In applications involving complex physics, such as plasmas and nanotechnologies,
numerical simulations serve as a prediction tool supplementing real experiments and are largely endorsed by engineers or researchers. Their performances rely
not only on computational power, but also on the efficiency of the underlying numerical method and the complexity of the underlying models. The contribution of applied mathematics is then required, on the one hand for
a better understanding of qualitative properties and a better identification of the different regimes present in the model, and on the other hand, for a more sounded construction of new models based on asymptotic analysis. This mathematical analysis is expected to greatly impact the design of *multiscale* numerical schemes.

The proposed research group MINGuS will be dedicated
to the mathematical and numerical analysis of (possibly stochastic) partial differential equations (PDEs),
originating from plasma physics and nanotechnologies,
with emphasis on
*multiscale* phenomena either of **highly-oscillatory**, of **dissipative** or **stochastic** types.
These equations can be also encountered in applications to rarefied gas dynamics, radiative transfer,
population dynamics or laser propagation, for which the
multiscale character is modelled by a scale physical parameter

Producing accurate solutions of multiscale equations is extremely challenging owing to severe restrictions to the numerical methods imposed by fast (or stiff) dynamics.
*Ad-hoc* numerical methods should aim at capturing the slow dynamics solely, instead of resolving finely the stiff dynamics at a formidable computational cost. At the other end of the spectrum, the separation of scales -as required for numerical efficiency- is envisaged in asymptotic techniques, whose purpose is to describe the model in the limit where the small parameter

To be more specific, MINGuS aims at finding, implementing and analysing new multiscale numerical schemes for the following physically relevant multiscale problems:

**Highly-oscillatory Schrödinger equation for nanoscale physics:**
In quantum mechanics, the Schrödinger equation describes
how the quantum state of some physical system changes with time.
Its mathematical and numerical study is of paramount
importance to fundamental and applied physics in general.
We wish to specifically contribute to the mathematical modeling
and the numerical simulation of confined quantum
mechanical systems (in one or more space dimensions) possibly involving stochastic terms.
Such systems are involved in quantum
semi-conductors or atom-chips, as well as in cold atom physics (Bose-Einstein condensates)
or laser propagation in optical fibers.

The prototypical equation is written

where the function

**Highly-oscillatory or highly-dissipative kinetic equations:**
Plasma is sometimes considered as the fourth state of matter,
obtained for example by bringing a gas to a very high temperature.
A globally neutral gas of neutral and charged particles, called plasma, is then obtained and is described by
a kinetic equation as soon as collective effects dominate
as compared to binary collisions. A situation of major importance is magnetic fusion in which collisions are not predominant.
In order to confine such a plasma in devices like tokamaks (ITER project) or stellarators, a large magnetic field is used to endow the charged particles with a cyclotronic motion around field lines. Note that kinetic models are also widely used for modeling plasmas in earth
magnetosphere or in rarefied gas dynamics.

Denoting

where

MINGuS project is the follow-up of IPSO, ending in december in 2017. IPSO original aim was to extend the analysis of geometric schemes from ODEs to PDEs. During the last evaluation period, IPSO also considered the numerical analysis of geometric schemes for (S)PDEs, possibly including multiscale phenomena. Breakthrough results , , , have been recently obtained which deserve to be deepened and extended. It thus appears quite natural to build the MINGuS team upon these foundations.

The objective of
MINGuS is twofold: the construction and the analysis of numerical schemes (such as “Uniformly Accurate numerical schemes", introduced by members of the IPSO project)
for multiscale (S)PDEs originating from physics. In turn, this requires

The MINGuS project is devoted to the mathematical and numerical analysis of models arising in plasma physics and nanotechnology.
The main goal is to construct and analyze numerical methods for the approximation of PDEs containing multiscale phenomena.
Specific multiscale numerical schemes
will be proposed and analyzed in different regimes (namely highly-oscillatory and dissipative).
The ultimate goal is to dissociate the physical parameters
(generically denoted by

Then, for a given stiff (highly-oscillatory or dissipative) PDE, the methodology of the MINGuS team will be the following

Mathematical study of the asymptotic behavior of multiscale models.

This part involves averaging and asymptotic analysis theory to derive asymptotic models, but also long-time behavior of the considered models.

Construction and analysis of multiscale numerical schemes.

This part is the core of the project and will be deeply inspired for the mathematical prerequisite. In particular,
our ultimate goal is the
design of *Uniformly Accurate* (UA) schemes, whose accuracy is independent of

Validation on physically relevant problems.

The last goal of the MINGuS project is to validate the new numerical methods, not only on toy problems, but also on realistic models arising in physics of plasmas and nanotechnologies. We will benefit from the Selalib software library which will help us to scale-up our new numerical methods to complex physics.

In the dissipative context, the asymptotic analysis is quite well understood in the deterministic case and multiscale numerical methods have been developed in the last decades. Indeed, the so-called Asymptotic-Preserving schemes has retained a lot of attention all over the world, in particular in the context of collisional kinetic equations. But, there is still a lot of work to do if one is interesting in the derivation high order asymptotic models, which enable to capture the original solution for all time. Moreover, this analysis is still misunderstood when more complex systems are considered, involving non homogeneous relaxation rates or stochastic terms for instance. Following the methodology we aim at using, we first address the mathematical analysis before deriving multiscale efficient numerical methods.

A simple model of dissipative systems is governed by the following differential equations

for given initial condition

Derivation of asymptotic problems

Our main goal is to analyze the asymptotic behavior of dissipative systems of the form () when *center manifold theorem*
is of great interest but is largely unsatisfactory from the following points of view

a constructive approach of

a better approximation of the transient phase is strongly required to capture the solution for small time: extending the tools developed in averaging theory, the main goal is to construct a suitable change of variable which enables to approximate the original solution for all time.

