3.1.1 Asymptotic analysis of dissipative PDEs (F. Castella, P. Chartier, A. Debussche, E. Faou, M. Lemou)

Derivation of asymptotic problems

Our main goal is to analyze the asymptotic behavior of dissipative systems of the form ((3)) when $\epsilon$ goes to zero. The center manifold theorem27 is of great interest but is largely unsatisfactory from the following points of view

• a constructive approach of $h$ and $x_0^{\epsilon }$ is clearly important to identify the high-order asymptotic models: this would require expansions of the solution by means of B-series or word-series 29 allowing the derivation of error estimates between the original solution and the asymptotic one.
• 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 (3). But, the same questions arise at the PDE level. Indeed, one certainly expects that dissipative terms occurring in collisional kinetic equations (2) 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 (2). The noise can describe creation or absorption (as in (2)), but it may also be a forcing term or a random magnetic field. In the parabolic scaling, one expects 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 by Debussche and Vovelle, 30, 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.

3.1.2 Numerical schemes for dissipative problems (All members)

The design of numerical schemes able to reproduce the transition from the microscopic to macroscopic scales largely matured with the emergence of Asymptotic Preserving schemes which have been developed initially for collisional kinetic equations (actually, for solving (2) when $\beta \to 0$). Several techniques have flourished in the last decades. As said before, AP schemes entail limitations which we aim at overcoming by deriving

• AP numerical schemes whose numerical cost diminishes as $\beta \to 0$,
• Uniformly accurate numerical schemes, whose accuracy is independent of $\beta$.

Time diminishing methods

The main goal consists in merging Monte-Carlo techniques 25 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 (3) whose error is uniform with respect to $\epsilon$. The construction of such a scheme requires a deep mathematical analysis as described above. Ideally one would like to develop schemes that preserve the center manifold (without computing the latter!) as well as schemes that resolve numerically the stiffness induced by the fast convergence to equilibrium (the so-called transient phase). First, our goal is to extend the strategy inspired by the central manifold theorem in the ODE case to the PDE context, in particular for collisional kinetic equations (2) when $\beta \to 0$. The design of Uniformly Accurate numerical schemes in this context would require to generalize two-scale techniques introduced by members of the team in the framework of highly-oscillatory problems 28.

Multiscale numerical methods for stochastic PDEs

AP schemes have been developed recently for kinetic equations with noise in the context of Uncertainty Quantification UQ 33. 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 developing AP schemes when the noise is also rescaled and the limit is a white noise driven SPDE, as described in section (3.1.1). 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.

3.2.1 Asymptotic analysis of highly-oscillatory PDEs (All members)

Derivation of asymptotic problems

We intend to study the asymptotic behavior of highly-oscillatory evolution equations of the form (4) posed in an infinite dimensional Banach space.

Recently, the stroboscopic averaging has been extended to the PDE context, considering nonlinear Schrödinger equation (1) in the highly-oscillatory regime. A very exciting way would be to use this averaging strategy for highly-oscillatory kinetic problem (2) 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 26) 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 (5).

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 34, 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 35, 31). 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 (1) with a time white noise in the regime $\epsilon <<1$), it is however possible to perform the analysis and to obtain averaged stochastic equations. We plan to go further by considering more difficult problems, such as the convergence of a stochastic Klein-Gordon-Zakharov system to as stochastic nonlinear Schrödinger equation.
• 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 36). 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.

3.2.2 Numerical schemes for highly-oscillatory problems (All members)

This section proposes to explore numerical issues raised by highly-oscillatory nonlinear PDEs for which (4) is a prototype. Simulating a highly-oscillatory phenomenon usually requires to adapt the numerical parameters in order to solve the period of size $\epsilon$ so as to accurately simulate the solution over each period, resulting in an unacceptable execution cost. Then, it is highly desirable to derive numerical schemes able to advance the solution by a time step independent of $\epsilon$. To do so, our goal is to construct Uniformly Accurate (UA) numerical schemes, for which the numerical error can be estimated by $C{h}^p$ ($h$ being any numerical parameters) with $C$ independent of $\epsilon$ and $p$ the order of the numerical scheme.

