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.

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,

The Selalib (SEmi-LAgrangian LIBrary) software library

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, organized 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.

The main goal is to characterize how spatial propagation phenomena (diffusion, transport, advection,

*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

**Analysis of PDEs and SPDEs**

for some

**Numerical schemes**

The efficient numerical solution of many kinetic models in plasma physics is impeded by the stiffness of these systems. Exponential integrators are attractive in this context as they remove the CFL condition induced by the linear part of the system, which in practice is often the most stringent stability constraint. In the literature, these schemes have been found to perform well, e.g., for drift-kinetic problems. Despite their overall efficiency and their many favorable properties, most of the commonly used exponential integrators behave rather erratically in terms of the allowed time step size in some situations. This severely limits their utility and robustness. Our goal in is to explain the observed behavior and suggest exponential methods that do not suffer from the stated deficiencies. To accomplish this we study the stability of exponential integrators for a linearized problem. This analysis shows that classic exponential integrators exhibit severe deficiencies in that regard. Based on the analysis conducted we propose to use Lawson methods, which can be shown not to suffer from the same stability issues. We confirm these results and demonstrate the efficiency of Lawson methods by performing numerical simulations for both the Vlasov-Poisson system and a drift-kinetic model of a ion temperature gradient instability.

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, we 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.

Contrat with RAVEL (onne year, budget 15000 euros): this is 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.

Participants: P. Chartier, M. Lemou and F. Méhats.

Contract with Cailabs (6 months, budget 3000 euros): This collaboration aims at exploring the possibility of deriving new fiber optics devices based on neural networks architecture.

Participants: P. Chartier, E. Faou, M. Lemou and F. Méhats.

M. Lemou and N. Crouseilles are 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 belong to this project.

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 (China).

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 assistant professor in the Wuhan University (China).

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.

The aim of this project is to treat multiscale models which are both infinite-dimensional and stochastic with a theoretic and computational approach. Multiscale analysis and multiscale numerical approximation for infinite-dimensional problems (partial differential equations) is an extensive part of contemporary mathematics, with such wide topics as hydrodynamic limits, homogenization, design of asymptotic-preserving scheme. Multiscale models in a random or stochastic context have been analysed and computed essentially in finite dimension (ordinary/stochastic differential equations), or in very specific areas, mainly the propagation of waves, of partial differential equations. The technical difficulties of our project are due to the stochastic aspect of the problems (this brings singular terms in the equations, which are difficult to understand with a pure PDE's analysis approach) and to their infinite-dimensional character, which typically raises compactness and computational issues. Our main fields of investigation are: stochastic hydrodynamic limit (for example for fluids), diffusion-approximation for dispersive equations, numerical approximation of stochastic multiscale equations in infinite dimension. Our aim is to create the new tools - analytical, probabilistic and numerical - which are required to understand a large class of stochastic multiscale partial differential equations. Various modelling issues require this indeed, and are pointing at a new class of mathematical problems that we wish to solve. We also intend to promote the kind of problems we are interested in, particularly among young researchers, but also to recognized experts, via schools, conference, and books.

The partners are ENS Lyon (coordinator J. Vovelle) and ENS Rennes (Coordinator A. Debussche).

We are involved in the national research multidisciplinary group around magnetic fusion activities. As such, we answer to annual calls.

A. Debussche and E. Faou are members of the IPL (Inria Project Lab) SURF: Sea Uncertainty Representation and Forecast. Head: Patrick Vidard.

This AdT started in october 2019 and will be finished in september 2021. An engineer has been hired (Y. Mocquard) to develop several packages in the Julia langage. The J-Plaff is shared with the Fluminance team.

Program: Eurofusion

Project acronym: MAGYK

Project title:

Duration: january 2019-december 2020

Coordinator: E. Sonnendrücker

Other partners: Switzerland, Germany, France, Austria, Finland.

