2024-2025

• Ali Assi - Homogenization of a Hamilton–Jacobi Equation with Junctions Modeling Traffic Lights with Phase Shift - mardi 17 juin 2025, 14h

We study a Hamilton–Jacobi equation with junctions modeling traffic flow on an infinite road with traffic lights placed at equal distances, located at each integer point. Each light switches periodically, with a fixed phase shift between consecutive ones. Using homogenization techniques, we derive an effective macroscopic equation and analyze how the phase shift influences large-scale traffic behavior. Our aim is to obtain an explicit formula for the effective Hamiltonian within the optimal control framework.

• Youssef Faouzi - On a class of unbalanced step-reinforced random walks - mardi 17 juin 2025, 14h45

In this presentation, I will introduce a new class of unbalanced step-reinforced random walks, a family of discrete-time stochastic processes with long-range dependence. These processes extend and unify several well-known models, including the elephant random walk of Schütz and Trimper (2004), as well as the positively and negatively step-reinforced random walks, introduced respectively by Simon (1955) and Bertoin (2024). In a step-reinforced random walk, each step is either an independent move (with probability 1−α) or a repetition of a previous step (with probability α), chosen uniformly at random from the walk’s history. In the positive case, the repeated step keeps its direction; in the negative case, the direction is reversed. Our new unbalanced model interpolates between these behaviors and includes asymmetric reinforcement. I will present the main results of our study: a strong law of large numbers and a central limit theorem for this new class, offering a unified theoretical framework for these step reinforced random walk models.

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• Averil Aussedat - Zajíček's theorem - Statement, proof and some consequences of a mathematical jewel - mardi 27 mai 2025, 14h

Zajíček's theorem characterizes the sets of non-differentiability of convex functions.

Its statement is sharp, elegant and tremendously powerful.

We provide the proof in a simple case and give some applications.

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• Léo Poirier - Converging properties of one-dimensionnal monotonous cellular automa  - Mardi 29 avril 2025, 14h

A cellular automaton is monotonous if it preserves the order on configurations inherited from an order on

states. If at first sight, the significantly sparse aspect of that product order lead us to believe that mono-

tonicity would not be that restrictive, the fact that cellular automata are characterised by a local rule

applied shift-invariantly makes the impact of this constraint more remarkable than expected. After intro-

ducing the basic concepts of our work, we first state general results about monotonous cellular automata

on Z with especially the fact that for any shift-ergodic measure of full support, the sequence converges in

Cesàro mean. Then, taking inspiration from previous constructions to show some (/un)computability re-

sults of the literature, and "monotonising" them, we get some upper bound on those "good behaviours".

Finally, we consider some toy example of majority cellular automata, on an asymmetrical neighbourhood,

to try to find a simple example to illustrate a result of ours linking directional convergence of all orbits

with convergence in measure. 

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• Georges Edde - Numerical Modeling of Air Flows in an Underground Cavity Connected to the Surface by a Shaft - Mardi 29 avril 2025, 14h

Underground cavities known as "marnières", typically consisting of a cavity 1 to 3 meters high and connected to the surface by a vertical shaft 20 to 40 meters deep and 1 to 2 meters in diameter, were historically dug by farmers—especially during the 19th century—to extract chalk used as fertilizer.

However, these structures were often not officially declared, due to the high costs associated with such declarations. As a result, their exact locations remain largely unknown. The issue today is that many of these shafts were either never properly sealed or only partially filled with heterogeneous materials. With the effects of climate change, the risk of collapse under buildings, roads, and other infrastructure has increased.

The objective of my thesis is to evaluate whether a novel detection method—using an infrared camera mounted on a drone—can identify these underground cavities. To support this, numerical simulations using FreeFem++ are carried out to analyze the thermal anomalies generated by such structures, based on real point cloud data from marnières in the Normandy region.

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• Emma Thulliez - Amélioration de cartes de concentrations en NO2 dans l'air, par combinaison de mesures de qualités différentes - Mardi 1er avril 2025, 14h
  Urban air quality is a major issue today. Pollutant concentrations, such as NO2, must be monitored to ensure that they do not exceed dangerous thresholds. Two recent techniques help map pollutant concentrations on a small scale. First, deterministic physicochemical models take into account the street network and calculate concentration estimates on a grid, providing a map. On the other hand, the advent of new low-cost technologies allows monitoring organizations to densify measurement networks. However, these devices are less reliable than reference devices and need to be corrected. We propose a new approach to improve maps generated using deterministic models by combining measurements from multiple sensor networks. To do this, we model the bias of deterministic models and estimate it using an MCMC method. This allows us to account for new variables and provide a corrected map in a forecasting context.

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• Nessim Dhaouadi - Surviving in a shifting and size changing environment in presence of selection - Mardi 4 mars 2025, 14h

In this talk, I will present to you a dummy proof talk intoducing our model to study the adaptation of a diffusing population facing two different dynamics.  On  one  hand, the population growth  is  time and space dependent, thus modelling strong heterogeneities of the environment. On the other hand, the environment itself is dynamic. It can both change size and shift over time.  The reasons for such moving range boundaries could be the consequences of flooding, forest fire, etc.