Recently, such numerical methods have been proposed by members of the team in the highly-oscillatory context. 28. They are mainly based on a separation of the fast and slow variables, as suggested by the decomposition (5). An additional ingredient to prove the uniformly accuracy of the method for (4) relies on the search for an appropriate initial data which enables to make the problem smooth with respect to $\epsilon$.

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 $1/\epsilon$ in comparison with standard methods, and finally averaged models are not able to capture the intermediate regime since they miss important information of the original problem. We are strongly convinced that this strategy should be further studied and extended to cope with some other problems. The ultimate goal being to construct a scheme for the original equation which degenerates automatically into a consistent approximation of the averaged model, without resolving it, the latter can be very difficult to solve.

• Space oscillations:

  When rapidly oscillating coefficients in space (i.e. terms of the form $a(x,x/\epsilon )$) occur in elliptic or parabolic equations, homogenization theory and numerical homogenization are usually employed to handle the stiffness. However, these strategies are in general not accurate for all $\epsilon \in ]0,1]$. Then, the construction of numerical schemes which are able to handle both regimes in an uniform way is of great interest. Separating fast and slow 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 $B$ depends on $t$ or $x$ in (2) but also for many other physical problems. We then have to deal with the delicate issue of space-time oscillations, which is known to be a very difficult problem from a mathematical and a numerical point of view. To take into account the space-time mixing, a periodic motion has to be detected together with a phase $S$ which possibly depends on the time and space variables. These techniques originate from 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 32). 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 32). 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 1 is still misunderstood although many complex problems involve multi-frequencies. Even if the quasi-periodic averaging is doable from a theoretical point of view in the ODE case, (see 37), it is unclear how it can be extended to PDEs. One of the main obstacle being the requirement, usual for ODEs like (4), for $ℱ$ to be analytic in the periodic variables, an assumption which is clearly impossible to meet in the PDE setting. An even more challenging problem is then the design of numerical methods for this problem.

• 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.

7.1.1 Selalib

• Name:
  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.
• URL:
  https://­selalib.­github.­io
• Contact:
  Philippe Helluy
• Participants:
  Edwin Chacon Golcher, Pierre Navaro, Sever Hirstoaga, Eric Sonnendrücker, Michel Mehrenberger
• Partners:
  Max Planck Insitute - Garching, Université de Strasbourg, CNRS, Université de Rennes 1

7.1.2 HOODESolver.jl

• Name:
  Julia package for solving numerically highly-oscillatory ODE problems
• Keywords:
  Ordinary differential equations, Numerical solver
• Functional Description:
  Julia is a programming language for scientists with a syntax and functionality similar to MATLAB, R and Python. HOODESolver.jl is a julia package allowing to solve ordinary differential equations with sophisticated numerical techniques resulting from research within the MINGUS project team. To use it, just install Julia on your workstation.
• Release Contributions:
  This is the first version of the package. It will evolve further because we want to have a better integration with the Julia organization on differential equations. This one already includes a lot of methods to numerically solve differential equations. This integration will allow us to have a larger audience and thus more feedback and possibly external collaborations.
• Contact:
  Nicolas Crouseilles
• Participants:
  Yves Mocquard, Pierre Navaro, Nicolas Crouseilles
• Partners:
  Université de Rennes 1, CNRS

8.2.1 Analysis of PDEs and SPDEs

In 4, we consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\epsilon$, for generic initial data of size $\epsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order ${\epsilon }^{M+1}$. This implies that for such initial data $u\left(0\right)$, we control the Sobolev norm of the solution $u\left(t\right)$ for time of order ${\epsilon }^{-M}$. Furthermore this property is locally stable: if $v\left(0\right)$ is sufficiently close to $u\left(0\right)$ (of order ${\epsilon }^{3/2}$) then the solution $v\left(t\right)$ is also controled for time of order ${\epsilon }^{-M}$.