Abstract: This proposal is aimed at developing new models and algorithms that will be instrumental in enabling the efficient and reliable simulation of the full tokamak including the edge and scrape-off layer up to the wall with gyrokinetic or full kinetic models. It is based on a collaboration between applied mathematicians and fusion physicists that has already been very successful in a previous enabling research project and brings new ideas and techniques into the magnetic fusion community. New modelling and theoretical studies to extend the modern gyrokinetic theory up to the wall including boundary conditions will be addressed, and the limits of gyrokinetics will be assessed. New multiscale methods will enable to efficiently and robustly separate time scales, which will on the one hand make gyrokinetic codes more efficient and on the other hand enable full implicit kinetic simulations. Difficult algorithmic issues for handling the core to edge transition, the singularities at the O- and X-points will be addressed. And finally, pioneering work based on recent (deep) machine learning techniques will be performed, on the one hand to automatically identify a Partial Differential Equation (PDE) from the data, which can be used for verification and sensitivity analysis purposes, and on the other hand to develop reduced order models that will define a low- cost low-fidelity model based on the original high-fidelity gyrokinetic or kinetic model that can be used for parameter scans and uncertainty quantification.

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

International Partner (Institution - Laboratory - Researcher):

University of Wisconsin-Madison, USA (United States)

Start year: 2018

See also: https://

The proposed associate team assembles the Inria team IPSO 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 (i) the development of uniformly accurate schemes for PDEs where space and time high oscillations co-exist and (ii) their extension to models with uncertainties. Applications to plasmas (Vlasov equations) and graphene (quantum models) are of paramount importance to the project.

The members of MINGuS have several interactions with the following partners

Europe: University of Geneva (Switzerland), University of Jaume I (Spain), University of Basque Country (Spain), University of Innsbruck (Austria), University of Ferrare (Italy), Max Planck Institute (Germany), SNS Pisa (Italy)

USA: Georgia Tech, University of Maryland, University of Wisconsin, NYU

Asia: Chinese Academy of Science (China), University of Wuhan (China), shanghai jiao tong university (China), National University of Singapore (Singapore)

SIMONS project.
Erwan Faou is one of the Principal investigators of the Simons Collaboration program *Wave Turbulence*. Head: Jalal Shatah (NYU).

Fernando Casas (University of Jaume I, Spain) was invited in the MINGuS team during 6 months (september 2018 to february 2019), funded by the Labex (CHL) Center Henri Lebesgue.

Yingzhe Li (University of Chinese Academy of Sciences, China) is visiting the IRMAR laboratory during one year (March 2019-February 2020) thanks to a chinese grant. He is currently a PhD student advised by Yajuan Sun, professor at CAS.

Xiaofei Zhao (University of Wuhan, China) was invited in the MINGuS team during 2 weeks (july 2019).

Yoshio Tsutsumi (Kyoto University, Japan) was invited in the IRMAR laboratory during 2 months (october-november 2019).

G. Barrué: Master 2 internship, A. Debussche.

Q. Chauleur: Master 2 internship, R. Carles (CNRS, Rennes) and E. Faou.

U. Léauté: Master 1 internship, B. Boutin (University Rennes I and N. Crouseilles).

A. V. Tuan: Master 2 internship, M. Lemou and F. Méhats.

P. Chartier was on a sabbatical visit from the 1st of February to the 30th of September 2019 at the University of the Basque Country, Spain.

P. Chartier was invited by G. Vilmart, University of Geneva, Geneva, Switzerland, January 2019.

P. Chartier was invited by F. Casas at the university of Jaume I, Castellon, Spain, July 2019.

P. Chartier was Invited by Q. Li at the university of Wisconsin, Madison, USA, September 2019.

P. Chartier was Invited by M. Tao at Georgia Tech, Atlanta, USA, August 2019.

A. Debussche was invited by G. Da Prato at Scuola Normale Superiore, Pise, Italy, April 2019.

E. Faou was a participant of the
Semester *Geometry, compatibility and structure preservation in computational differential equations*, Isaac Newton Institute, Cambridge, UK (3 months stay, September-December 2019).

