E. Cancès, V. Ehrlacher and A. Levitt, together with D. Gontier (Dauphine) and D. Lombardi (Inria REO), have studied the convergence of properties of periodic systems as the size of the computing domain is increased. This convergence is known to be difficult in the case of metals. They have characterized in the speed of convergence for a number of schemes in the metallic case, and have studied the properties of a widely used numerical method that adds an artificial electronic temperature.

A. Levitt has continued his study of Wannier functions in periodic systems. With A. Damle (Cornell, USA) and L. Lin (Berkeley, USA), they have proposed an efficient numerical method for the computation of maximally-localized Wannier functions in metals, and have showed on the example of the free electron gas that they are not in general exponentially localized . With D. Gontier (Dauphine) and S. Siraj-Dine, they proposed a new method for the computation of Wannier functions which applies to any insulator, and in particular to the difficult case of topological insulators .

The objective of computational statistical physics is to compute macroscopic properties of materials starting from a microscopic description, using concepts of statistical physics (thermodynamic ensembles and molecular dynamics). The contributions of the team can be divided into four main topics: (i) the development of methods for sampling the configuration space; (ii) the numerical analysis of such methods; (iii) the efficient computation of dynamical properties which requires to sample metastable trajectories; (iv) coarse-graining techniques to reduce the computational cost of molecular dynamic simulations and gain some insights on the models.

New numerical methods in order to sample probability measures on the configuration space have been developed: either measures supported on submanifolds, or stationary states of stochastic dynamics. First, in , T. Lelièvre and G. Stoltz, together with M. Rousset (Inria Rennes, France) have studied how to sample probability measures supported on submanifolds, by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. However, the usual method has to be generalized using a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for large timesteps and hence avoid biases in the invariant measure. A full mathematical analysis of such procedures is provided, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples. Second, the work by J. Roussel and G. Stoltz focuses on the use of control variates for non-equilibrium systems. Whereas most variance reduction methods rely on the knowledge of the invariant probability measure, this latter is not explicit out of equilibrium. Control variates offer an attractive alternative in this framework. J. Roussel and G. Stoltz have proposed a general strategy for constructing an efficient control variate, relying on physical simplifications of the dynamics. The authors provide an asymptotic analysis of the variance reduction in a perturbative framework, along with extensive numerical tests on three different systems.

In terms of applications of such sampling techniques, members of the project-team have been working on two different subjects: random matrices models and adaptive techniques to compute large deviation rate functionals. The paper was written by G. Ferré and D. Chafaï (Université Paris Dauphine, France), following the simple idea: the eigenvalues of random matrices are distributed according to Boltzmann–Gibbs measures, but researchers in this field do not use techniques from statistical physics for numerical investigations. The authors therefore used a Hamiltonian Monte Carlo algorithm to investigate numerically conjectures about random matrices and related Coulomb gases. The next step is to add constraints to these systems to understand better the behavior of random matrices with constraints and the large size limit of their spectra (the algorithm mentioned above to sample probability measures supported on submanifolds may be useful in this context). The work focuses on computing free energies and entropy functions, as they arise in large deviations theory, through adaptive techniques. It is actually in the spirit of techniques used in mathematical finance, adapted to the statistical mechanics context, and enriched with new estimators based on variational representations of entropy functions. These tools have been pioneered by H. Touchette (Stellenbosch University, South Africa), with whom the paper was written by G. Ferré.

Concerning the numerical analysis of sampling techniques of probability measures on the configuration space, let us mention three works.

First, in , G. Ferré and G. Stoltz study the numerical errors that arise when a stochastic differential equation (SDE) is discretized in order to compute scaled cumulant functions (or free energy) and ergodic properties of Feynman–Kac semigroups. These quantities naturally arise in large deviations theory, for estimating probabilities of rare events. This analysis is made difficult by the nonlinear (mean field) feature of the dynamics at hand. The obtained estimates generalize previous results on the numerical analysis of ergodic properties of discretized SDEs. As a theoretical extension of the previous work, the purpose of the work by G. Ferré and G. Stoltz, in collaboration with M. Rousset (Inria Rennes, France), is to provide further theoretical investigations on the long time behavior of Feynman–Kac semigroups. More precisely, it aims at giving practical criteria for these nonlinear semigroups to have a limit, and makes precise in which sense this limit is to be understood. This was an open problem so far for systems evolving in unbounded configuration spaces, which was addressed through Lyapunov function techniques. Although theoretical, these results are of practical importance since, if these dynamics do not have a well-defined long time behavior, it is hopeless to try to compute rare events.

Finally, together with C. Andrieu (Univ. Bristol, United-Kingdom), A. Durmus (ENS Saclay, France) and N. Nüsken (Univ. Potsdam, Germany), J. Roussel derived in spectral gap estimates for several Piecewise Deterministic Markov Processes (PDMPs), namely the Randomized Hamiltonian Monte Carlo, the Zig-Zag process and the Bouncy Particle Sampler. The hypocoercivity technique provides estimates with explicit dependence on the parameters of the dynamics. Moreover the general framework considered allows to compare quantitatively the bounds found for the different methods. Such PDMDs are currently more and more used as efficient sampling tools, but their theoretical properties are still not yet well understood.