In 5, we consider the nonlinear Schrödinger-Langevin equation for both signs of the logarithmic nonlinearity. We explicitly compute the dynamics of Gaussian solutions for large times, which is obtained through the study of a particular nonlinear differential equation of order 2. We then give the asymptotic behavior of general energy weak solutions under some regularity assumptions. Some numerical simulations are performed in order to corroborate the theoretical results.

In 8, we study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.

In 9, we consider the Nonlinear Schrödinger (NLS) equation and prove that the Gaussian measure with covariance ${(1-{\partial }_x^2)}^{-\alpha }$ on $L^2\left(T\right)$ is quasi-invariant for the associated flow for $\alpha >1/2$. This is sharp and improves a previous result obtained in a former work where the values $\alpha >3/4$ were obtained. Also, our method is completely different and simpler, it is based on an explicit formula for the Radon-Nikodym derivative. We obtain an explicit formula for this latter in the same spirit as former results in the literature. The arguments are general and can be used to other Hamiltonian equations.

In 10, we study the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion (Penrose criterion). We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear Landau damping effect with an algebraic rate of damping.

In 14, we propose and analyze a new asynchronous rumor spreading protocol to deliver a rumor to all the nodes of a large-scale distributed network. This spreading protocol relies on what we call a $k-$pull operation, with $k\ge 2$. Specifically a $k-$pull operation consists, for an uninformed node $s$, in contacting $k-1$ other nodes at random in the network, and if at least one of them knows the rumor, then node $s$ learns it. We perform a thorough study of the total number $T_{k,n}$ of $k-$pull operations needed for all the $n$ nodes to learn the rumor. We compute the expected value and the variance of $T_{k,n}$, together with their limiting values when $n$ tends to infinity. We also analyze the limiting distribution of $(T_{k,n}-E\left(T_{k,n}\right))/n$ and prove that it has a double exponential distribution when $n$ tends to infinity. Finally, we show that when $k>2$, our new protocol requires less operations than the traditional $2-$push-pull and $2-$push protocols by using stochastic dominance arguments. All these results generalize the standard case $k=2$.

In 20, we consider the logarithmic Schrödinger equations with damping, also called Schrödinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.

In 21, we consider the isothermal Euler system with damping. We rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $\gamma \to 1$, and we explicitly compute the propagation and the behavior of Gaussian initial data. We then show the weak $L^1$ convergence of the density as well as the asymptotic behavior of its first and second moments.

In 23, we consider the two-dimensional stochastic Gross-Pitaevskii equation, which is a model to describe Bose-Einstein condensation at positive temperature. The equation is a complex Ginzburg-Landau equation with a harmonic potential and an additive space-time white noise. We study the well-posedness of the model using an inhomogeneous Wick renormalization due to the potential, and prove the existence of an invariant measure and of stationary martingale solutions.

In 7, we consider multiscale stochastic spatial gene networks involving chemical reactions and diffusions. The model is Markovian and the transitions are driven by Poisson random clocks. We consider a case where there are two different spatial scales: a microscopic one with fast dynamic and a macroscopic one with slow dynamic. At the microscopic level, the species are abundant and for the large population limit a partial differential equation (PDE) is obtained. On the contrary at the macroscopic level, the species are not abundant and their dynamic remains governed by jump processes. It results that the PDE governing the fast dynamic contains coefficients which randomly change. The global weak limit is an infinite dimensional continuous piecewise deterministic Markov process (PDMP). Also, we prove convergence in the supremum norm.

A biochemical network can be simulated by a set of ordinary differential equations (ODE) under well stirred reactor conditions, for large numbers of molecules, and frequent reactions. This is no longer a robust representation when some molecular species are in small numbers and reactions changing them are infrequent. In this case, discrete stochastic events trigger changes of the smooth deterministic dynamics of the biochemical network. Piecewise-deterministic Markov processes (PDMP) are well adapted for describing such situations. Although PDMP models are now well established in biology, these models remain computationally challenging. Previously we have introduced the push-forward method to compute how the probability measure is spread by the deterministic ODE flow of PDMPs, through the use of analytic expressions of the corresponding semigroup. In this paper 11, we provide a more general simulation algorithm that works also for non-integrable systems. The method can be used for biochemical simulations with applications in fundamental biology, biotechnology and biocomputing.