M. Lemou was invited by J. Joudioux and L. Anderson, at the Albert Einstein Institute, Golm, Germany, February 2019.

M. Lemou was invited by A. M. M. Luz at the Universidade Federal Fluminense, Rio de Janeiro, Brazil, April 2019.

M. Lemou was invited by S. Jin at Shanghai Jiao Tong University, Shanghai, China, April 2019.

M. Lemou was invited by J. Ben-Artzi at the university of Cardiff, Cardiff, UK, May 2019.

M. Lemou was invited by G. Vilmart, University of Geneva, Geneva, Switzerland, January 2019.

M. Lemou was Invited by Q. Li at the university of Wisconsin, Madison, USA, September 2019.

M. Lemou was Invited by M. Tao at Georgia Tech, Atlanta, USA, August 2019.

F. Méhats was invited by A. de la Luz at the Universidade Federal Fluminense, Rio de Janeiro, Brazil, April 2019.

F. Méhats was invited by G. Vilmart, University of Geneva, Geneva, Switzerland, January 2019.

F. Castella organized the MINGuS team meeting, Dinard, december 2019. [15 participants]

P. Chartier, N. Crouseilles, M. Lemou and F. Méhats organized a workshop on "Asymptotic methods and numerical approximations of multi-scale evolution problems, and uncertainty quantification", ENS Rennes, May 2019. [30 participants]

N. Crouseilles is co-organizer of the weekly seminar "Mathematic and applications" at ENS Rennes.

P. Chartier and M. Lemou co-organized (with M. Thalhammer, university of Innsbruck) the mini-symposium "Advanced numerical methods for differential equations" in the ICIAM 2019 conference, Valencia, Spain, July 2019.

F. Méhats co-organized (with W. Bao, National University of Singapore) the mini-symposium "Multiscale methods and analysis for oscillatory PDEs" in the Scicade conference, Innsbruck, Austria, July 2019.

N. Crouseilles was member of the scientific committee of the SMAI-19 Conference, June 2019.

E. Faou was member of the scientific committee of the Scicade Conference, July 2019.

P. Chartier is member of the editorial board 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 member of the editorial board of the collection "Mathématiques

A. Debussche is associate editor of Differential and Integral Equations (2002-19).

A. Debussche is associate editor of Potential Analysis (2011-2019).

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

A. Debussche is associate editor of Applied Mathematics

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

The members of the MINGuS 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 gave a talk in the seminar of Columbia university, New York, USA, June 2019.

P. Chartier gave a talk in the workshop *Nonlinear Evolution Equations: Analysis and Numerics*, organized by M. Hochbruck, H. Koch, S.-J. Oh, and A. Ostermann, Oberwolfach, Germany, February 2019.

P. Chartier gave a talk in the workshop HaLu, Gran Sasso Science Institute (GSSI) School of Advanced Studies, L'Aquila, Italy, June 2019.

P. Chartier gave a talk in the ICIAM conference, Valencia, Spain, July 2019.

P. Chartier gave a talk in the Scicade conference, University of Innsbruck, Austria, July 2019.

N. Crouseilles gave a talk at the workshop *Quantum and Kinetic Transport*, Shanghai, China, April 2019.

N. Crouseilles gave a talk in the workshop *Numerical Methods for Multiscale Models arising in Physics and Biology*, University of Nantes, France, June 2019.

N. Crouseilles gave a talk in the Scicade conference, University of Innsbruck, Austria, July 2019.

A. Debussche gave a talk in the conference *Partial Differential Equations: from theory to applications*,
Nancy, France, March 2019.

A. Debussche gave a talk in the workshop *Numerical Methods for SPDE: 20 Successful Years and Future Challenges*,
Reims, France, June 2019.

A. Debussche gave a mini-course in the workshop *PROPAL : propagation d'ondes en milieux aléatoires*,
Mittag-Leffler Institute, Stockholm, Sweden, May 2019

A. Debussche gave a talk in the workshop *Recents Trends in Stochastic Analysis and SPDEs*,
University of Pisa, Italy, July 2019.

A. Debussche gave a talk in the workshop *Touch down of Stochastic Analysis*,
University of Bielefeld, Germany, September 2019.

A. Debussche gave a talk in the conference *Challenges and New Perspectives in Mathematics*,
Hassan II Academy of Sciences and Technology, Morocco, November 2019.