Obviously, even at the ODE level, a deep mathematical analysis has to be performed to understand the asymptotic behavior of the solution of (). But, the same questions arise at the PDE level. Indeed, one certainly expects that dissipative terms occurring in collisional kinetic equations () may be treated theoretically along this perspective. The key new point indeed is to see the center manifold theorem as a change of variable in the space on unknowns, while the standard point of view leads to considering the center manifold as an asymptotic object.

Stochastic PDEs

We aim at analyzing the asymptotic behavior of stochastic collisional kinetic problems, that is equation of the type (). The noise can describe creation or absorption (as in ()), but it may also be a forcing term or a random magnetic field. In the parabolic scaling, one expect to obtain parabolic SPDEs at the limit. More precisely, we want to understand the fluid limits of kinetic equations in the presence of noise. The noise is smooth and non delta correlated. It contains also a small parameter and after rescaling converges formally to white noise. Thus, this adds another scale in the multiscale analysis. Following the pioneering work , some substantial progresses have been done in this topic.

More realistic problems may be addressed such as high field limit describing sprays, or even hydrodynamic limit. The full Boltzmann equation is a very long term project and we wish to address simpler problems such as convergences of BGK models to a stochastic Stokes equation.

The main difficulty is that when the noise acts as a forcing term, which is a physically relevant situation, the equilibria are affected by the noise and we face difficulties similar to that of high field limit problems. Also, a good theory of averaging lemma in the presence of noise is lacking. The methods we use are generalization of the perturbed test function method to the infinite dimensional setting. We work at the level of the generator of the infinite dimensional process and prove convergence in the sense of the martingale problems. A further step is to analyse the speed of convergence. This is a prerequisite if one wants to design efficient schemes. This requires more refined tools and a good understanding of the Kolmogorov equation.

The design of numerical schemes able to reproduce the transition from the
microscopic to macroscopic scales largely matured with the emergence of
the Asymptotic Preserving schemes which have been developed initially for
collisional kinetic equations (actually, for solving () when

AP numerical schemes whose numerical cost diminishes as

Uniformly accurate numerical schemes, whose accuracy is independent of

Time diminishing methods

The main goal consists in merging Monte-Carlo techniques
with AP methods for handling *automatically* multiscale phenomena.
As a result, we expect that the cost of the so-obtained method decreases when the asymptotic
regime is approached; indeed, in the collisional (i.e. dissipative) regime, the deviational part
becomes negligible so that a very few number of particles will be generated to sample it.
A work in this direction has been done by members of the team.

We propose to build up a method which permits to realize the transition from the microscopic to the macroscopic description without domain decomposition strategies which normally oblige to fix and tune an interface in the physical space and some threshold parameters. Since it will permit to go over domain decomposition and AP techniques, this approach is a very promising research direction in the numerical approximation of multiscale kinetic problems arising in physics and engineering.

Uniformly accurate methods

To overcome the accuracy reduction observed in AP schemes for intermediate regimes, we intend to construct and analyse
multiscale numerical schemes for () whose error is uniform with respect to

Multiscale numerical methods for stochastic PDEs

AP schemes have been developed recently for kinetic equations with noise in the context of Uncertainty Quantification UQ . These two aspects (multiscale and UQ) are two domains which usually come within the competency of separate communities. UQ has drawn a lot of attention recently to control the propagation of data pollution; undoubtedly UQ has a lot of applications and one of our goals will be to study how sources of uncertainty are amplified or not by the multiscale character of the model. We also wish to go much further and by developing AP schemes when the noise is also rescaled and the limit is a white noise driven SPDE, as described in section (). For simple nonlinear problem, this should not present much difficulties but new ideas will definitely be necessary for more complicated problems when noise deeply changes the asymptotic equation.

As a generic model for highly-oscillatory systems, we will consider the equation

for a given *averaging* theory -
allow to decompose

into a fast solution component, the *averaged* differential equation. Although equation () can be satisfied only up to a small remainder, various methods have been recently introduced in situations where () is posed in

In the asymptotic behavior *per se* but it also paves the way of the construction of multiscale numerical methods.

Derivation of asymptotic problems

We intend to study the asymptotic behavior of highly-oscillatory evolution equations of the form () posed in an infinite dimensional Banach space.

Recently, the stroboscopic averaging has been extended to the PDE context, considering nonlinear Schrödinger equation () in the highly-oscillatory regime. A very exciting way would be to use this averaging strategy for highly-oscillatory kinetic problem () as those encountered in strongly magnetized plasmas. This turns out to be a very promising way to re-derive gyrokinetic models which are the basis of tokamak simulations in the physicists community. In contract with models derived in the literature (see ) which only capture the average with respect to the oscillations, this strategy allows for the complete recovery of the exact solution from the asymptotic (non stiff) model. This can be done by solving companion transport equation that stems naturally from the decomposition ().

Long-time behavior of Hamiltonian systems

The study of long-time behavior of nonlinear Hamiltonian systems have received a lot of interest during the last decades. It enables to put in light some characteristic phenomena in complex situations, which are beyond the reach of numerical simulations. This kind of analysis is of great interest since it can provide very precise properties of the solution. In particular, we will focus on the dynamics of nonlinear PDEs when the initial condition is close to a stationary solution. Then, the long-time behavior of the solution is studied through mainly three axis

*linear stability*: considering the linearized PDE, do we have stability of a stationary solution ? Do we have linear Landau damping around stable non homogeneous stationary states?

*nonlinear stability*: under a criteria, do we have stability of a stationary solution in energy norm like in ,
and does this stability persist under numerical discretization? For example one of our goals is to address the question of the existence and stability of discrete travelling wave in space and time.

do we have existence of damped solutions for the full nonlinear problem ? Around homogeneous stationary states, solutions scatter towards a modified stationary state (see , ). The question of existence of Landau damping effects around non homogeneous states is still open and is one of our main goal in the next future.

