Transport

in Physics, Biology and Urban traffic

CIRM, Marseille, France
July 18 - August 26, 2022

Research projects


List of the projects


Modèles cinétiques pour les plasmas de bords
Supervisors: Mehdi Badsi, Anaïs Crestetto, Nicolas Crouseilles, Michel Mehrenberger
Team: Valentin Ayot, Averil Prost, Christian Tayou Fotso

L'interaction d'un plasma avec un bord matériel est un problème très étudié en physique. Lorsqu'un plasma interagit avec une paroi, une couche mince (appelée "gaine de Debye") se forme près du bord due au déplacement rapide des électrons (plus léger que les ions) qui sont absorbés par la paroi, créant un déséquilibre de charge près du bord et ainsi la formation d'une couche limite pour le potentiel électrique solution de l'équation de Poisson. La modélisation de ces phénomènes requiert la prise en compte d'équations dites cinétiques pour chaque espèce (électrons et ions). Dans ce cadre, une théorie des états d'équilibre a été établie récemment dans le cas collisionnel [1] et une exploration numérique du cas dynamique a été effectuée dans [2]. L'objectif de ce projet est d'étendre ces travaux récents en prenant en compte les effets collisionnels ou de type source volumique au modèle double espèce (ions et électrons) de Vlasov-Poisson, ce qui s'avère nécessaire pour modéliser les cas pertinents d'un point de vue de la physique.
Le coeur du projet sera de construire et d'analyser des méthodes numériques adaptées pour étudier la dynamique de la formation de gaine en prenant en compte les effets collisionnels.

[1] M. Badsi, M. Campos-Pinto, B. Després, A minimization formulation of a bi-kinetic sheath, 2016, Kinetic Related Models.
[2] M. Badsi, M. Merhenberger, L. Navoret, Numerical stability of plasma sheath, 2018, ESAIM Proc.

Modelling, analysis, and simulation of traffic jam in colonies of Myxococcus xanthus
Supervisors: Vincent Calvez (ICJ, Lyon), Sylvain Faure (LMO, Orsay), Loïc Gouarin (CMAP, Palaiseau), Aline Lefebvre (CMAP, Palaiseau), Tâm Mignot (LCB, Marseille)
Team: Hélène Bloch, Michèle Romanos, Jean-Baptiste Saulnier, Benoît Gaudeul

Myxococcus xanthus is a social bacterium that experiences a fascinating life cycle, with a transi- tion from uni-cellular to multi-cellular stages. One step of this developmental transition has raised great modelling efforts, the so-called rippling phase, when the population of bacteria synchronizes in accordion waves with high activity, but very limited net motion of each individual (see, e.g. [2, 3] and more references therein). This fascinating pattern is results from repeated reversions of individual bacteria embedded in a dense population. The reversion events are known to be finely tuned by biochemical processes at the intra-cellular level [1]. However, the onset of collective motion is still the subject of current investigation.
The goal of the project is to revisit these previous modelling studies based on high-resolution data of collective motion of bacteria [4]. More precisely, we aim to investigate the relationship between modulation of cell reversion and local congestion among the population, following [5] with active particles.

[1] M. Guzzo et al, A gated relaxation oscillator mediated by FrzX controls morphogenetic movements in Myxococcus xanthus. Nature Microbiology, 2018.
[2] O. Igoshin et al, Waves and aggregation patterns in myxobacteria. PNAS, 2004.
[3] A. Manhart, Counter-propagating wave patterns in a swarm model with memory. J. Math. Biol., 2019.
[4] S. Panigrahi et al, Misic, a general deep learning-based method for the high-throughput cell segmentation of complex bacterial communities. eLife, 2021.
[5] SA. Seguin et al, Clustering and flow around a sphere moving into a grain cloud. Eur. Phys. J. E, 2016.

