arXiv subjects

Thomas Rey

Publications and source records attributed to Thomas Rey.

At least 19 recordsLinked to original sources

arXiv2026

Swarm sign language: motion-based communication between drones

In stealth-constrained swarm robotics, visual communication provides a critical alternative to active radio transmissions, which might be jammed. This research investigates motion-based communication for non-active information exchange, utilizing modular, dynamically feasible planar trajectories as visual cues. On the receiver drone end, a pose estimator tracks the transmitting drone's pose, feeding it into our custom 3DTrajDecoder. The decoder is designed to classify and segment the spatiotemporal sequence while simultaneously regressing its size and normal vector. To robustly train the decoder on both communicative and non-communicative trajectories, we developed a configurable online procedural generation pipeline. We validate our system through real-world testing and simulation to define its operating domain, supported by an extensive ablation study detailing our architectural choices and system limitations.

cs.RO↗

arXiv2026

How opinions shape epidemics: a graphon-based kinetic approach

Understanding the mutual influence between social behavior and physical health is crucial for designing effective epidemic mitigation strategies. Individual interactions drive the evolution of opinions, which in turn shape how infectious diseases are perceived and consequently how they spread within a population, for instance through the adoption or rejection of preventive measures. At the same time, the distribution and dynamics of physical contacts play a fundamental role in determining transmission patterns. To this end, we develop a mathematical framework to analyze the coupled dynamics of opinion formation, disease transmission, and physical contacts by employing graphon-based networks, which capture heterogeneous and large-scale connectivity patterns typical of realistic social structures. The epidemic compartmental model further incorporates a kinetic description of microscopic level physical contacts, allowing for a consistent multiscale representation of interaction patterns. Starting from a microscopic description governed by interpersonal compromise and intrinsic self-thinking processes, we derive a kinetic compartmental epidemic model on graphons via a mean-field limit. This formulation allows us to investigate the joint evolution of the disease state and the opinion distribution, with a particular focus on the role of social networks and physical contacts. Numerical experiments demonstrate that the graphon-kinetic approach provides a comprehensive representation of the coupled opinion-epidemic dynamics, revealing new possibilities for controlling disease spread by shaping population opinion patterns.

physics.soc-ph↗

arXiv2025

High-Order Asymptotic-Preserving IMEX schemes for an ES-BGK model for Gas Mixtures

In this work we construct a high-order Asymptotic-Preserving (AP) Implicit-Explicit (IMEX) scheme for the ES-BGK model for gas mixtures introduced in [Brull, Commun. Math. Sci., 2015]. The time discretization is based on the IMEX strategy proposed in [Filbet, Jin, J. Sci. Comput., 2011] for the single-species BGK model and is here extended to the multi-species ES-BGK setting. The resulting method is fully explicit, uniformly stable with respect to the Knudsen number and, in the fluid regime, it reduces to a consistent and high-order accurate solver for the limiting macroscopic equations of the mixture. The IMEX structure removes the stiffness associated with the relaxation term so that the time step is constrained only by a hyperbolic CFL condition. The full solver couples a high-order space and velocity discretization that includes third-order time integration, a CWENO3 finite-volume reconstruction in space, exact conservation of macroscopic moments in the discrete velocity space, and a multithreaded implementation. The proposed approach can handle an arbitrary number of species. Its accuracy and robustness are demonstrated on a set of multidimensional kinetic tests for gas mixtures, where the AP property and the correct asymptotics are numerically verified across different regimes.

math.NA↗

arXiv2025

Discrete $H$-theorem for a finite volume discretization of a nonlinear kinetic system: application to hypocoercivity

In this article, we study the long-time behavior of a finite-volume discretization for a nonlinear kinetic reaction model involving two interacting species. Building upon the seminal work of [Favre, Pirner, Schmeiser, ARMA, 2023], we extend the discrete exponential convergence to equilibrium result established in [Bessemoulin-Chatard, Laidin, Rey, IMAJNA, 2025], which was obtained in a perturbative framework using weighted $L^2$ estimates. The analysis applies to a broader class of exponentially decaying initial data, without requiring proximity to equilibrium, by exploiting the properties of the Boltzmann entropy. The proof relies on the propagation of the initial $L^\infty$ bounds, derived from monotonicity properties of the scheme, allowing controlled linearizations within the nonlinear entropy estimates. Moreover, we show that the time-discrete dissipation inherent to the numerical scheme plays a crucial stabilizing role, providing control over the nonlinear terms.

math.NA↗

arXiv2025

Formal derivation of an isentropic two-phase flow model from the multi-species Boltzmann equation