A. Debussche gave a talk in the conference *Paths between probability, PDEs and physics*, Imperial College, July 2019.

E. Faou gave a talk in the workshop *The future of structure-preserving algorithms*, ICMS, Edinburgh, UK, October 2019.

E. Faou gave a talk in the Analysis seminar, CMS University of Cambridge, UK, October 2019.

E. Faou gave a talk in the *Plasma day* session, at the Isaac Newton Institute, Cambridge, UK, October 2019.

E. Faou gave a Colloquium talk at the University of Bielefeld, Germany, April 2019.

E. Faou gave a talk in the workshop *Dynamics of nonlinear dispersive PDEs*,
La Thuile, Italy, February 2019.

E. Faou gave a talk in the workshop *Nonlinear Evolution Equations: Analysis and Numerics*, organized by M. Hochbruck, H. Koch, S.-J. Oh, and A. Ostermann, Oberwolfach, Germany, February 2019.

Y. Li gave a talk in the Scicade conference, University of Innsbruck, July 2019.

Y. Li gave a talk in the NumKin conference, Max Planck Institute, Garching, October 2019.

M. Lemou gave a talk at the Albert Einstein Institute, Golm, Germany, February 2019.

M. Lemou gave a talk at the workshop *Quantum and Kinetic Transport*, Shanghai, China, April 2019.

M. Lemou gave a talk at the university of Cardiff seminar, Cardiff, UK, May 2019.

M. Lemou gave a talk in the ICIAM conference, Valencia, Spain, July 2019.

M. Lemou gave a talk in the Scicade conference, Innsbruck, Austria, July 2019.

M. Lemou gave a talk at the university of Wisconsin seminar, Madison, USA, September 2019.

M. Lemou gave a talk at the Georgia Tech seminar, Atlanta, USA, September 2019.

M. Lemou gave a talk in the workshop *Recent Progress and Challenge in Quantum and Kinetic Problems*, Singapore, October 2019.

J. Massot gave a talk in the NumKin conference, Max Planck Institute, Garching, October 2019.

F. Méhats gave a talk in the workshop *Recent Progress and Challenge in Quantum and Kinetic Problems*, Singapore, October 2019.

F. Méhats gave a talk in the NumKin conference, Max Planck Institute, Garching, October 2019.

P. Navaro participated at the conference JuliaCon 2019, Baltimore, USA, August 2019.

A. Rosello gave a talk in the conference *Paths between probability, PDEs and physics*, Imperial College, July 2019.

N. Crouseilles was member of the committee of the Blaise Pascal prize (GAMNI-SMAI), 2019.

N. Crouseilles was member of the committee of the best PhD talk in the Scicade conference, 2019.

N. Crouseilles was member of the committee of the expert reviewers for the European Doctoral programme of the University of Innsbruck.

A. Debussche was reviewer for ERC projects.

E. Faou was member of the committee of the PhD prize SMAI-GAMNI, 2019. 2019).

F. Castella is member of the UFR mathématiques council, University Rennes 1.

N. Crouseilles is responsible of Fédération Recherche Fusion for the University of Rennes I.

N. Crouseilles is member of the IRMAR laboratory council, University Rennes 1.

N. Crouseilles is member of the scientific council of ENS Rennes (until september 2019).

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 is scientific vice-deputy and international relations of ENS Rennes.

A. Debussche is co-director of the Henri Lebesgue Center (Excellence laboratory of the program investissement d'avenir).

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

E. Faou is co-director of the Henri Lebesgue Center (Excellence laboratory of the program investissement d'avenir).

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 Henri Lebesgue Center.

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

Master :

F. Castella, Numerical methods for ODEs and PDEs, 60 hours, Master 1, University of Rennes.

N. Crouseilles, Numerical methods for PDEs, 24 hours, Master 1, ENS Rennes.

E. Faou, Normal forms, 24 hours, Master 2, University of Rennes.

M. Lemou, Numerical methods for kinetic equations, 18 hours, Master 2, University of Rennes.

M. Lemou, elliptic PDEs, 36 hours, Master 1, University of Rennes.

P. Navaro, Python courses, Master 2 Smart Data, ENSAI.

P. Navaro, Scientific computing tools for big data, Master 2, University of Rennes.

PhD : G. Barrué, Approximation diffusion pour des équations dispersives, University of Rennes I, started in september 2019, A. Debussche.

