Doctorant F/H Parareal multi-scale methods for highly oscillatory Vlasov equations (Campagne Cordi S)



May 24, 2022


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.

Main activities

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.

References :

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, 2021.

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, 815–843, 2001.


Skills : The profile is in applied mathematics/scientific computing.

Knowledge in mathematical modelling and numerical methods for partial differential equations.

Programming in Python, Fortran or C/C++.

Benefits package
  • Restauration subventionnée
  • 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)
  • General Information
  • 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
  • Contacts
  • Inria Team : ALPINES
  • PhD Supervisor : Hirstoaga Sever /
  • About Inria

    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.

    Recruitment Policy : As part of its diversity policy, all Inria positions are accessible to people with disabilities.

    Warning : you must enter your e-mail address in order to save your application to Inria. Applications must be submitted online on the Inria website. Processing of applications sent from other channels is not guaranteed.

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