The sampling of dynamical properties along molecular dynamics
trajectories is crucial to get access to important quantities such as
transition rates or reactive paths. This is difficult numerically
because of the metastability of trajectories. Members of the project-team are following two
numerical approaches to sample metastable trajectories: the accelerated
dynamics *à la* A.F. Voter and the adaptive multilevel splitting
(AMS) technique to sample reactive paths between metastable states.

Concerning the mathematical analysis of the accelerated dynamics, in , T. Lelièvre reviews the recent mathematical approaches to justify these numerical methods, using the notion of quasi-stationary distribution. Moreover, in , T. Lelièvre together with D. Le Peutrec (Université de Paris Saclay, France) and G. Di Gesu and B. Nectoux (TU Wien, Austria) give an overview of the results obtained during the PhD of B. Nectoux. Using the quasi-stationary distribution approach and tools from semi-classical analysis, one can justify the use of kinetic Monte Carlo models parametrized by the Eyring-Kramers formulas to describe exit events from metastable states, for the overdamped Langevin dynamics. Concerning the implementation, in , F. Hédin and T. Lelièvre test the Generalized Parallel Replica algorithm to biological systems, and obtain strong linear scalability, providing up to 70% of the maximum possible speedup on several hundreds of CPUs. The “Parallel Replica” (ParRep) dynamics is known for allowing to simulate very long trajectories of metastable Langevin dynamics in the materials science community, but it relies on assumptions that can hardly be transposed to the world of biochemical simulations. The later developed “Generalized ParRep” variant solves those issues, and it had not been applied to significant systems of interest so far. Finally, let us mention the work where T. Lelièvre together with J. Reygner (Ecole des Ponts, France) and L. Pillaud-Vivien (Inria Paris, France) analyze mathematically the Fleming-Viot particle process in the simple case of a finite state space. This Fleming-Viot particle process is a key ingredient of the Generalized ParRep algorithm mentioned above, in order to both approximate the convergence time to the quasi-stationary distribution, and to efficiently sample it.

In two related works, members of the project-team have studied the
quality of the effective dynamics derived from a high dimension
stochastic differential equation on a few degrees of freedom, using a
projection approach *à la Mori-Zwanzig*. More precisely,
in , F. Legoll, T. Lelièvre and U. Sharma
obtain precise error bounds in the case of non reversible dynamics.
This analysis also aims at discussing what is a good notion of mean
force for non reversible systems. In , T.
Lelièvre together with W. Zhang (ZIB, Germany) extend previous results
on pathwise error estimates for such effective dynamics to the case of
nonlinear vectorial reaction coordinates.

Once a good coarse-grained model has been obtained, one can try to use it in order to get a better integrator of the original dynamic in the spirit of a predictor-corrector method. In , T. Lelièvre together with G. Samaey and P. Zielinski (KU Leuven, Belgium) analyze such a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest.

In homogenization theory, members of the project-team have pursued their ongoing systematic study of perturbations of periodic problems (by local and nonlocal defects). This has been done in two different directions.

For linear elliptic equations, C. Le Bris has written, in collaboration with X. Blanc (Paris Diderot, France) and P.-L. Lions (Collège de France, France), two manuscripts that present a more versatile proof of the existence of a corrector function for periodic problems with local defects, and also extend the results: the first manuscript addresses the case of an equation (or a system) in divergence form, while the second manuscript extends the analysis to advection-diffusion equations.

Second, they have also provided more details on the quality of
approximation achieved by their theory. The fact that a corrector
exists with suitable properties allows one to quantify the rate of
convergence of the two-scale expansion using that corrector to the
actual exact solution, as the small homogenization
parameter

Also in the context of homogenization theory, C. Le Bris and F. Legoll have initiated a collaboration with R. Cottereau (Ecole Centrale and now CNRS Marseille, France). The topic is in some sense a follow-up on both an earlier work of R. Cottereau and the series of works completed by C. Le Bris and F. Legoll in collaboration with K. Li and next S. Lemaire over the years. Schematically, the purpose of the work is to determine the homogenized coefficient for a medium without explicitly performing a homogenization approach nor using a MsFEM type approach. In earlier works, an approximation approach, somewhat engineering-style, was designed. The purpose now is to examine the performance of this approach in the context of the so-called Arlequin method, a very popular method in the mechanical engineering community. One couples a sub-region of the medium where a homogeneous model is employed, along with a complementary sub-region where the original multiscale model is solved explicitly. The coupling is performed using the Arlequin method. Then, one optimizes a suitable criterion so that optimization leads to an homogeneous sub-region indeed described by the homogenized coefficient seeked for. Some numerical analysis questions, together with practical perspectives for computational enhancements of the approach, are currently examined.

Finally, C. Le Bris has informally participated into the supervision of the master thesis of S. Wolf (Ecole Normale Supérieure, Paris, France), and in this context performed some works in interaction with the student and X. Blanc. The purpose is to investigate perturbations of periodic homogenization problems when the perturbation is geometric in nature. The test case considered is that of a domain perforated by holes the locations of which are not necessarily periodic, but only periodic up to a local perturbation. The results proven, on the prototypical Poisson equation, are natural extensions of the celebrated results by J.-L. Lions published in the late 1960s for the periodic case. This provides a proof of concept, showing that perturbations of a periodic geometry are also possible, a fact that will be more thoroughly investigated in the near future within the above mentioned collaboration.