Asymptotic behavior of stochastic PDEs

The study of SPDEs has known a growing interest recently, in particular with the fields medal of M. Hairer in 2014. In many applications such as radiative transfer, molecular dynamics or simulation of optical fibers, part of the physical interactions are naturally modeled by adding supplementary random terms (the noise) to the initial deterministic equations. From the mathematical point of view, such terms change drastically the behavior of the system.

In the presence of noise, highly-oscillatory dispersive equations presents new problems.
In particular, to study stochastic averaging of the solution, the analysis of the long time behavior of stochastic dispersive equations is required,
which is known to be a difficult problem in the general case. In some cases (for instance highly-oscillatory Schrödinger equation () with a time white noise in the regime

The long-time behavior of stochastic Schrödinger equations is of great interest to analyze mathematically the validity of the Zakharov theory for wave turbulence (see ). The problem of wave turbulence can be viewed either as a deterministic Hamiltonian PDE with random initial data or a randomly forced PDEs where the stochastic forcing is concentrated in some part of the spectrum (in this sense it is expected to be a hypoelliptic problem). One of our goals is to test the validity the Zakharov equation, or at least to make rigorous the spectrum distribution spreading observed in the numerical experiments.

This section proposes to explore numerical issues raised by
highly-oscillatory nonlinear PDEs for which () is a prototype.
Simulating a highly-oscillatory phenomenon usually requires to
adapt the numerical parameters in order to solve the period of size *Uniformly Accurate* (UA) numerical schemes, for which
the numerical error can be estimated by

Recently, such numerical methods have been proposed by members of the team in the highly-oscillatory context .
They are mainly based on a separation of the fast and slow variables, as suggested by the decomposition ().
An additional ingredient to prove the uniformly accuracy of the method for () relies on the search
for an appropriate initial data which enables to make the problem smooth with respect to

Such an approach is assuredly powerful since it provides a numerical method which
enables to capture the high oscillations in time of the solution
(and not only its average) even with a large time step. Moreover, in the asymptotic regime,
the potential gain is of order

Space oscillations:

When rapidly oscillating coefficients in **space** (*i.e.* terms of the form *spatial* scales merits to be explored in this context.
The delicate issue is then to extend the choice suitable initial
condition to an *appropriate choice of boundary conditions* of the augmented problem.

Space-time oscillations:

For more complex problems however, the recent proposed approaches fail
since the main oscillations cannot be identified explicitly.
This is the case for instance when the magnetic field **geometric optics** which is a very popular technique to handle highly-frequency waves.

Geometrical properties:

The questions related to the geometric aspects of multiscale numerical schemes are of crucial importance, in particular when long-time simulations are addressed (see ). Indeed, one of the main questions of geometric integration is whether intrinsic properties of the solution may be passed onto its numerical approximation. For instance, if the model under study is Hamiltonian, then the exact flow is symplectic, which motivates the design of symplectic numerical approximation. For practical simulations of Hamiltonian systems, symplectic methods are known to possess very nice properties (see ). It is important to combine multiscale techniques to geometric numerical integration. All the problems and equations we intend to deal with will be addressed with a view to preserve intrinsic geometric properties of the exact solutions and/or to approach the asymptotic limit of the system in presence of a small parameter. An example of a numerical method developed by members of the team is the multi-revolution method.

Quasi-periodic case:

So far, numerical methods have been proposed for the periodic case with single frequency. However,
the quasi-periodic case

extension to stochastic PDEs:

All these questions will be revisited within the stochastic context. The mathematical study opens the way to the derivation of efficient multiscale numerical schemes for this kind of problems. We believe that the theory is now sufficiently well understood to address the derivation and numerical analysis of multiscale numerical schemes. Multi-revolution composition methods have been recently extended to highly-oscillatory stochastic differential equations The generalization of such multiscale numerical methods to SPDEs is of great interest. The analysis and simulation of numerical schemes for highly-oscillatory nonlinear stochastic Schrödinger equation under diffusion-approximation for instance will be one important objective for us. Finally, an important aspect concerns the quantification of uncertainties in highly-oscillatory kinetic or quantum models (due to an incomplete knowledge of coefficients or imprecise measurement of datas). The construction of efficient multiscale numerical methods which can handle multiple scales as well as random inputs have important engineering applications (see ()).

The MINGuS project aims at applying the new numerical methods on realistic problems arising for instance in physics of nanotechnology and physics of plasmas. Therefore, in addition to efforts devoted to the design and the analysis of numerical methods, the inherent large size of the problems at hand requires advanced mathematical and computational methods which are hard to implement. Another application is concerned with population dynamics for which the main goal is to understand how the spatial propagation phenomena affect the demography of a population (plankton, parasite fungi, ...).

Our activity is mostly at an early stage in the process of transfer to industry. However, all the models
we use are physically relevant and all have applications in many areas (ITER,
Bose-Einstein condensate, wave turbulence,
optical tomography, transport phenomena, population dynamics,

In order to tackle as realistic physical problems as possible, a fundamental aspect will consist in working on the realization of numerical methods and algorithms which are able to make an efficient use of a large number of processors. Then, it is essential for the numerical methods developed in the MINGuS project to be thought through this prism. We will benefit from the strong expertise of P. Navaro in scientific computing and more precisely on the Selalib software library (see description below).

Below, we detail our main applications: first, the modeling and numerical approximation of magnetized plasmas is our major application and will require important efforts in terms of software developments to scale-up our multiscale methods; second, the transport of charged particles in nanostructures has very interesting applications (like graphene material), for which our contributions will mainly focus on dedicated problems; lastly, applications on population dynamics will be dedicated to mathematical modeling and some numerical validations.

The Selalib (SEmi-LAgrangian LIBrary) software library

Its main objective is to develop a documented library implementing several numerical methods for the numerical approximation of kinetic models of the form (). Another objective of the library is to provide physicists with easy-to-use gyrokinetic solvers. It has been originally developed by E. Sonnendrücker and his collaborators in the past CALVI Inria project, and has played an important role in the activities of the IPL FRATRES. P. Navaro is one of the main software engineer of this library and as such he played an important daily role in its development and its portability on supercomputers. Though Selalib has reached a certain maturity some additional works are needed to make available by the community. There are currently discussions for a possible evolution of Selalib, namely the writing of a new release which will be available for free download. Obviously, the team will be involved in this process.