In 13, we analyse average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value which means that they all know a property of the global system. Using a well chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds of the convergence time. We apply these general results to both the proportion problem and the system size problem.

10.1.1 Inria associate team not involved in an IIL or an international program

• 2018-2022: associated team ANTIPODE with university of Wisconsin-Madison (US), headed by P. Chartier (2018-2021) and N. Crouseilles (2021-2022). 15000 euro per year.

  The project focuses on the development of multi-scale numerical schemes for PDEs with uncertain data. The project is in partnership with university of Wisconsin-Madison and with Shanghai Jiao Tong University. Since the leader left Inria and due to the pandemia, this future of this project is a bit uncertain.

10.1.2 Participation in other International Programs

11.1.1 Scientific events: organisation

• P. Navaro: Organization of "Action Nationale de Formation sur le langage Julia" for CNRS (Fréjus, France), 13-17 September.
• P. Navaro: Formation "Python pour le calcul scientifique" for IMT members (Rennes, France), 13-17 June.
• P. Navaro: Organisation of the mini-symposium "Julia, a langage for mathematicians" SMAI conference (la Grande Motte, France), 21-25 June.
• P. Navaro: Organisation of 6 webinars "Café Calcul".
• E. Faou: Organization of the semester Hamiltonian Methods in Dispersive and Wave Evolution Equations, ICERM, Brown University, USA. Fall 2021.
• E. Faou: Organization of the workshop Numerics, Modeling, and Experiments in Wave Phenomena, with D. Cordoba, E. Dormy, T. Sapsis and L. Vega, ICERM, Brown University, USA.

11.1.2 Journal

11.1.3 Invited talks

• J. Massot: invitation to the working group CMAP Polytechnique (Palaiseau), 8 October.
• J. Massot: invitation: invitation to the Workshop Models and methods for kinetic equations (Bordeaux), 20 November.
• J. Massot: invitation to the seminar (Nantes), 9 November.
• A. Mouzard: invitation to "Colloque des Jeunes Probabilistes et Statisticiens" (Oléron, France), October.
• A. Mouzard: invitation to "Rough path techniques in stochastic analysis and mathematical probability" (Oslo, Norway), November.
• A. Mouzard: invitation to "Simons Collaboration on Wave Turbulence Annual Meeting 2021" (online with NYU and Lyon), December.
• A. Mouzard: invitation to Colloquium Singular and Random PDEs (Nancy, France), December.
• E. Faou: Paris-Moscow seminar "Dynamical systems and PDEs" managed by S. Kuksin and D. Treschev (online), September.
• E. Faou: Seminar at the University of Nantes, September.
• E. Faou: Wave turbulence seminar, New-York University. Online, January.
• E. Faou: lecture series on wave turbulence, New-York University. Online, January.
• N. Crouseilles: invited to the workshop "Numerical methods for kinetic equations" (CIRM Marseille, France), June

11.1.4 Scientific expertise

• A. Crestetto: member of the recruitment committee of maitre de conference, university of Nantes.
• A. Crestetto: member of the recruitment committee of maitre de conference, university of Nice.
• A. Crestetto: member of the CNU, section 26.
• A. Debussche: member of the EAB (External Advisory Board) of the Synergy ERC STUOD.
• A. Debussche: reviewer for ERC, Austrian Science Fund.
• N. Crouseilles: member of the Inria evaluation committee (and then jury for promotions DR, admissibility jury CR-Saclay, admission jury CR).
• E. Faou: Member of the ICIAM Maxwell Prize.
• N. Crouseilles: member of the recruitment committee of professor, university of Bordeaux.

