2022-04884 - Doctorant F/H Parareal multi-scale methods for highly
oscillatory Vlasov equations (Campagne Cordi S)
Contract type : Fixed-term contract
Level of qualifications required : Graduate degree or equivalent
Fonction : PhD Position
Context : The PhD will take place at the Inria Paris center in the team
ALPINES. We plan to work in collaboration with Laura Grigori (Inria, ALPINES
team), an expert in applied mathematics and computer science.
ALPINES is a joint research group between Inria (Paris) and Jacques-Louis
Lions Laboratory (Sorbonne University and CNRS), which focuses on
scientific computing. The research work consists in the development of novel
parallel numerical algorithms and tools suitable for state-of-the-art
mathematical models used in complex and large scale scientific applications,
and in particular numerical simulations.
The general scientific context : The modelling and the numerical
simulation of plasma (a gas of charged particles) is of great importance
from a physical and mathematical point of view. In this context, the Vlasov-
Maxwell equations provide a kinetic modelling approach of the dynamics of
charged particles under the influence of an electro-magnetic field.
Difficulties in solving numerically such equations come from the existence of
several scales in space and time of the solutions.
Proposed work : The specific problem to be treated is the Vlasov-Poisson
system with an additional strong external magnetic field, which has several
applications in plasma physics, for example the confinement of particles. The
multi-scale behaviour due to high frequency oscillations in time imposes tiny
time steps to the discretizations and therefore, the computational cost of
long time simulations is prohibitive. A solution for avoiding this problem is
to use reduced models, based on averaging, whose solutions are free of
oscillations. An example is the two-scale limit model (4). Nevertheless,
in some applications this model does not cover the general situation where the
rapid motion around the magnetic field line is not periodic. In addition, the
model only incorporates a two scale behaviour, whereas realistic phenomena may
contain more than two scales.
Task 1 : A first direction of research will be the development of new
reduced models for Vlasov-Poisson problems. More precisely, our first aim is
to improve existing results on homogenization (see 4) to a broader
framework, which is free of periodicity and can deal with three time scales or
more. Such general results exist in the literature (3) only for
diffusion equations and do not seem to be derived for Vlasov-like equations.
Secondly, we plan to develop first-order homogenized models to increase
accuracy, following the strategy developed in 4 for standard models.
Task 2 : The objective is to efficiently implement the previous models in
order to perform simulations for several applications in plasma physics. We
therefore aim to develop, analyze and implement a parareal method (see
2) for solving the previous equations. This algorithm is an efficient
method which performs real time simulations by means of parallel-in-time
integration. We will follow a strategy where a different (reduced) model
than the original one is used for the coarse solver. This method was
successfully applied in 1 for solving highly oscillatory differential
equations with plasma physics applications.
1 L. Grigori, S.A. Hirstoaga, V. T. Nguyen, J. Salomon: “Reduced model-
based parareal simulations of oscillatory singularly perturbed ordinary
differential equations”, Journal of Computational Physics, vol. 436, 110282,
2 J.-L. Lions, Y. Maday, and G. Turinici: “A parareal in time
discretization of PDE's”, Comptes Rendus de l'Académie des Sciences - Series I
- Mathematics, 332, 661–668, 2001.
3 G. Allaire, M. Briane: “Multiscale convergence and reiterated
homogenisation”, Proceedings of the Royal Society of Edinburgh Section A:
Mathematics 126.2, 297-342, 1996.
4 E. Frénod, P.-A. Raviart, E. Sonnendrücker: “Two-scale expansion of a
singularly perturbed convection equation”, J. Math. Pures Appl., vol. 80,
Skills : The profile is in applied mathematics/scientific computing.
Knowledge in mathematical modelling and numerical methods for partial
Programming in Python, Fortran or C/C++.
Transports publics remboursés partiellement
Congés: 7 semaines de congés annuels + 10 jours de RTT (base temps
plein) + possibilité d'autorisations d'absence exceptionnelle (ex :
enfants malades, déménagement)
Possibilité de télétravail et aménagement du temps de travail
Équipements professionnels à disposition (visioconférence, prêts de
matériels informatiques, etc.)
Prestations sociales, culturelles et sportives (Association de gestion
des œuvres sociales d'Inria)
Theme/Domain : Distributed and High Performance Computing
Scientific computing (BAP E)
Town/city : Paris
Inria Center : CRI de Paris
Starting date : 2022-09-01
Duration of contract : 3 years
Deadline to apply : 2022-05-24
Inria Team : ALPINES
PhD Supervisor :
Hirstoaga Sever / Sever.Hirstoaga@inria.fr
Inria is the French national research institute dedicated to digital science
and technology. It employs 2,600 people. Its 200 agile project teams,
generally run jointly with academic partners, include more than 3,500
scientists and engineers working to meet the challenges of digital technology,
often at the interface with other disciplines. The Institute also employs
numerous talents in over forty different professions. 900 research support
staff contribute to the preparation and development of scientific and
entrepreneurial projects that have a worldwide impact.
Instruction to apply
Defence Security :
This position is likely to be situated in a restricted area (ZRR), as
defined in Decree No. 2011-1425 relating to the protection of national
scientific and technical potential (PPST).Authorisation to enter an area is
granted by the director of the unit, following a favourable Ministerial
decision, as defined in the decree of 3 July 2012 relating to the PPST. An
unfavourable Ministerial decision in respect of a position situated in a ZRR
would result in the cancellation of the appointment.
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