Collective motion of biofilms
Supervisors: Vincent Calvez (ICJ, Lyon), Florence Hubert (I2M, Marseille), Julien Olivier (I2M, Marseille) and Magali Tournus (I2M, Marseille)
Team: Maxime Estavoyer, Adil El Abdouni, Ignacio Madrid

Collective organisms in nature are capable of a wide range of motility patterns to find food, reproduce or avoid predation. Remarkably, multicellular systems can also display collective move- ment to achieve similar goals. Myxococcus xanthus, a soil predatory bacterium, assembles multi- cellular biofilms to collectively predate on other microorganisms. During the hunting phase, the bacterial biofilm is composed with isolated bacteria and largers clusters of various sizes. In her PhD thesis, S. Rombouts conducted experiments and analysis about various collective be- haviours occurring during predation. One of the main outcome of her study is the evidence of synergistic effects when various motility apparatus are expressed within the cell population, with the consequence of increasing the rate of predation.
The goal of the project is to explore the fascinating biological issues raised in [4] from a mathematical point of view.

Hybrid semi-Lagrangian and skew-symmetric discretization of the gyrokinetic equations
Supervisors : Martin Campos Pinto, Michel Mehrenberger, Eric Sonnendrücker.
Team: Dominik Bell, Xue Hong, Davor Kumozec, Frederik Schnack

Méthode de tenseurs pour la résolution de modèles de jeux à champ moyen.
Supervisors : Virginie Erlacher, Luca Nenna.
Team: Laila Baroukh, Damien Prel, Léopold Trémant

Les modèles de jeux à champs moyen sont utilisés dans de nombreux contextes, en particulier pour la simulation de mouvement de foules. Dans ce type de modèle, l'objectif est de déterminer m(t,x) la densité d'agents en un point de l'espace x à un instant de temps t. Cependant, la résolution numérique de ce type de modèles est extrêmement coûteuse, voire impossible, d'un point de vue numérique car elle nécessite de résoudre des problèmes globaux espace-temps. L'objectif de ce projet est d'investiguer le potentiel des méthodes dites de faible rang (ou méthodes de tenseurs) pour contourner ce problème dans le cas où le nombre de degrés de liberté du système considéré est trop grand pour que des méthodes numériques classiques soient envisageables. Les approches de faible rang ont été utilisées avec succès pour la résolution de nombreux problèmes en grande dimension, mais leur applicabilité pour la résolution de modèles de jeux à champs moyens n'a à ce jour pas été explorée. Ces dernières se basent sur le principe de la séparation de variables et consistent à rechercher une approximation de la fonction m(t,x) sous la forme \sum_{k=1}^r r_k(t) s_k(x) où le nombre de termes r (appelé rang) est aussi petit que possible, et où les fonctions r_k et s_k ne dépendent que de la variable temporelle ou de la variable spatiale respectivement.

Réduction ordre en vitesse et apprentissage pour les équations de Vlasov-Maxwell
Supervisors : Emmanuel Franck, Laurent Navoret
Team: Ibtissem Lannabi, Youssouf Nasseri, Giuseppe Parasiliti Rantone, Guillaume Steimer

Etude de l’apport de schémas volumes finis composites d’ordre élevé prenant en compte les termes sources.
Supervisors: Emmanuel Franck, Philippe Hoch, Clément Lasuen.
Team: Mohamed Boujoudar, Yoann Le Hénaff, Paul Paragot

Schémas cinétiques : aspects théoriques et optimisations informatiques
Supervisors: Philippe Helluy
Team: Clément Flint, Kévin Guillon, Romane Hélie

Modeling Compartmentalization within Intracellular Signaling Pathway
Supervisors: Erwan Hingant, Béatrie Laroche, Juan Calvo Yagüe, Romain Yvinec
Team: Claire Alamichel, Nathan Quiblier, Saoussen Latrach

Recent biological imaging have shown that membrane receptors involved in intracellular signaling pathways are capable of inducing cascades of biochemical reactions from extremely dynamic pools of intracellular compartments. The subcellular trafficking of receptors generates spatio-temporal heterogeneous cellular compartments with a critical role on physiological functions, and profound consequences on the search for new therapeutic strategies. In this project, we aim to develop new modeling formalism, combining biochemical reaction networks, coagulation-fragmentation and advection-diffusion dynamics to fully represent the complexity of signaling pathways. We will design efficient numerical schemes to explore the role of spatio-temporal heterogeneity of intracellular compartments in the response of cells to extracellular signals.