11.1.5 Research administration

• A. Crestetto: member of the scientific council of UFR Sciences et Techniques University of Nantes.
• A. Crestetto: locale coordinator SMAI for mathematical laboratory of university of Nantes.
• A. Debussche: member of the scientific council of Fédération Denis Poisson (Orleéans-Tours) (2012-).
• A. Debussche: vice-head of research at ENS Rennes.
• A. Debussche: member of the CA ENS Paris-Saclay.
• E. Faou: Director of the Henri Lebesgue Center (Excellence laboratory of the program investissement d'avenir).
• E. Faou: Member of the organization committee of the Agence Lebesgue de Mathématiques pour l'innovation.
• N. Crouseilles: member of the IRMAR laboratory council.
• N. Crouseilles: responsible for the university of Rennes 1 of the Federation de recherche pour la fusion confinement magnétique.

11.2.1 Teaching

All the members of team teach. We list below the Master courses only.

• F. Castella, Numerical methods for ODEs and PDEs, 60 hours, Master 1, University of Rennes 1.
• N. Crouseilles, Numerical methods for PDEs, 24 hours, Master 1, ENS Rennes.
• A. Debussche, Distribution and functional analysis, 30 hours, Master 1, ENS Rennes.
• E. Faou, Numerical transport, 24 hours, Master 2, University of Rennes 1.
• P. Navaro (course Python-Fortran, Bordeaux). Cancelled.
• P. Navaro, Python courses, 20 hours, Master 2 Smart Data, ENSAI.
• P. Navaro, Scientific computing tools for big data, 20 hours, Master 2, University of Rennes.
• E. Faou: Numerical transport, 24 hours, Master 2, University of Rennes.

11.2.2 Supervision

• E. Faou and N. Crouseilles: PhD of Y. Le Hénaff.
• E. Faou: PhD of Q. Chauleur (with R. Carles, CNRS IRMAR, Rennes).
• A. Debussche: PhD of G. Barrué (with A. de Bouard, Ecole Polytechnique).
• A. Debussche: PhD of B. Hug (with E. Mémin, Inria Rennes).
• A. Crestetto and N. Crouseilles: PhD of D. Prel.

11.2.3 Juries

• A. Crestetto: member of the PhD defense jury of Mme Rihab Daadaa, 13 December, University Lorraine.
• A. Crestetto: member of the PhD defense jury of J. Massot, University Rennes, December.
• A. Debussche: member of the PhD defense jury of A. Mouzard, 24 June, University Rennes.
• A. Debussche: member of the PhD defense jury of P. M. Boulvard, 5 May, University Paris.
• A. Debussche: member of the PhD defense jury of L. Li, 23 March, University Rennes.
• A. Debussche: member of the HDR defense jury of C.-E. Bréhier, 17 June, ENS Lyon.
• F. Castella: member of the PhD defense jury of L. Trémant, 8 December, University Rennes.
• E. Faou: reviewer for the PhD defense of P. Brun, University of Bordeaux, December.
• E. Faou: president of the HDR defense jury of C. Scheid, University of Nice, December.
• E. Faou: president of the HDR defense jury of V. Duchêne, University of Rennes, December.
• N. Crouseilles: president of the PhD defense jury of V. Pages, Sorbonne University, December.
• N. Crouseilles: member of the PhD defense jury of J. Massot, University Rennes, December.

11.3.1 Interventions

• N. Crouseilles gave a conference for master students for University of Nantes and Ecole Centrale of Nantes to explain research opportunities at Inria, online, January.
• A. Crestetto: participation to the "fête de la science", Nantes.
• J. Massot: talk for the Nantes Society of Astronomy, 26 December.