The project-team has pursued its efforts in the field of stochastic homogenization of elliptic equations, aiming at designing numerical approaches that are practically relevant and keep the computational workload limited.

Using standard homogenization theory, one knows that the homogenized tensor, which is a deterministic matrix, depends on the solution of a stochastic equation, the so-called corrector problem, which is posed on the whole space

In collaboration with B. Stamm (Aachen University, Germany) and S. Xiang (now also at Aachen University, Germany), E. Cancès, V. Ehrlacher and F. Legoll have studied, both from a theoretical and a numerical standpoints, new alternatives for the approximation of the homogenized matrix. They all rely on the use of an embedded corrector problem, previously introduced by the authors, where a finite-size domain made of the highly oscillatory material is embedded in a homogeneous infinite medium whose diffusion coefficients have to be appropriately determined. In , they have shown that the different approximations introduced all converge to the homogenized matrix of the medium when the size of the embedded domain goes to infinity. In , they present an efficient algorithm for the resolution of such problems for particular heterogeneous materials, based on the reformulation of the embedded corrector problem as an integral equation, which is discretized using spherical harmonics and solved using the fast multipole method.

Besides the averaged behavior of the oscillatory solution

In collaboration with T. Hudson (University of Warwick, United Kingdom), F. Legoll and T. Lelièvre have considered in a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using stochastic two-scale convergence, they have obtained the homogenized limit of the evolution, and have demonstrated that, under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. This work is motivated by rate-independent stress-strain hysteresis observed in filled rubber.

From a numerical perspective, the Multiscale Finite Element Method (MsFEM) is a classical strategy to address the situation when the homogenized problem is not known (e.g. in difficult nonlinear cases), or when the scale of the heterogeneities, although small, is not considered to be zero (and hence the homogenized problem cannot be considered as a sufficiently accurate approximation).

During the year, several research tracks have been pursued in this general direction.

The final writing of the various works performed in the context of the PhD thesis of F. Madiot is still ongoing. The issues examined there are on the one hand the application (and adequate adjustment) of MsFEM approaches to the case of an advection-diffusion equation with a dominating convection term posed in a perforated domain, and on the other hand some more general study of a numerical approach based, again in the case of convection-dominated flows, on the introduction of the invariant measure associated to the problem. The final version of the two manuscripts describing the efforts in each of these directions should be completed in a near future.

The MsFEM approach uses a Galerkin approximation of the problem on a pre-computed basis, obtained by solving local problems mimicking the problem at hand at the scale of mesh elements, with carefully chosen right-hand sides and boundary conditions. The initially proposed version of MsFEM uses as basis functions the solutions to these local problems, posed on each mesh element, with null right-hand sides and with the coarse P1 elements as Dirichlet boundary conditions. Various improvements have next been proposed, such as the *oversampling* variant, which solves local problems on larger domains and restricts their solutions to the considered element. In collaboration with U. Hetmaniuk (University of Washington in Seattle, USA), C. Le Bris, F. Legoll and P.-L. Rothé have introduced and studied a MsFEM method improved differently. They have considered a variant of the classical MsFEM approach with enrichments based on Legendre polynomials, both in the bulk of the mesh elements and on their interfaces. A convergence analysis of this new variant has been performed. Promising numerical results have been obtained. These results are currently being collected in a manuscript in preparation.

One of the perspectives of the team, through the PhD thesis of A. Lesage, is the development of Multiscale Finite Element Methods for thin heterogeneous plates. The fact that one of the dimension of the domain of interest scales as the typical size of the heterogeneities within the material induces theoretical and practical difficulties that have to be carefully taken into account. The first steps of the work of V. Ehrlacher, F. Legoll and A. Lesage, in collaboration with A. Lebée (École des Ponts) have consisted in studying the homogenized limit (and the two-scale expansion) of problems posed on thin heterogeneous plates. The case of a diffusion equation has been first dealt with, while the more challenging case of elasticity is currently under study.

The aim of the research performed in the project-team about complex fluids is

to guide the mathematical modeling with PDEs of real materials flows, multi-phase fluids such as suspensions of particles or stratified air-water flows in particular, and

to propose efficient algorithms for the computation of flow solutions, mainly for the many applications in the hydraulic engineering context.

Concerning the first point, new results have been obtained in collaboration with A. Caboussat (HEG, Switzerland) and M. Picasso (EPFL, Switzerland),
in the framework of the SEDIFLO project (funded by ANR) and of Arwa Mrad PhD thesis at EPFL.
In , they have shown numerically inability of some classical incompressible density-dependent Navier-Stokes equations to take into account some
multiphase concentration effects in a prototypical set-up of fluvial erosion (in comparison with physical experiments).
Hence the need for *new* models, that better describe complex flows associated with heterogeneities in the fluid microstructure.
Concerning the second point, new results have been obtained in
collaboration with M. Grepl and K. Veroy (Aachen, Germany)
regarding the numerical reduction of transport models for data assimilation , in the framework of M. Kaercher PhD thesis at Aachen.