Numerical methods for mean field optimal transport
Supervisors : Mathieu Lauriere
Team: Sebastian Baudelet, Brieuc Fresnais, Amal Machtalay

Mean field games correspond to the limit of finite population games when the number of players grows to infinity. In the setting introduced by Lasry and Lions [1], and Caines, Huang and Malhammé [2] around 2006, one generally looks for a Nash equilibrium. Alternatively, one can look for a social optimum, which corresponds to the situation where the players cooperate in order to optimize a joint objective [3,4]. When the objective is formulated with a terminal constraint on the population distribution instead of a terminal cost, the problem can be viewed as a generalization of standard optimal transport, in which the dynamics and the running cost involve mean field interactions. An example is the so-called mean field Shrödinger problem [5]. The goal of this project is to develop efficient numerical methods for such problems, based either on traditional tools such as finite difference schemes, or on machine learning tools such as deep neural networks.

[1] Lasry, J. M., & Lions, P. L. (2007). Mean field games. Japanese journal of mathematics, 2(1), 229-260.
[2] Huang, M., Malhamé, R. P., & Caines, P. E. (2006). Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communications in Information & Systems, 6(3), 221-252.
[3] Bensoussan, A., Frehse, J., & Yam, P. (2013). Mean field games and mean field type control theory (Vol. 101). New York: Springer.
[4] Carmona, R., Delarue, F., & Lachapelle, A. (2013). Control of McKean–Vlasov dynamics versus mean field games. Mathematics and Financial Economics, 7(2), 131-166.
[5] Backhoff, J., Conforti, G., Gentil, I., & Léonard, C. (2020). The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities. Probability Theory and Related Fields, 178(1), 475-530.

Advanced numerical schemes for the simulation of complex ecosystems
Supervisors: Bastien Polizzi, Sébastian Minjeaud, Olivier Bernard, Thierry Goudon
Team: Mickaël Bestard, Léo Meyer, Florent Noisette

The goal of this project is to implement new schemes with unknowns stored on staggered grids, for the simulation of mixture flows arising in the modeling of biofilms formation. These models involve PDE coupled to constraints. In order to handle these constraints with accuracy, the construction of the numerical scheme should be consistent with the various equivalent formulations of the constraint. The staggered framework turns out to be well adapted to this purpose. This will allow us to test the impact of several constitutive laws on the behavior of the solutions, and to discuss their biological interpretation.

Planning of the presentations


Wednesday 24 August
14h: Advanced numerical schemes for the simulation of complex ecosystems, Mickaël Bestard, Léo Meyer, Florent Noisette
14h40: Modelling, analysis, and simulation of traffic jam in colonies of Myxococcus xanthus, Hélène Bloch, Michèle Romanos, Jean-Baptiste Saulnier, Benoît Gaudeul
15h20: break
15h40: Kinetic schemes: theoretical aspects and optimization, Clément Flint, Kévin Guillon, Romane Hélie
16h20: Modeling Compartmentalization within Intracellular Signaling Pathway, Claire Alamichel, Nathan Quiblier, Saoussen Latrach

Thursday 25 August
9h: Collective motion of biofilms, Maxime Estavoyer, Adil El Abdouni, Ignacio Madrid
9h40: A reduced model for the Vlasov-Poisson Fokker-Planck model, Ibtissem Lannabi, Youssouf Nasseri, Giuseppe Parasiliti Rantone, Guillaume Steimer
10h20: break
10h40: Hybrid semi-Lagrangian and skew-symmetric discretization of the gyrokinetic equations, Dominik Bell, Xue Hong, Davor Kumozec, Frederik Schnack
11h20: Tensor method for solving mean field game model, Laila Baroukh, Damien Prel, Léopold Trémant

14h: Numerical methods for mean field optimal transport, Sebastian Baudelet, Brieuc Fresnais, Amal Machtalay
14h40: Study of the contribution of high order composite finite volume schemes taking into account the source terms, Mohamed Boujoudar, Yoann Le Hénaff, Paul Paragot
15h20: Kinetic methods for edge plasma, Valentin Ayot, Averil Prost, Christian Tayou Fotso