At the scientific level, Selalib is of great interest for us since it provides a powerful
tool with which we can test, validate and compare our new methods and algorithms (*users level*).
Besides numerical algorithms the library provides low-level utilities,
input-output modules as well as parallelization strategies dedicated to kinetic problems.
Moreover, a collection of simulations for typical test cases (of increasing difficulties)
with various discretization schemes supplements the library.
This library turns out to be the ideal complement of our activities
and it will help us to scale-up our numerical methods to high-dimensional kinetic problems.
During the last years, several experiments have been successfully performed in this direction
(especially with PhD students) and it is important for us that this approach remains throughout.

Then, we intend to integrate several of the numerical methods developed by the team
within the Selalib library, with the strong help of P. Navaro (*contributors level*).
This work has important advantages:

Nowadays, a great challenge consists in the downscaling at the nanometer scale of electronic components in order to improve speed and efficiency of semiconductor materials. In this task, modeling and numerical simulations play an important role in the determination of the limit size of the nanotransistors At the nanoscale, quantum effects have to be considered and the Schrödinger equation () is prominent equation in this context. In the so-called semiclassical regime or when the transport is strongly confined, the solution endows space-time highly oscillations which are very difficult to capture numerically.

An important application is the modeling of charged particles transport in graphene. Graphene is a sheet of carbone made of a single layer of molecule, organised in a bidimensional honeycomb crystal. The transport of charged particles in this structure is usually performed by Dirac equation (which is the relativistic counterpart of the Schrödinger equation). Due to the unusual properties of graphene -at room temperature, electrons moving in graphene behave as massless relativistic particles- physicists and compagnies are nowadays actively studying this material. Here, predicting how the material properties are affected by the uncertainties in the hexagonal lattice structure or in external potentials, is a major issue. Then, one can wonder how to combine multiscale UA schemes with some well-known UQ numerical methods (as stochastic Galerkin (SG) approaches) ?

The main goal is to characterize how spatial propagation phenomena (diffusion, transport, advection, ...) affect the time evolution of the demography of a population. In collaboration with Y. Lagadeuc (ECOBIO, Rennes), this question has been studied for plankton. In this context, mathematical models have been proposed and it has been shown that the spatial dynamic (in this context, due to the marine current) which is fast compared to demographic scales, can strongly modify the demographic evolution of the plankton.

In collaboration with Ecole d'Agronomie de Rennes, a mathematical study on the demography of a parasite fungi of plants has been performed. In this context, the demography is specific: the fungi can proliferate through sexual reproduction or through parthenogenesis. This two ways of reproduction give rise mathematically to quadratic and linear growth rates with respect to the population variable. The demography is then coupled with transport (transport of fungi spore by wind). Here, the goal is characterize the propagation of the fungi population by finding travelling waves solutions which are well adapted to describe the evolution of invasive fronts. Moreover, this approach enables to recover with a good agreement realistic examples (infection of ash or banana tree) for which experimental data are available.

In these contexts, mathematical models are a powerful tool for biologists since measurements are very complicated to obtain and laboratory experiments hardly reproduce reality. The models derived are multiscale due to the nature of the underlying phenomena and the next step is to provide efficient numerical schemes.

Arnaud Debussche has been awarded the senior IUF.

New associated team ANTIpODE with the University of Wisconsin.

New contract with the startup RAVEL.

New contract with the startup CAILABS.

*SEmi-LAgrangian LIBrary*

Keywords: Plasma physics - Semilagrangian method - Parallel computing - Plasma turbulence

Scientific Description: The objective of the Selalib project (SEmi-LAgrangian LIBrary) is to develop a well-designed, organized and documented library implementing several numerical methods for kinetic models of plasma physics. Its ultimate goal is to produce gyrokinetic simulations.

Another objective of the library is to provide to physicists easy-to-use gyrokinetic solvers, based on the semi-lagrangian techniques developed by Eric Sonnendrücker and his collaborators in the past CALVI project. The new models and schemes from TONUS are also intended to be incorporated into Selalib.

Functional Description: Selalib is a collection of modules conceived to aid in the development of plasma physics simulations, particularly in the study of turbulence in fusion plasmas. Selalib offers basic capabilities from general and mathematical utilities and modules to aid in parallelization, up to pre-packaged simulations.

Partners: Max Planck Insitute - Garching - Université de Strasbourg

Contact: Philippe Helluy

In the analysis of highly-oscillatory evolution problems, it is commonly assumed that a single frequency is present and that it is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. This work is a first step towards the resolution of these difficulties. In particular, P. Chartier, M. Lemou, F. Méhats and G. Vilmart show that it is still possible in this situation to infer the asymptotic behaviour of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.

Recently

**Contract with RAVEL (01/09/2018-31/08/2019, budget 15000 euros) :** P. Chartier, M. Lemou and F. Méhats initiated a collaboration with the startup RAVEL on a one-year basis (with possible renewal at the end of the year). The objective is to study the mathematical fondations of artificial intelligence and in particular machine learning algorithms for data anonymized though homomorphic encryption.

**Contract with CaiLabs (20/12/2018-20/06/2019, budget 8774 euros) :** E. Faou initiated a collaboration with the startup CaiLabs on a six-months basis (with possible renewal at the end of the contract).
The collaboration between CaiLabs and MINGuS aims at modelling optical devices allowing the recognition of simple objects. The structure of the devices combines quantum propagation phenomena, reflection mirrors and frequency absorbers and possesses a deep neural networks structure.