PhD : J. Bernier, Study of some perturbation of equations which involve symmetries: resonancy and stability, University of Rennes I, defended in july 2019, E. Faou and N. Crouseilles.

PhD : Q. Chauleur, Equation de Vlasov singulière et équations reliées, University of Rennes I, started in september 2019, R. Carles (CNRS, Rennes) and E. Faou.

PhD: Y. Li (Chinese Academy of Sciences), Structure preserving methods for Vlasov equations, march 2019-february 2020, Y. Sun (Chinese Academy of Sciences) and N. Crouseilles.

PhD in progress : J. Massot, Exponential methods for hybrid kinetic models, started in october 2018, N. Crouseilles.

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

PhD in progress : L. Trémant, Asymptotic analysis methods and numerical of dissipative multi-scale models: ODE with central manifold and kinetic models, started in october 2018, P. Chartier and M. Lemou.

F. Castella was referee for the PhD thesis of H. Moundoyi (Laboratoire de biologie marine de Roscoff, France), supervised by P. Cormier and B. Sarels.

F. Castella was referee for the PhD thesis of F. Patout (ENS Lyon, France), supervised by V. Calvez and J. Garnier.

N. Crouseilles was referee for the PhD thesis of B. Fedele (University Toulouse 3, France), supervised by C. Negulescu and M. Ottaviani (CEA).

A. Debussche was referee for the PhD of E. Altmann (Sorbonne Université, France), supervised by L. Zambotti.

A. Debussche was member of the defense committee of the PhD of B. Kouegou Kamen (Aix-Marseille Université, France), supervised by E. Pardoux.

A. Debussche was referee for the PhD of T. Yeo (Aix-Marseille université, France) supervised by E. Pardoux.

E. Faou was referee for the Habilitation degree of Karolina Kropielnicka (Univ. Gdansk, Poland).

M. Lemou was referee of the PhD thesis of X. Li (University Paris-Dauphine, France), supervised by J. Dolbeault.

F. Méhats was referee of the PhD thesis of T. Dolmaire (university Paris Diderot, France), supervised by L. Desvillettes and I. Gallagher.

F. Castella was member of the HCERES committee for the evaluation of the laboratory "Mathématiques et Informatique pour la Complexité et les Systèmes (MICS)", CentraleSupelec, Gif-sur-Yvette.

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

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

N. Crouseilles is member of the Inria Evaluation Committe (2019-2023).

N. Crouseilles was member of the Inria hiring committee for the following Inria promotions: CRHC, DR2-DR1, DR1-DR0, DR0-DR02.

N. Crouseilles is member of the hiring committee of the professor position, ENS Rennes.

E. Faou was member of the CNU 26 until the summer 2019.

E. Faou was member of the HCERES committee for the evaluation of the Mathematics Institute of Toulouse, University Paul Sabatier.

E. Faou is AMIES correspondent (Agency for Interaction in Mathematics with Business and Society) for Inria Rennes Bretagne atlantique and IRMAR.

N. Crouseilles: participation to high school students internship at IRMAR laboratory (one week), June 2019.

N. Crouseilles: interview by a first year student of University Rennes I (in order to inform the different ways to become Inria researcher).

J. Massot: talk at the N. Mandela high school (terminal S classes) about the links between astronomy and mathematics, April 2019.

J. Massot: participation to high school students internship at IRMAR laboratory (one week), June 2019.

P. Navaro: participation to the Julia day, Lyon, France, January 2019.

P. Navaro: participation to the Julia day, Nantes, France, June 2019.

A. Rosello: participation to "MATHC2

L. Trémant: participation to "Maths en Jean", June 2019, University of Rennes.

J. Massot: writing of a Python library *ponio* (Python Objects for Numerical IntegratOr) https://

P. Navaro: Python training at the IRMAR laboratory.

P. Navaro: Julia training at the IRMAR laboratory.

P. Navaro: R training at the Finist'R internal workshop, Roscoff, France, August 2019.

P. Navaro, Python courses, within the exchange program between ENSAI and Hong Kong university.