Many research activities of the project-team are conducted in close collaboration with private or public companies: CEA, SANOFI, EDF. The project-team is also supported by the Office of Naval Research and the European Office of Aerospace Research and Development, for multiscale simulations of random materials. All these contracts are operated at and administrated by the École des Ponts.

The project-team is involved in several ANR projects:

S. Boyaval is the PI of the ANR JCJC project SEDIFLO (2016-2020) to investigate new numerical models of solid transport in rivers.

V. Ehrlacher is a member of the ANR project ADAPT (2018-2022), PI: D. Lombardi, Inria REO team-project. This project is concerned with the parallelization of tensor methods for high-dimensional problems.

F. Legoll is a member of the ANR project CINE-PARA (2015-2019), PI: Y. Maday, Sorbonne Université. This project is concerned with parallel-in-time algorithms.

G. Stoltz is the PI of the ANR project COSMOS (2014-2019) which focuses on the development of efficient numerical techniques to simulate high-dimensional systems in molecular dynamics and computational statistics. It includes research teams from Institut Mines-Telecom, Inria Rennes and IBPC Paris.

Members of the project-team are participating in the following GdR:

AMORE (Advanced Model Order REduction),

CORREL (correlated methods in electronic structure computations),

DYNQUA (time evolution of quantum systems, with applications to transport problems, nonequilibrium systems, etc.),

EGRIN (gravity flows),

MANU (MAthematics for NUclear applications),

MASCOT-NUM (stochastic methods for the analysis of numerical codes),

MEPHY (multiphase flows)

REST (theoretical spectroscopy),

CHOCOLAS (experimental and numerical study of shock waves).

The project-team is involved in two Labex: the Labex Bezout (started in 2011) and the Labex MMCD (started in 2012).

The ERC consolidator Grant MSMATH (ERC Grant Agreement number 614492, PI T. Lelièvre) is running (it started in June 2014).

T. Lelièvre, G. Stoltz and F. Legoll participate in the Laboratoire International Associé (LIA) CNRS / University of Illinois at Urbana-Champaign on complex biological systems and their simulation by high performance computers. This LIA involves French research teams from Université de Nancy, Institut de Biologie Structurale (Grenoble) and Institut de Biologie Physico-Chimique (Paris). The LIA has been renewed for 4 years, starting January 1st, 2018.

E. Cancès

is the director of CERMICS, the Applied Mathematics department at École des Ponts,

is a member of the editorial boards of Mathematical Modelling and Numerical Analysis (2006-), SIAM Journal of Scientific Computing (2008-), SIAM Multiscale Modeling and Simulation (2012-), and the Journal of Computational Mathematics (2017-),

has co-organized an Oberwolfach workshop (March), an IMA workshop (March), the 2018 SIAM MS conference (July), and an ISCD summer school (July - August),

was a member of the DFG Review Panel “Mathematics” for Clusters of Excellence, Cologne, April.

V. Ehrlacher

is a member of the “Conseil d'Enseignement et de Recherche” of Ecole des Ponts,

has co-organized the GdR MASCOT-NUM Working meeting on “Uncertainty quantification in materials science”, at IHP, May (with J. Baccou, J. Reygner and G. Perrin).

G. Ferré and J. Roussel have co-organized the working group J-PSI (Jeunes chercheurs en physique statistique et interactions, until July) at IHP. The working group was provided financial support from the SMAI through a BOUM grant, and ended with a one-day conference in June at Inria Paris.

C. Le Bris is a managing editor of Networks and Heterogeneous Media. He is a member of the editorial boards of Annales mathématiques du Québec (2013-), Archive for Rational Mechanics and Analysis (2004-), COCV (Control, Optimization and Calculus of Variations) (2003-), Mathematics in Action (2008-), Nonlinearity (2005-), Journal de Mathématiques Pures et Appliquées (2009-), Pure and Applied Analysis (2018-). He is a member of the editorial boards of the monograph series Mathématiques & Applications, Series, Springer (2008-), Modelling, Simulations and Applications, Series, Springer (2009-), Springer Monographs in Mathematics, Springer (2016-). He is a member of

the Cabinet of the High Commissioner for Atomic Energy (until September),

the “International Scientific Advisory Committee” of the Centre de Recherche Mathématique, Université de Montréal (until mid-2018),

the “Advisory Board” of the DFG Cluster of Excellence Engineering of Advanced Materials, Erlangen,

the “International Scientific Advisory Board” of the DFG research center Matheon, Berlin,

the “Conseil scientifique de la SMAI” (Scientific Council of the French Applied Maths Society),

the International Mathematical Union Circle,

the “Conseil de la Faculté des sciences et ingénierie”, Sorbonne Université.

He is the president of the scientific advisory board of the Institut des Sciences du calcul et des données, Sorbonne Université. He has held a regular position of Visiting Professor at the University of Chicago.

F. Legoll

is a member of the editorial board of SIAM MMS (2012-) and of ESAIM: Proceedings and Surveys (2012-),

is a member of the ANR committee CES-40 "mathématiques et informatique".