A. Crestetto is a member of the regional initiative "Pari Scientifique Exprodil".

M. Lemou is the head of the project "MUNIQ" of ENS Rennes. This two-years project (2018-2019) intends to gather multiscale numerical methods and uncertainty quantification techniques. The MINGuS members are P. Chartier, N. Crouseilles, M. Lemou and F. Méhats and colleagues from university of Madison-Wisconsin also belongs to this project.

**ANR MOONRISE: 2015-2019**

The project *Moonrise* submitted by Florian Méhats has been funded by the ANR for 4 years, for the period 2015-2019. This project aims at exploring modeling, mathematical and numerical issues originating from the presence of high-oscillations in nonlinear PDEs from the physics of nanotechnologies (quantum transport) and from the physics of plasmas (magnetized transport in tokamaks). The partners of the project are the IRMAR (Rennes), the IMT (Toulouse) and the CEA Cadarache. In the MINGuS team, François Castella, Philippe Chartier, Nicolas Crouseilles and Mohammed Lemou are members of the project Moonrise.

**Postdocs**

Loïc Le Treust has been hired as a Postdoc, under the supervision of Philippe Chartier and Florian Méhats. His contract started in september 2015 and ended in august 2016. Loïc Le Treust is now assistant professor at the university of Marseille.

Yong Zhang has been hired as a Postdoc, under the supervision of Philippe Chartier and Florian Méhats. His contract started in september 2015 and ended in august 2016. Yong Zhang is now professor at the Tianjin university.

Xiaofei Zhao has been hired as a Postdoc from september 2015 to september 2016 under the supervision of Florian Méhats. Xiaofei Zhao is now postdoc in the MINGuS team.

**ANR MFG: 2016-2020**

Mean Field Games (MFG) theory is a new and challenging mathematical topic which analyzes the dynamics of a very large number of interacting rational agents. Introduced ten years ago, the MFG models have been used in many areas such as, e.g., economics (heterogeneous agent models, growth modeling,...), finance (formation of volatility, models of bank runs,...), social sciences (crowd models, models of segregation) and engineering (data networks, energy systems...). Their importance comes from the fact that they are the simplest (“stochastic control"-type) models taking into account interactions between rational agents (thus getting beyond optimization), yet without entering into the issues of strategic interactions. MFG theory lies at the intersection of mean field theories (it studies systems with a very large number of agents), game theory, optimal control and stochastic analysis (the agents optimize a payoff in a possibly noisy setting), calculus of variations (MFG equilibria may arise as minima of suitable functionals) and partial differential equations (PDE): In the simplest cases, the value of each agent is found by solving a backward Hamilton-Jacobi equation whereas the distribution of the agents' states evolves according to a forward Fokker-Planck equation. The “Master" equation (stated in the space of probability measures) subsumes the individual and collective behaviors. Finally, modeling, numerical analysis and scientific computing are crucial for the applications. French mathematicians play a world-leading role in the research on MFG: The terminology itself comes from a series of pioneering works by J.-M. Lasry and P.-L. Lions who introduced most of the key ideas for the mathematical analysis of MFG; the last conference on MFG was held last June in Paris and organized by Y. Achdou, P. Cardaliaguet and J.-M. Lasry. As testifies the proposal, the number of researchers working on MFG in France (and also abroad) is extremely fast-growing, not only because the theoretical aspects are exciting and challenging, but also because MFG models find more and more applications. The aim of the project is to better coordinate the French mathematical research on MFG and to achieve significant progress in the theory and its applications.

The partners of the project are the CEREMADE laboratory (Paris Dauphine), the IRMAR laboratory (Rennes I), the university of Nice and of Tours.

**ANR MoHyCon: 2017-2021**

The MoHyCon project is related to the analysis and simulation of numerical methods for multiscale models of semiconductors. As almost all current electronic technology involves the use of semiconductors, there is a strong interest for modeling and simulating the behavior of such devices, which was recently reinforced by the development of organic semiconductors used for example in solar panels or in mobile phones and television screens (among others).

There exists a hierarchy of semiconductors models, including mainly three classes, which correspond to different scales of observation: microscopic, mesoscopic and macroscopic. At the microscopic scale, the particles are described one by one, leading to a huge system almost impossible to study, both theoretically and numerically. Within MoHyCon, we are then rather interested in the two other scales. The considered models at the mesoscopic scale are kinetic, of Boltzmann type, describing a distribution of particles submitted to an electric field. These models describe accurately the behavior of the semiconductor, but can be intricate and highly time and resource consuming to solve numerically. Thus, when the mean free path becomes small, it is preferable to consider fluid models, describing macroscopic quantities. Depending on the considered number of moments, various models can be obtained. The more common ones are the energy transport model, describing the densities of electron and energy, and the more simple drift-diffusion model, where the temperature is assumed to be a given function of the electron density.

In this project and provided this context, our aim is to construct and study rigorously numerical methods for these multiscale models. To this end, we will consider two distinct approaches: “Asymptotic Preserving” (AP) methods and coupling methods. The idea of AP methods is to design only one scheme which will be able to treat accurately every scale, without imposing restrictive stability conditions on the discretization parameters. Regarding the coupling methods, they consist in decomposing the domain into different regions on which the more relevant model (kinetic or macroscopic) will be considered. After locating the kinetic and fluid domains, the main difficulty is to obtain correct coupling conditions at each interface between two regions.

Considering the AP approach, our aim is to construct schemes for the linear Boltzmann equation for semiconductors, asymptotic preserving at the limit given by the drift-diffusion model. We will start with a very simplified model, with a linearized BGK collision operator. The constructed scheme will tend to an implicit discretization of Scharfetter-Gummel type for the drift-diffusion equation. The main objective will be then to lead a complete and rigorous study of the AP property by adapting to the discrete framework some continuous techniques: establish a discrete dissipation property yielding uniform estimates on the approximate solution which will allow to pass to the diffusion limit in the scheme.