We will first investigate the fixed domain case, in particular estimating the principal eigenvalue of the underlying periodic parabolic operator. This estimate is crucial to construct sub and supersolutions on the moving domain. We then address the problem of extinction vs. persistence, taking into account the interplay between the moving habitat and the selection. Finally if we have time, to explore these dynamics, we construct a stable space-time finite elements scheme using upwind test functions in order to gain some insight on the dynamics of this problem. These will unravel some significant differences with classical results on fixed domains. 

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• Piero Visconti - Optimal Control of Monotone Nonlinear Stochastic PDE - Mardi 28 janvier 2025, 14h
  Many physical systems are modeled by monotone PDE, for example, diffusive processes occuring in porous media are often described with this type of equation. We place our attention on optimal control problems whose dynamics are governed by systems of this type which are subject so stochastic forcing. We provide necessary conditions of optimality for such control systems in the case where the nonlinearity is sub/superlinear in terms of co-state variables which satisfy a backward stochastic evolution equation.

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• Hexa-Atelier du 10 décembre 2024 - Lien vers les abstracts 

  • 14h-14h30 - Nicolas Prevost (LMRS) - Systèmes de particules en interaction : limites hydrodynamiques et grandes déviations dynamiques.
    14h30-15h00 - Guillaume Sallé (LMI) - Algorithmes de Newton stochastiques.
    15h00-15h30 - Maxence Poutrel (LMRS) - Ergodicité de l'automate cellulaire probabiliste du modèle à sphères dures.

  • 16h00-16h30 - Fernanda Urrea (LMI) - Optimality conditions for optimal control theory.
    16h30-17h00 - Silvio Bove (LMRS) - Homogenization of a nonlinear monotone problem in a domain with oscillating boundary.
    17h00-17h30 - Averil Aussedat (LMI) - Monge-Ampère: a newbie's understanding of the story of this equation.

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• Augustin Leclerc - Calculs de modes électromagnétiques guidés dans des guides d'ondes torsadés et ouverts - Lundi 25 novembre 2024, 14h (À l'INSA)

Cette thèse explore la modélisation et le calcul des modes électromagnétiques (EM) dans des guides d'ondes de géométries complexes, notamment dans des câbles électriques torsadés en milieux ouverts. L'objectif est de développer des méthodes numériques pour résoudre les équations de Maxwell afin de mieux comprendre la propagation des champs électromagnétiques dans des configurations réalistes. Cette étude est motivée par les enjeux liés à la réduction du rayonnement électromagnétique et à l'amélioration des performances des câbles en termes de confinement des champs.

     Nous traitons deux principales configurations : les guides d’ondes droits et les guides d'ondes torsadés. Pour les guides droits, des méthodes semi-analytiques sont mises en œuvre, notamment pour les câbles coaxiaux, permettant de tester les modèles numériques. Nous étendons ces méthodes aux guides d'ondes ouverts, où des conditions aux limites absorbantes (CLA) sont introduites pour modéliser un environnement infini en limitant les réflexions parasites. Dans les câbles torsadés, une géométrie hélicoïdale est exploitée afin de reformuler les équations de propagation des ondes EM dans un cadre numérique adapté, permettant d'obtenir des simulations en basse fréquence. Un travail sur la construction de CLA dans ce cadre a également été initié.

     Les résultats obtenus fournissent une meilleure compréhension des phénomènes électromagnétiques à basse fréquence  et ouvrent des perspectives pour la conception de dispositifs plus performants et l'étude des systèmes électromagnétiques dans des environnements hétérogènes réels.

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• Houssem Dahbi - Statistical inferences for ergodic double Heston model based on continuous time observations - Mardi 22 octobre 2024, 14h 

The double Heston model is one of the most popular option pricing models in the financial theory. It is applied to several issues such that risk management and volatility surface calibration. The talk deals with the problem of global parameter estimations in this model. The main stochastic results are about the stationarity and the ergodicity of the double Heston process. The statistical part of the talk is about the maximum likelihood and the conditional least squares estimations based on continuous time observations; then for each estimation method, we study the asymptotic properties of the resulted estimators in the ergodic case. 

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• Boris Gnamah Tchamie - Problème inverse de sources dans un système d'équations couplées - Mardi 1 octobre 2024, 14h
  Cette étude traite un problème de source inverse non linéaire dans un système de deux équations aux dérivées partielles d'advection-dispersion-réaction elliptiques en 2D couplées. Plus spécifiquement, dans un tel système, nous abordons la tâche d'identifier plusieurs sources ponctuelles inconnues définissant le côté droit de sa première équation à partir de certaines observations locales liées à la solution d'état de sa deuxième équation couplée. Pour surmonter ce lien indirect difficile entre observations et inconnues, nous développons des fonctions adjointes appropriées à partir de potentiels scalaires pilotés par les directions du champ de vitesse et les vecteurs propres orthogonaux du tenseur de dispersion symétrique. Cela transforme l'identification des sources inconnues apparaissant en déterminant les racines d'un écart de réciprocité défini à partir d'observations liées aux deux états couplés le long des interfaces séparant chaque région suspectée dans laquelle une source inconnue pourrait se produire. En utilisant les données disponibles liées au deuxième état couplé le long de ces interfaces, nous reconstruisons les données similaires requises associées au premier état couplé. Quelques expériences numériques sur l'identification de sources de pollution dans l'eau des rivières à partir du modèle couplé DBO-OD sont présentées.

2023-2024