T. Lelièvre

is editor-in-chief of ESAIM: Proceedings and Surveys (with D. Chafai, C. Imbert and P. Lafitte),

is a member of the editorial boards of IMA: Journal of Numerical Analysis and SIAM/ASA Journal of Uncertainty Quantification,

is a member of the “Conseil d'Administration” of SMAI and École des Ponts,

Together with G. Stoltz, they have

co-organized the Workshop “Advances in Computational Statistical Physics”, CIRM, September (with G. Pavliotis),

co-organized the CECAM discussion meeting “Coarse-graining with Machine Learning in molecular dynamics”, Sanofi Campus Gentilly, December (with P. Gkeka, P. Monmarché).

G. Stoltz

is a member of the scientific council of UNIT (Université Numérique Ingénierie et Technologie),

co-organized with C. Robert the workshop “Computational Statistics and Molecular Simulation: A Practical Cross-Fertilization” (BIRS-Oaxaca, November),

co-organizes the working group “Machine learning and optimization” of the Labex Bezout (with W. Hachel and R. Elie).

Members of the project-team have delivered lectures in the following seminars, workshops and conferences:

S. Boyaval, weekly seminar of Laboratoire Jean Kuntzmann, Grenoble, February,

S. Boyaval, GDR EGRIN annual meeting, Clermont-Ferrand, June,

S. Boyaval, La Trobe University – Kyushu University joint Industrial Math seminar, Melbourne, September,

E. Cancès, Energy and forces workshop, Cambridge, UK, January,

E. Cancès, workshop “Mathematical models and computation of nonlinear problems”, China, January,

E. Cancès, weekly seminar of the mathematics department, Sapienza University of Rome, February,

E. Cancès, weekly seminar of Maison de la Simulation, Saclay, March,

E. Cancès, 2D materials workshop, Minneapolis, March,

E. Cancès, Fields Institute workshop, Toronto, May,

E. Cancès, workshop on computational mathematics, Suzhou, China, June,

E. Cancès, Centre Henri Lebesgue workshop, Rennes, June,

E. Cancès, SIAM Materials Science conference, Portland, July,

E. Cancès, IPAM workshop, Los Angeles, August,

E. Cancès, GAMM workshop, Aachen, September,

E. Cancès, Franco-German Meeting Workshop on Mathematical Aspects in Computational Chemistry, Aachen, September,

E. Cancès, CECAM workshop, Lausanne, November,

E. Cancès, workshop “Big data challenges for predictive modeling of complex systems”, Hong Kong, November,

V. Ehrlacher, Groupe de travail ENS Rennes, January,

V. Ehrlacher, workshop on “Mathematical Methods in Quantum Chemistry”, Oberwolfach, Germany, March,

V. Ehrlacher, Séminaire DEFI-MEDISIM-POEMS, October,

G. Ferré, CERMICS PhD Seminar, Paris, February,

G. Ferré, Les probabilités de demain, IHP, Paris, March,

G. Ferré, Congrès National d'Analyse Numérique, Cap d'Agde, May,

G. Ferré, International Conference in Monte Carlo and Quasi Monte Carlo Methods in Scientific Computing, Rennes, July,

G. Ferré, SIAM Materials Science conference, Portland, July (two talks),

G. Ferré, Franco-German Meeting Workshop on Mathematical Aspects in Computational Chemistry, Aachen, September,

G. Ferré, Student Probability Seminar, Courant Institute of Mathematical Science, New-York, December,

M. Josien, CANUM conference, Cap d'Agde, May,

M. Josien, SIAM Materials Science, Portland, USA, July,

F. Hédin, “PinT 7th Workshop on Parallel-in-Time methods”, Roscoff Marine Station, France, May,

F. Hédin, “CECAM Workshop, Frontiers of coarse graining in molecular dynamics”, Zuse Institute Berlin, Germany, July,

F. Hédin, CIRM Conference “Advances in Computational Statistical Physics”, September,

C. Le Bris, Séminaire Pierre-Louis Lions, Collège de France, January,

C. Le Bris, Applied Mathematics Colloquium of the University of Maryland, February,

C. Le Bris, PDE seminar, University of Chicago, April,

C. Le Bris, Journées de l'Ecole Doctorale Carnot-Pasteur, Université de Besançon, June,

C. Le Bris, Journées Scientifiques de Marcoule, CEA Marcoule, June,

C. Le Bris, Journées de Cadarache, CEA Cadarache, June,

C. Le Bris, (plenary lecture) 25th International Conference on Domain Decomposition Methods, St. John's, Canada, July,

C. Le Bris, LMS Durham Research Symposium on Homogenization in Disordered Media, Durham, UK, August,

C. Le Bris, Groupe de travail Calcul des Variations Paris-Ile de France, November

F. Legoll, EMMC conference, Nantes, March,

F. Legoll, University of Chicago, CAMP seminar, Chicago, USA, May,

F. Legoll, AIMS conference, Taipei, Taiwan, July,

F. Legoll, NumDiff conference, Halle, Germany, September,

T. Lelièvre, Journée de l'ANR CINE-PARA, Université Paris 13, January,

T. Lelièvre, Workshop “Interplay of Analysis and Probability in Applied Mathematics”, Oberwolfach, Feburary,

T. Lelièvre, Séminaire de la Maison de la Simulation, Saclay, March,

T. Lelièvre, Séminaire du LJK, Grenoble, March,

T. Lelièvre, Séminaire Statistical Machine Learning in Paris, Paris, April,

T. Lelièvre, Workshop “Data-driven modelling of complex systems”, ATI, London, May,

T. Lelièvre, Workshop “Uncertainty quantification in materials science”, IHP, Paris, May,

T. Lelièvre, Séminaire Mathématiques pour l'Industrie et la Physique, Toulouse, May,

T. Lelièvre, Fields Institute, “Focus Program on Nanoscale Systems and Coupled Phenomena: Mathematical Analysis, Modeling, and Applications”, Toronto, May,