As regards the second approach, the aim is to build a hybrid model coupling the kinetic equation on weakly collisional regions with macroscopic models on the remaining domain. We will first study specifically discretizations for energy-transport model, since we are then going to couple them with kinetic and other macroscopic schemes in our hybrid method. A particular attention will be paid to the implementation in order to obtain a robust and efficient code.

This ANR project is headed by M. Bessemoulin (CNRS, Nantes).

**IPL FRATRES**
MINGuS is associated to IPL FRATRES which started in june 2015. The aim of this project is
to organize Inria teams activities which develop mathematical and numerical tools in magnetically confined nuclear fusion.
The ambition is to prepare the next generation of numerical modeling methodologies
able to use in an optimal way the processing capabilities of modern massively parallel architectures.
This objective requires close collaboration between a) applied mathematicians and physicists that
develop and study mathematical models of PDE; b)
numerical analysts developing approximation schemes; c) specialists of algorithmics
proposing solvers and libraries using the many levels of parallelism offered by the modern architecture
and d) computer scientists. The project road map ambitions to contribute in
close connection with National and European initiatives devoted to nuclear Fusion
to the improvement and design of numerical simulation technologies applied to plasma
physics and in particular to the ITER project for magnetic confinement fusion.

The IPL FRATRES ends at the end of the year. The final meeting was organized in november during which X. Zhao (former IPL postdoc), G. Morel (IPL postdoc) and N. Crouseilles have presented the kinetic activities of the IPL project.

**Postdoc**

Xiaofei Zhao has been hired as a postdoc, under the supervision of Nicolas Crouseilles and Sever Hirstoaga (Inria-Nancy). His contract started in october 2016 and ended in september 2017. Xiaofei Zhao is now postdoc in the MINGuS team.

Guillaume Morel has been hired as a postdoc, under the supervision of Nicolas Crouseilles and Michel Mehrenberger (AMU). His contract started in october 2018 and ended in september 2019.

**ANTIpODE**

Title: Asymptotic Numerical meThods for Oscillatory partial Differential Equations with uncertainties

International Partner (Institution - Laboratory - Researcher):

University of Wisconsin-Madison, USA (United States) - DEPARTMENT MATHEMATICS - Shi Jin

Start year: 2018

See also: https://

The proposed associate team assembles the Inria team MINGuS and the research group led by Prof. Shi Jin from the Department of Mathematics at the University of Wisconsin, Madison. The main scientific objective of ANTIpODE consists in marrying uniformly accurate and uncertainty quantification techniques for multi-scale PDEs with uncertain data. Multi-scale models, as those originating e.g. from the simulation of plasma fusion or from quantum models, indeed often come with uncertainties. The main scope of this proposal is thus

**Informal International Partners**
The members of the MINGuS have several interactions with the following partners

the group of W. Bao (university of Singapore).

university of Maryland (S. Cerrai)

university of Ferrare (G. Dimarco)

university of Madison-Wisconsin (S. Jin and Q. Li)

university of Geneva (G. Vilmart)

The following scientists came in Rennes to visit some MINGuS members

Fernando Casas (Professor, University of Jaume I, Castellon, Spain): invited professor for five months, from september 2018 to january 2019, funded by the Labex (CHL) Center Henri Lebesgue, collaboration on dissipative equations, splitting methods and gradient descent methods with P. Chartier, N. Crouseilles, M. Lemou and F. Méhats.

Molei Tao (Associate professor, Georgia Tech, Atlanta, USA): eigth-days visit in november, funded by the ANR Moorise, collaboration on highly-oscillatory differential equations with P. Chartier, M. Lemou and F. Méhats.

Di Fang (PhD student, University of Wisconsin, USA): ten-days visit in december 2019, funded by the associated team ANTIpODE, collaboration on highly-oscillatory differential equations with P. Chartier, M. Lemou and F. Méhats.

Here we list the members of the team who did some research stays abroad

P. Chartier, N. Crouseilles, M. Lemou, F. Méhats and P. Navaro: invited working visit (Eric Sonnendrücker), Max Planck Institute, Garching, Germany, March 19th-23th, 2018.

P. Chartier, M. Lemou and F. Méhats: working visit, university of Madison-Wisconsin, USA, october 10th-20th, 2018.

P. Chartier: invited working visit (Christophe Besse), University of Toulouse 3, Toulouse, February 26th-28th, 2018.

P. Chartier: invited working visit (Gilles Vilmart), University of Geneva, Geneva, Switzerland, January 19th-26th, 2018.

P. Chartier: invited working visit (Ander Murua), University of the Basque Country, San-Sebastian, Spain, January 15th-19th, 2018.

N. Crouseilles: invited working visit (Giacomo Dimarco), University of Ferrare, Italy, october 2018.

A. Debussche: invited working visit (Sandra Cerrai), university of Maryland, USA, april 2018.

M. Lemou and F. Méhats: invited working visit (Ana Maria Luz), University Federal of Fluminence, Niteroi, Brazil, April 3th-8th, 2018.

F. Castella organized the workshop "Multiscale numerical methods", Saint-Malo, december 13-15 2017. [15 participants]

P. Chartier, M. Lemou and F. Méhats organized the MOONRISE Workshop on "Numerical methods for multi-scale PDEs", Institut d'études scientifiques de Cargèse, Cargèse, France, September 3th-7th, 2018.

P. Chartier, N. Crouseilles, M. Lemou and F. Méhats organized the workshop Henri Lebesgue Center on "Geometric and multi-scale methods for kinetic equations", Rennes, June 12th-15th, 2018.

A. Crestetto organized the "journée d'analyse Rennes-Nantes", january 2018, ENS Rennes.

A. Crestetto organized the workshop Henri Lebesgue Center on "mathematical models in health sciences".

N. Crouseilles co-organized the weekly seminar "Mathematic and applications", ENS Rennes.

A. Debussche organized the workshop on Stochastic partial differential equations, may 2018, CIRM, Marseille, France.

N. Crouseilles was member of the scientific committee of the SMAI-2019 conference organized by the university of Nantes.

E. Faou was head of organization of the semester *scientific computing* sponsored by the Labex Lebesgue (2 international summer schools, 7 workshops and international conferences), january-july 2018.