Starting from the multi-species Boltzmann equation for a gas mixture, we propose the formal derivation of the isentropic two-phase flow model introduced in [Romenski, E., and Toro, E. F., Comput. Fluid Dyn. J., 13 (2004)]. We examine the asymptotic limit as the Knudsen numbers approach zero, in a regime characterized by resonant intra-species collisions, where interactions between particles of the same species dominate. This specific regime leads to a multi-velocity and multi-pressure hydrodynamic model, enabling the explicit computation of the coefficients for the two-phase macroscopic model. Our derivation also accounts for the inclusion of the evolution of the volume fraction, which is a key variable in many macroscopic multiphase models

math.AP↗

arXiv2025

Hierarchical dynamic domain decomposition for the multiscale Boltzmann equation

In this work, we present a hierarchical domain decomposition method for the multi-scale Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga in \cite{lev-mor-nad-1998}. This criterion is used to dynamically partition the two-dimensional spatial domain into three regimes: the Euler regime, an intermediate kinetic regime governed by the ES-BGK model, and the full Boltzmann regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, the ES-BGK model in moderately non-equilibrium regions where a fluid description is insufficient but full kinetic resolution is not yet necessary, and the full Boltzmann solver where strong non-equilibrium effects dominate, such as near shocks and boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver and the ES-BGK models are considerably cheaper than the full kinetic Boltzmann model. To ensure accurate and efficient coupling between regimes, we employ asymptotic-preserving (AP) numerical schemes and fast spectral solvers for evaluating the Boltzmann collision operator. Among the main novelties of this work are the use of a full 2D spatial and 3D velocity decomposition, the integration of three distinct physical regimes within a unified solver framework, and a parallelized implementation exploiting CPU multithreading. This combination enables robust and scalable simulation of multiscale kinetic flows with complex geometries.

math.NA↗

arXiv2025

A Parareal in time numerical method for the collisional Vlasov equation in the hyperbolic scaling

We present the design of a multiscale parareal method for kinetic equations in the fluid dynamic regime. The goal is to reduce the cost of a fully kinetic simulation using a parallel in time procedure. Using the multiscale property of kinetic models, the cheap, coarse propagator consists in a fluid solver and the fine (expensive) propagation is achieved through a kinetic solver for a collisional Vlasov equation. To validate our approach, we present simulations in the 1D in space, 3D in velocity settings over a wide range of initial data and kinetic regimes, showcasing the accuracy, efficiency, and the speedup capabilities of our method.

math.NA↗

arXiv2024

The Boltzmann equation for a multi-species inelastic mixture

A granular gas is a collection of macroscopic particles that interact through energy-dissipating collisions, also known as inelastic collisions. This inelasticity is characterized by a collision mechanics in which mass and momentum are conserved and kinetic energy is dissipated. Such a system can be described by a kinetic equation of the Boltzmann type. Nevertheless, due to the macroscopic aspect of the particles, any realistic description of a granular gas should be written as a mixture model composed of M different species, each with its own mass. We propose in this work such a granular multi-species model and analyse it, providing Povzner-type inequalities, and a Cauchy theory in general Orlicz spaces. We also analyse its large time behavior, showing that it exhibits a mixture analogue of the seminal Haff's Law.

math.AP↗

arXiv2024

Discrete hypocoercivity for a nonlinear kinetic reaction model

In this article, we propose a finite volume discretization of a one dimensional nonlinear reaction kinetic model proposed in [Neumann, Schmeiser, Kint. Rel. Mod. 2016], which describes a 2-species recombination-generation process. Specifically, we establish the long-time convergence of approximate solutions towards equilibrium, at exponential rate. The study is based on an adaptation for a discretization of the linearized problem of the $L^2$ hypocoercivity method introduced in [Dolbeault, Mouhot, Schmeiser, 2015]. From this, we can deduce a local result for the discrete nonlinear problem. As in the continuous framework, this result requires the establishment of a maximum principle, which necessitates the use of monotone numerical fluxes.

math.NA↗

arXiv2023

Hybrid Kinetic/Fluid numerical method for the Vlasov-Poisson-BGK equation in the diffusive scaling

This short note presents an extension of the hybrid, model-adaptation method introduced in [T.~Laidin, \textit{arXiv 2202.03696}, 2022] for linear collisional kinetic equations in a diffusive scaling to the nonlinear mean-field Vlasov-Poisson-BGK model. The aim of the approach is to reduce the computational cost by taking advantage of the lower dimensionality of the asymptotic model while reducing the overall error. It relies on two criteria motivated by a perturbative approach to obtain a dynamic domain adaptation. The performance of the method and the conservation of mass are illustrated through numerical examples.

math.NA↗

arXiv2022

Continuous limits of large plant-pollinator random networks and some applications