T. Lelièvre, Workshop “Simulation and probability: recent trends”, Rennes, June,

T. Lelièvre, Workshop “Particle based methods”, ICMS, Edinburgh, July,

T. Lelièvre, CECAM workshop “Frontiers of coarse graining in molecular dynamics”, Berlin, July,

T. Lelièvre, Franco-German Workshop on mathematical aspects in computational chemistry, Aachen, September,

T. Lelièvre, Séminaire “Simulation, Incertitudes et Méta-modèles”, CEA Saclay, October,

T. Lelièvre, Workshop “Computational Statistics and Molecular Simulation: A Practical Cross-Fertilization”, Oaxaca, November,

T. Lelièvre, Groupe de travail Évolution de Populations et Systèmes de Particules en Interaction, Ecole Polytechnique, December,

A. Levitt, Mathematical Methods in Quantum Chemistry, Oberwolfach, March,

A. Levitt, Analytical & Numerical Methods in Quantum Transport, Aalborg, May,

A. Levitt, Beijing Normal University seminar, June,

A. Levitt, Chinese Academy of Sciences seminar, June,

A. Levitt, Franco-German Meeting Workshop on Mathematical Aspects in Computational Chemistry, Aachen September,

P.-L. Rothé, PhD seminar, Inria Paris, June,

J. Roussel, SIAM Materials Science conference, Portland, July,

J. Roussel, Monte Carlo & Quasi-Monte Carlo Methods conference, Rennes, France, July,

L. Silva Lopes, CECAM Coarse Graining Workshop, Berlin, Germany, July,

L. Silva Lopes, Advances in Computational Statistical Physics, Marseille, September,

S. Siraj-Dine, SIAM Materials Science conference, Portland, July,

G. Stoltz, Seminar of the polymer physics group, ETH Zürich, February,

G. Stoltz, Applied mathematics seminar Duke University, Durham, North Carolina, USA, February,

G. Stoltz, Statistical Machine Learning in Paris seminar, Paris, April,

G. Stoltz, Focus Program on Nanoscale Systems and Coupled Phenomena: Mathematical Analysis, Modeling, and Applications, Fields institute, Toronto, Canada, May,

G. Stoltz, Journées scientifiques Inria, Bordeaux, France, June,

G. Stoltz, Applied mathematics seminar Courant Institute of Mathematical Sciences, New York, October,

G. Stoltz, Inria-LJLL seminar, December,

P. Terrier, Minerals, Metals & Materials Society Annual Meeting & Exhibition, Phoenix, March,

P. Terrier, CANUM, Cap d'Adge, June.

Members of the project-team have delivered the following series of lectures:

E. Cancès, Fourier transform and applications in quantum physics and chemistry, 9h, GDR CORREL spring school, Paris, April,

E. Cancès, Optimization problems in molecular simulation, 12h, ISCD summer school, Roscoff, July,

E. Cancès, Mathematical methods and numerical algorithms for quantum chemistry, 12h, MWM autumn school, Gelsenkirchen, October,

C. Le Bris, Aziz Lectures, University of Maryland, College Park, February,

C. Le Bris, Fields Institute Coxeter Lecture Series, Toronto, May,

T. Lelièvre, Mini-school math/chemistry GDR CORREL, 9h, April,

T. Lelièvre, Lectures on “Stochastic numerical methods and molecular dynamics simulations” (15h), Ecole d'été ISCD (Sorbonne Université), Roscoff, August.

Members of the project-team have presented posters in the following seminars, workshops and international conferences:

A. Lesage, Fifth workshop on thin structures, Naples, Italy, September,

J. Roussel, Advances in Computational Statistical Physics, CIRM, Marseille, France, September,

G. Ferré, Data-driven modelling of Complex Systems, Alan Turing Institute, London,

G. Ferré, Simulation Aléatoire : problèmes actuels, Inria Rennes,

G. Ferré, Advances in Computational Statistical physics, CIRM.

Members of the team have benefited from long-term stays in institutions abroad:

G. Ferré, Courant Institute of Mathematical Science, New York University, New York, USA, October-November,

P.-L. Rothé, Department of Applied Mathematics, University of Washington, Seattle, USA, April-May.