P. Chartier is member of the editorial committee of Mathematical Modelling and Numerical Analysis (M2AN).

A. Debussche is editor in chief of Stochastic Partial Differential Equations: analysis and computations (2013-).

A. Debussche is associate editor of Differential and Integral Equations (2002-2018).

A. Debussche is associate editor of Potential Analysis (2011-2018).

A. Debussche is associate editor of ESAIM:PROC (2012-).

A. Debussche is associate editor of Journal of Evolution Equation (2014-).

A. Debussche is associate editor of Applied Mathematics & Optimization (2014-).

A. Debussche is associate editor of SIAM JUQ (2017-).

A. Debussche is member of the editorial board of the collection "Mathématiques & Applications de la SMAI", Springer.

M. Lemou is member of the editorial committee of "Communications in Mathematical Sciences" (CMS).

The members of the IPSO team are reviewers for almost all the journals in which they publish (SIAM, JCP, CPDE, CMP, ARMA, JSP, JSC, JMAA, ANM, JCAM, NMPDE, Numer. Math.,

J. Bernier: invited speaker at the workshop "dynamics of Hamiltonian PDEs", february 2018, La Thuile, Italy.

J. Bernier: invited speaker at the workshop "NABUCO", february 2018, Toulouse, France.

P. Chartier: invited speaker at the 12th AIMS Conference on Dynamical Systems, Differential Equations and Application, July 5 - July 9, 2018 Taipei, Taiwan.

A. Crestetto: invited speaker at the conference ABPDEIII, august 2018, Lille, France.

A. Crestetto: invited speaker at the workshop on "Numerical Methods for Multiscale PDEs", september 2018, Cargèse, France.

N. Crouseilles: invited talk at the Workshop Numerical Aspects of Hyperbolic Balance Laws and Related Problems, University of Ferrara, Italy, April 2018.

N. Crouseilles: invited talk at the "Journée Scientifique Inria", june 27-28, Bordeaux, France.

N. Crouseilles: invited speaker at the 12th AIMS Conference on Dynamical Systems, Differential Equations and Application, July 5 - July 9, 2018 Taipei, Taiwan.

N. Crouseilles: invited speaker at the worskhop "Integrating the Integrators for Nonlinear Evolution Equations: from Analysis to Numerical Methods, High-Performance-Computing and Applications", december 2018, BIRS, Banff, Canada.

A. Desbussche: invited speaker "Aziz lecture", university of Maryland, april 2018, USA.

A. Desbussche: invited speaker at the "school on PDEs", Riemann international school of mathematics, july 2018, Varese, Italy.

E. Faou: invited plenary speaker at the conference on "Mathematics of Wave Phenomena", august 2018, KIT Karlsruhe, Germany.

E. Faou: invited speaker at the worskhop "Integrating the Integrators for Nonlinear Evolution Equations: from Analysis to Numerical Methods, High-Performance-Computing and Applications", december 2018, BIRS, Banff, Canada.

M. Lemou: invited speaker at the 12th AIMS Conference on Dynamical Systems, Differential Equations and Application, July 5 - July 9, 2018 Taipei, Taiwan.

M. Lemou: invited speaker at the Ki-Net Conference on "Mathematical and Numerical Aspects of Quantum Dynamics", June 19-21, 2018. CSCAMM, University of Maryland, USA.

M. Lemou: invited speaker at "Premier Congrès Franco-Marocain de Mathématiques Appliquées", april 16-20 2018, Marrakech, Maroc.

F. Méhats: invited talk at the Workshop Numerical Aspects of Hyperbolic Balance Laws and Related Problems, University of Ferrara, Italy, April 2018.

F. Méhats: 7 hours mini course on Averaging Techniques, Colloque Inter' Actions, Lyon, May 2018.

F. Méhats: talk at the day Space-Time Multiscale Methods (Atelier MePhy), Paris, June 2018.

F. Méhats: talk at the University of Wisconsin, October 2018.

F. Méhats: talk at a workshop on dispersive equations, October 2018.

G. Morel: invited speaker at the Numkin workshop, october 2018, Max Planck Institute, Garching, Germany.

P. Navaro: invited speaker at the CANUM 2018, France.

P. Navaro: invited speaker at the IPL meeting, France.

P. Navaro: invited speaker at the Finist'R meeting, France.

X. Zhao: invited speaker at the workshop on "Numerics for cosmology - the Schrödinger method", may 22nd-25th 2018, WPI Vienna, Austria.

X. Zhao: invited speaker at the workshop on "weak turbulence, kinetic problems and related", july 14-15 2018, Wuhan university, China.

X. Zhao: invited speaker at the "Forum on modern challenges in modelling and numerics", july 20th-21st 2018, Beijing CSRC, China.

X. Zhao: invited speaker at the "Modern Numerics on Nonlinear Wave and Dispersive Equations and Related Problems", august 8th-9th 2018, Chengdu university, China.

X. Zhao: invited speaker at the Numkin workshop, october 2018, Max Planck Institute, Garching, Germany.

X. Zhao: invited speaker at the workshop on "Numerical Methods for Multiscale PDEs", september 2018, Cargèse, France.

P. Chartier was member of the hiring committee CR2-Inria (Bordeaux).

P. Chartier was member of the hiring committee the Inria promotion committee (DR1-DR0).

A. Crestetto was member of the hiring committee AGPR-2018, ENS Rennes, France.

A. Crestetto was the vice-head of the hiring committee of the MCF position, university of Nantes, France.

A. Debussche was reviewer for Austrian Science Fundation (FWF).

E. Faou is the member of the CNU 26.

E. Faou was member of the hiring committee of the professor position, university of Dijon, France.

F. Castella was member of the evaluation HCERES committee of the MICS laboratory (CentraleSupélec).

P. Chartier is the vice-head of science (DSA) of the Rennes Inria-Center until the end of 2018,

P. Chartier is member of the national evaluation committee (CE) of Inria until the end of 2018.

P. Chartier is member of the direction committee (ED) of the Rennes Inria-Center until the end of 2018

P. Chartier is member of the G6-committee (bureau of the direction committee) Inria-Rennes-Bretagne-Atlantique center until the end of 2018.