International journals

• 3 articleJ.Joackim Bernier, N.Nicolas Crouseilles and Y.Yingzhe Li. Exact splitting methods for kinetic and Schrödinger equations.Journal of Scientific Computing8612021
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• 4 articleJ.Joackim Bernier, E.Erwan Faou and B.Benoit Grebert. Rational normal forms and stability of small solutions to nonlinear Schrödinger equations.Annals of PDE622021, 1-53
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• 5 articleQ.Quentin Chauleur. Dynamics of the Schrödinger-Langevin equation.Nonlinearity3442021, 1943-1974
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• 6 articleN.Nicolas Crouseilles, P.-A.Paul-Antoine Hervieux, Y.Yingzhe Li, G.Giovanni Manfredi and Y.Yajuan Sun. Geometric Particle-in-Cell methods for the Vlasov-Maxwell equations with spin effects.Journal of Plasma Physics873May 2021, article n° 825870301
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• 7 articleA.Arnaud Debussche and M. J.Mac Jugal Nguepedja Nankep. A Piecewise Deterministic Limit for a Multiscale Stochastic Spatial Gene Network.Applied Mathematics and Optimization2021
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• 8 articleA.Arnaud Debussche, A.Angelo Rosello and J.Julien Vovelle. Diffusion-approximation for a kinetic spray-like system with markovian forcing.Discrete and Continuous Dynamical Systems - Series S1482021, 2751-2803
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• 9 articleA.Arnaud Debussche and Y.Yoshio Tsutsumi. Quasi-Invariance of Gaussian Measures Transported by the Cubic NLS with Third-Order Dispersion on T.Journal of Functional Analysis28132021, article n° 109032
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• 10 articleE.Erwan Faou, R.Romain Horsin and F.Frédéric Rousset. On Linear Damping around Inhomogeneous Stationary States of the Vlasov-HMF Model.Journal of Dynamics and Differential Equations33 - special issue32021, 1531-1577
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• 11 articleG. C.Guilherme C. P. Innocentini, A.Arran Hodgkinson, F.Fernando Antoneli, A.Arnaud Debussche and O.Ovidiu Radulescu. Push-forward method for piecewise deterministic biochemical simulations.Theoretical Computer Science8932021, 17-40
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• 12 articleY.Yves Mocquard, P.Pierre Navaro and N.Nicolas Crouseilles. HOODESolver.jl: A Julia package for highly oscillatory problems.Journal of Open Source Software6612021
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• 13 articleY.Yves Mocquard, B.Bruno Sericola, F.Frédérique Robin and E.Emmanuelle Anceaume. Stochastic Analysis of Average Based Distributed Algorithms.Journal of Applied Probability582June 2021, 394 - 410
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• 14 articleF.Frédérique Robin, B.Bruno Sericola, E.Emmanuelle Anceaume and Y.Yves Mocquard. Stochastic analysis of rumor spreading with $k$-pull operations.Methodology and Computing in Applied ProbabilityOctober 2021
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Reports & preprints

• 15 miscI.Ibrahim Almuslimani, P.Philippe Chartier, M.Mohammed Lemou and F.Florian Méhats. Uniformly accurate schemes for oscillatory stochastic differential equations.October 2021
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• 16 miscP.Paul Alphonse and J.Joackim Bernier. Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects.December 2021
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• 17 miscF.Fernando Casas, P.Philippe Chartier, A.Alejandro Escorihuela-Tomàs and Y.Yong Zhang. Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations.February 2021
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• 18 miscP.Philippe Chartier, M.Mohammed Lemou, F.Florian Méhats and X.Xiaofei Zhao. Derivative-free high-order uniformly accurate schemes for highly-oscillatory systems.February 2021
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• 19 miscP.Philippe Chartier, M.Mohammed Lemou and L.Léopold Trémant. A uniformly accurate numerical method for a class of dissipative systems.April 2021
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• 20 miscQ.Quentin Chauleur and E.Erwan Faou. Around plane waves solutions of the Schrödinger-Langevin equation.October 2021
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• 21 miscQ.Quentin Chauleur. The isothermal limit for the compressible Euler equations with damping.September 2021
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• 22 miscA.Anaïs Crestetto, N.Nicolas Crouseilles, Y.Yingzhe Li and J.Josselin Massot. Comparison of high-order Eulerian methods for electron hybrid model.November 2021
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• 23 miscA.Anne De Bouard, A.Arnaud Debussche and R.Reika Fukuizumi. Two dimensional Gross-Pitaevskii equation with space-time white noise.February 2021
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• 24 miscE.Erwan Faou and B.Benoît Grébert. Discrete pseudo-differential operators and applications to numerical schemes.September 2021
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