We study a stochastic individual-based model of interacting plant and pollinator species through a bipartite graph: each species is a node of the graph, an edge representing interactions between a pair of species. The dynamics of the system depends on the between- and within-species interactions: pollination by insects increases plant reproduction rate but has a cost which can increase plant death rate, depending on the densities of pollinators. Pollinators reproduction is increased by the resources harvested on plants. Each species is characterized by a trait corresponding to its degree of generalism. This trait determines the structure of the interaction graph and the quantities of resources exchanged between species. Our model includes in particular nested or modular networks. Deterministic approximations of the stochastic measure-valued process by systems of ordinary differential equations or integro-differential equations are established and studied, when the population is large or when the graph is dense and can be replaced with a graphon. The long-time behaviors of these limits are studied and central limit theorems are established to quantify the difference between the discrete stochastic individual-based model and the deterministic approximations. Finally, studying the continuous limits of the interaction network and the resulting PDEs, we show that nested plant-pollinator communities are expected to collapse towards a coexistence between a single pair of species of plants and pollinators.

math.PR↗

arXiv2021

On Deterministic Numerical Methods for the Quantum Boltzmann-Nordheim Equation. I. Spectrally Accurate Approximations, Bose-Einstein Condensation, Fermi-Dirac Saturation

Spectral methods, thanks to their high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the collisional kinetic equations of Boltzmann type, such as the Boltzmann-Nordheim equation. This equation, modeled on the seminal Boltzmann equation, describes using a statistical physics formalism the time evolution of a gas composed of bosons or fermions. Using the spectral-Galerkin algorithm introduced in [F. Filbet, J. Hu, and S. Jin, ESAIM: Math. Model. Numer. Anal., 2011], together with some novel parallelization techniques, we investigate some of the conjectured properties of the large time behavior of the solutions to this equation. In particular, we are able to observe numerically both Bose-Einstein condensation and Fermi-Dirac relaxation.

math.NA↗

arXiv2021

Projective and Telescopic Projective Integration for Non-Linear Kinetic Mixtures

We propose fully explicit projective integration and telescopic projective integration schemes for the multispecies Boltzmann and Bhatnagar-Gross-Krook (BGK) equations. The methods employ a sequence of small forward-Euler steps, intercalated with large extrapolation steps. The telescopic approach repeats said extrapolations as the basis for an even larger step. This hierarchy renders the computational complexity of the method essentially independent of the stiffness of the problem, which permits the efficient solution of equations in the hyperbolic scaling with very small Knudsen numbers. We validate the schemes on a range of scenarios, demonstrating its prowess in dealing with extreme mass ratios, fluid instabilities, and other complex phenomena.

math.NA↗

arXiv2021

Moment preserving Fourier-Galerkin spectral methods and application to the Boltzmann equation

Spectral methods, thanks to the high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier-Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term, and prove spectral consistency and stability of the resulting Fourier-Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings.

math.NA↗

arXiv2020

On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation

Spectral methods, thanks to the high accuracy and the possibility to use fast algorithms, represent an effective way to approximate the Boltzmann collision operator. On the other hand, the loss of some local invariants leads to the wrong long time behavior. A way to overcome this drawback, without sacrificing spectral accuracy, has been proposed recently with the construction of equilibrium preserving spectral methods. Despite the ability to capture the steady state with arbitrary accuracy, the theoretical properties of the method have never been studied in details. In this paper, using the perturbation argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation, we prove stability, convergence and spectrally accurate long time behavior of the equilibrium preserving approach.

math.NA↗

arXiv2020

Recent development in kinetic theory of granular materials: analysis and numerical methods

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. We will discuss both theoretical and numerical developments. We will finally showcase some important open problems and conjectures by means of numerical experiments based on spectral methods.

math.AP↗

arXiv2019

Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

We present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutire, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are 10 treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.

math.AP↗

arXiv2019

Convergence of knowledge in a cultural evolution model with population structure, random social learning and credibility biases

Understanding how knowledge is created and propagates within groups is crucial to explain how human populations have evolved through time. Anthropologists have relied on different theoretical models to address this question. In this work, we introduce a mathematically oriented model that shares properties with individual based approaches, inhomogeneous Markov chains and learning algorithms, such as those introduced in [F. Cucker, S. Smale, Bull. Amer. Math. Soc., 39 (1), 2002] and [F. Cucker, S. Smale and D.~X Zhou, Found. Comput. Math., 2004]. After deriving the model, we study some of its mathematical properties, and establish theoretical and quantitative results in a simplified case. Finally, we run numerical simulations to illustrate some properties of the model.

math.DS↗