Members of the project-team have participated (without giving talks nor presenting posters) in the following seminars, workshops and international conferences:

G. Ferré YES'X Workshop, Scalable Statistics: Accuracy and computational complexity, March,

M. Josien, Coxeter Lecture Series, Seminar talks, Toronto, Canada, May

A. Lesage, CANUM conference, Cap d'Agde, May,

A. Lesage, 6th European conference on computational mechanics, Glasgow, United Kingdom, June,

M. Ramil, Perspectives en physique statistique computationnelle au CIRM (Centre International de Recherche Mathématiques), Marseille, September,

M. Ramil, Journées Kolmogorov, Evry, September,

M. Ramil, ANR EFI workshop, Lyon, November,

P.-L. Rothé, 6th European conference on computational mechanics, Glasgow, United Kingdom, June,

P.-L. Rothé, FreeFem++ days, Paris, December,

S. Siraj-Dine, Workshop Mathematical Challenges in Quantum Mechanics, Rome, February,

S. Siraj-Dine, Oberwolfach Workshop on Mathematical Methods in Quantum Chemistry, March,

S. Siraj-Dine, ICMP XIX Congress on Mathematical Physics, Montréal, July.

A. Levitt has implemented a method to construct
maximally-localized Wannier functions for metals. A. Levitt and S.
Siraj-Dine have implemented a method for the computation of
Wannier functions of topological insulators. Both these methods
are available at https://

J. Roussel and G. Stoltz have restructured the SIMOL code, in particular
separating core functions, routines for quantum simulations and advanced
features for molecular dynamics, in order to obtain a simpler and more
accessible base code. The code is available at https://

A first implementation of the Generalized Parallel Replica algorithm,
developed by F. Hédin and T. Lelièvre, is available at https://

The members of the project-team have taught the following courses.

At École des Ponts 1st year (equivalent to L3):

Analyse et calcul scientifique, 30h (A. Levitt, G. Stoltz),

Équations aux dérivées partielles et éléments finis, 15h (F. Legoll, P.-L. Rothé),

Hydraulique numérique, 15h (S. Boyaval),

Mécanique quantique, 10h (E. Cancès, A. Levitt),

Méthodes numériques pour les problèmes en grande dimension, 17h30 (V. Ehrlacher, S. Boyaval),

Optimisation, 15h, L3 (A. Lesage, A. Levitt),

Outils mathématiques pour l'ingénieur, 15h (E. Cancès, G. Ferré, F. Legoll, T. Lelièvre, P-L. Rothé),

Probabilités, 27h (M. Ramil)

Projets de première année, 15h (J. Roussel, P. Terrier),

At École des Ponts 2nd year (equivalent to M1):

Analyse de Fourier, 15h (A. Levitt),

Analyse spectrale et application aux Équations aux dérivées partielles, 36h (F. Legoll, V. Ehrlacher),

Contrôle de systèmes dynamiques et équations aux dérivées partielles, 18h (E. Cancès),

Projet du département IMI, 12h (G. Ferré, M. Ramil, J. Roussel, L. Silva Lopes),

Projets Modéliser Programmer Simuler (T. Lelièvre),

Simulation moléculaire en sciences des matériaux, 6h (A. Levitt),

Statistics and data sciences, 24h (G. Stoltz).

At École des Ponts 3rd year (equivalent to M2):

Méthodes de quantification des incertitudes en ingénierie, 18h (V. Ehrlacher),

Remise à niveau: outils mathématiques, 6h (A. Lesage).

At the M2 “Mathématiques de la modélisation” of Sorbonne Université:

Introduction à la physique statistique computationnelle, 20h (G. Stoltz),

Méthodes numériques probabilistes, 24h (T. Lelièvre),

Problèmes multiéchelles, aspects théoriques et numériques, 24h (F. Legoll),

Théorie spectrale et variationnelle, 10h (E. Cancès).

At other institutions:

Analyse variationnelle des équations aux dérivées partielles, 32h, École Polytechnique (T. Lelièvre),

Aléatoire, 32h, École Polytechnique (T. Lelièvre),

Maths 1 et 2, 9h, L3, École des Mines (A. Levitt, G. Stoltz),

Mathématiques pour l'ingénieur, 36h, L2, UPEC (S. Siraj-Dine),

Numerical methods for partial differential equations, 21h, University of Chicago (C. Le Bris).

The following PhD theses have been defended in the group at École des Ponts:

Amina Benaceur, Réduction de modèles en thermique et mécanique non-linéaires, Université Paris-Est, École des Ponts, defended on December 21th, 2018, supervised by A. Ern (CERMICS), co-supervised by V. Ehrlacher,

Marc Josien, Etude mathématique et numérique de quelques modèles multi-échelles issus de la mécanique des matériaux, Université Paris-Est, École des Ponts, defended on November 20th, 2018, supervised by C. Le Bris,

Julien Roussel, Analyse théorique et numérique de dynamiques non-réversibles en physique statistique computationnelle, Université Paris-Est, École des Ponts, defended on November 27th, 2018, supervised by G. Stoltz,

Pierre Terrier, Reduced models for defect migration in metals, Université Paris-Est, École des Ponts and CEA Saclay, defended on December 19th, supervised by G. Stoltz and M. Athènes (CEA).