A. Crestetto is member of the mathematics department council, university of Nantes.

A. Crestetto is member of the scientific council of the sciences unity, university of Nantes.

A. Crestetto is the SMAI correspondent for the mathematics laboratory of Nantes.

N. Crouseilles is member of the scientific council of ENS Rennes.

N. Crouseilles is member of the mathematic laboratory (IRMAR) council.

N. Crouseilles is member of the Fédération de Fusion council (University of Rennes representative).

A. Debussche is member of the scientific council of the Fédération Denis Poisson.

A. Debussche is member of the administrative council of ENS Paris-Saclay.

A. Debussche if scientific vice-deputy and international relations of ENS Rennes.

A. Debussche is vice-head ot the Centre Henri Lebesgue.

A. Debussche is vice-head of the Lebesgue agency for Mathematic and Innovation.

E. Faou is the AMIES correspondent for Inria Bretagne Atlantique.

E. Faou is member of the scientific council of the Pôle Universitaire Léonard de Vinci.

M. Lemou is the head of the IRMAR team "Analyse numérique" composed of 48 members.

M. Lemou is member of the scientific council of ENS Rennes.

M. Lemou is member of the scientific council of the LABEX Lebesgue.

P. Navaro is member of the national network "calcul" http://

Licence:

P. Chartier: lecture "Ordinary differential equations" of the first year (Licence) of the Magistère, Ecole Normale Supérieure de Rennes (24 hours).

N. Crouseilles: lecture on "Numerical methods" of L3, ENS Rennes (30h).

Master:

F. Castella: lecture on "Numerical methods" of M1, University of Rennes 1 (50 hours).

P. Chartier: lecture on "Numerical methods for kinetic equations" of the third year (M2) of the Magistère, University of Rennes 1 (12 hours).

A. Crestetto: lecture on "Numerical methods for incompressible fluids" of M2, university of Nantes (62h).

N. Crouseilles: lecture on "Numerical methods for PDEs" of M1, ENS Rennes (30h).

E. Faou: lecture on "numerical integration of Hamiltonian PDEs" of M2, university of Rennes 1 (30h).

M. Lemou: lecture on "elliptic PDEs" of the second year (M1) of the Magistère, University of Rennes 1 (36 hours).

M. Lemou: lecture on "fundamental tools for PDEs" of the third year (M2) of the Magistère, University of Rennes 1 (12 hours).

M. Lemou: lecture on "Numerical tools for kinetic equations" of the third year (M2) of the Magistère, University of Rennes 1 (12 hours).

P. Navaro: lecture on "numerical tools for Big Data", master, university of Rennes 2.

P. Navaro: lecture on "Python programming", master, ENSAI.

Doctorat:

F. Casas: lecture for PhD students " A (very) Short Introduction to Geometric Numerical Integration and Splitting Methods", (4h).

PhD: M. Fontaine, Modèles mathématiques de type HMF : stabilité et méthodes numériques autour d'états staionnaires, ENS Rennes, july 13th, advisors: M. Lemou and F. Méhats.

PhD: M. jugal Nguepedja Nankep, Modélisation stochastique de systèmes biologiques multi-échelles et inhomogènes en espace, ENS Rennes, march 22th, advisor: A. Debussche.

PhD in progress: J. Bernier, Mathematical and numerical anaysis of nonlinear transport equations, started in september 2016, advisors: N. Crouseilles and E. Faou.

PhD in progress: A. Rosello, "Approximation-diffusion pour des équations cinétiques pour les modèles de type spray", started in september 2016, advisors: A. Debussche and J. Vovelle (CNRS, Lyon).

PhD in progress: J. Massot, "Modélisation et simulation de plasmas chauds", started in october 2018, advisors: A. Crestetto (Nantes) and N. Crouseilles.

PhD in progress: L. Trémant, "Méthodes d'analyse asymptotique et d'approximation numérique de modèles dissipatifs multi-échelles : EDO à variété centrale et modèles cinétiques", started in october 2018, advisors: P. Chartier and M. Lemou.

F. Castella: reviewer of the thesis and member of the PhD committee of P. Terrier (university of Paris Est and CEA).

A. Crestetto: member of the PhD committee of the thesis of M. Stauffert (university of Versailles).

N. Crouseilles: reviewer of the thesis and member of the PhD comittee of N. Bouzat (university of Strasbourg, Inria and CEA).

N. Crouseilles: member of the PhD comittee of M. Malo (ENS Rennes).

A. Debussche: reviewer of the HDR and member of the HDR committee of G. Dujardin (Inria, university of Lille).

A. Debussche: member of the PhD committee of M. Tomasevic (university of Nice).

A. Debussche: member of the PhD committee of P. Tsatsoulis (university of Warwick).

E. Faou: reviewer of the thesis and member of the PhD committee of M.-S. Dupuy (university of Paris Diderot).

E. Faou: member of the HDR committee of G. Dujardin (Inria, university of Lille).

A. Crestetto: talks on "traffic flows" in several educational institutions within the framework of "Essentielles", university of Nantes.

N. Crouseilles: welcoming of two schoolchildren at IRMAR laboratory.

N. Crouseilles: welcoming of a student within the framework of the modulus "visit a researcher" (3 meetings).

P. Navaro: Python training (3 days) at the CNRS delegation of Lille, april 2018.

P. Navaro: Python training (3 days) for the Rennes DSI, june 2018.