The following PhD theses are ongoing in the group at École des Ponts:

Zineb Belkacemi, Machine learning techniques in molecular simulation, Université Paris-Est, Thèse CIFRE Sanofi, started November 1st, 2018, co-supervised by T. Lelièvre and G. Stoltz,

Robert Benda, Multiscale modeling of functionalized nanotube networks for sensor applications, Ecole Polytechnique, started September 1st, 2018, supervised by E. Cancès and B. Lebental (École Polytechnique),

Raed Blel, Monte Carlo methods and model redcution, started October 1st, 2018, supervised by V. Ehrlacher and T. Lelièvre,

Lingling Cao, Mathematical analysis of models of thermo-electronic transport, Université Paris-Est, École des Ponts, started November 1st, 2016, supervised by E. Cancès and G. Stoltz,

Rafaël Coyaud, Méthodes numériques déterministes et stochastiques pour le transport optimal, Université Paris-Est, École des Ponts, started October 1st, 2017, supervised by A. Alfonsi (CERMICS) and co-supervised by V. Ehrlacher,

Qiming Du, Mathematical analysis of splitting methods, École Doctorale Sciences Mathématiques de Paris Centre, started September 1st, 2016, supervised by A. Guyader (Sorbonne Université) and T. Lelièvre,

Grégoire Ferré, Efficient sampling methods for nonequilibrium systems, Université Paris-Est, École des Ponts started October 1st, 2016, supervised by G. Stoltz,

Adrien Lesage, Multi-scale methods for calculation and optimization of thin structures, started October 1st, 2017, supervised by F. Legoll, co-supervised by V. Ehrlacher and A. Lebée (École des Ponts),

Sofiane Martel, Modélisation de la turbulence par mesures invariantes d'EDPS, Université Paris-Est, École des Ponts, started January 1st, 2017, supervised by S. Boyaval and co-supervised by J. Reygner (CERMICS),

Pierre-Loïk Rothé, Numerical methods for the estimation of fluctuations in multi-scale materials and related problems, started October 1st, 2016, supervised by F. Legoll,

Mouad Ramil, Metastability for interacting particle systems, started October 1st, 2017, supervised by T. Lelièvre and J. Reygner (CERMICS),

Laura Silva Lopes, Numerical methods for simulating rare events in molecular dynamics, started October 1st, 2016, supervised by J. Hénin (IBPC) and T. Lelièvre,

Sami Siraj-Dine, Modélisation mathématique des matériaux 2D, École des Ponts, started October 1st, 2017, supervised by E. Cancès, C. Fermanian and co-supervised by A. Levitt.

Project-team members have participated in the following PhD juries:

E. Cancès, PhD of Marco Vanzini (“Auxiliary systems for observables: dynamical local connector approximation for electron addition and removal spectra”), defended at Ecole Polytechnique in January 2018,

E. Cancès, PhD of Giovanna Marcelli (“A mathematical analysis of spin and charge transport in topological insulators”), defended at Sapienza University of Rome in February 2018,

E. Cancès, PhD of Mi-Song Dupuy (“Analyse de la méthode projector augmented-wave pour les calculs de structure électronique en géométrie périodique”), defended at Université Paris Diderot in September 2018,

E. Cancès, PhD of Carlo Marcati (“Discontinuous hp finite element methods for elliptic eigenvalue problems with singular potentials, with applications in quantum chemistry”), defended at Sorbonne Université in October 2018,

V. Ehrlacher, PhD of Mi-Song Dupuy, (“Analyse de la méthode projector augmented-wave pour les calculs de structure électronique en géométrie périodique”), defended at Université Paris-Diderot in September 2018.

V. Ehrlacher, PhD of Nicolas Cagniart, (“Quelques approches non linéaires en réduction de complexité”), defended at Sorbonne Université in November 2018,

V. Ehrlacher, PhD of Jules Fauque, (“Modèle d’ordre réduit en mécanique du contact. Application à la simulation du comportement des combustibles nucléaires”), defended at Ecole des Mines de Paris in November 2018,

V. Ehrlacher, PhD of Ahmad Al-Takash, (“Development of numerical methods to accelerate the prediction of the behavior of multiphysics under cyclic loading”), defended at ENSMA in November 2018,

F. Legoll, PhD of Brian Staber (“Stochastic analysis, simulation and identification of hyperelastic constitutive equations”), defended at Université Paris-Est in June 2018,

T. Lelièvre, PhD of Bob Pépin (“Time Averages of Diffusion Processes and Applications to Two-Timescale Problems”), défended at Université du Luxembourg, April 2018,

T. Lelièvre, PhD of Michel Nowak (“Accelerating Monte Carlo particle transport with adaptively generated importance maps”), defended at Université Paris Saclay, September 2018,

T. Lelièvre, PhD of Ze Lei (“Irreversible Markov Chains for Particle Systems and Spin Models: Mixing and Dynamical Scaling”), défended at Ecole Normale Supérieure, December 2018,

G. Stoltz, PhD of Sabri Souguir (“Simulation numérique de l'initiation de la rupture à l'échelle atomique”), defended at Ecole des Ponts in November 2018.

A. Levitt is a member of the editorial board of Interstices, Inria's popularization website.

E. Cancès has been interviewed in “La Jaune et La Rouge”, the journal of the alumni of Ecole Polytechnique, in January.

C. Le Bris organized an open day at CERMICS in June for the administrative staff of École des Ponts.