Publications

2025

• Stochastic Dynamics of Incoherent Branched Flows
  
  • Garnier Josselin
  • Picozzi Antonio
  • Torres Theo
  Physical Review Letters, American Physical Society, 2025, 134 (22), pp.223803. (10.1103/PhysRevLett.134.223803)
  DOI : 10.1103/PhysRevLett.134.223803
• Estimation of extreme risk measures with neural networks
  
  • Girard Stéphane
  • Allouche Michaël
  • Gobet Emmanuel
  , 2025, pp.1-6. We propose new parameterizations for neural networks in order to estimate extreme risk measures, such as conditional tail moments, in heavy-tailed settings. The proposed neural network estimator is able to extrapolate in the distribution tails thanks to an extension of the usual extreme-value second-order condition to an arbitrary order. The convergence rate of the uniform error between the log-conditional tail moment and its neural network approximation is established. The finite sample performance of the neural network estimator is compared to bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. Finally, the neural network estimator is tested on real data to investigate the behavior of cryptocurrency extreme loss returns.
• Automatically generated cardiovascular digital twin in critical care: a proof of concept study
  
  • Kimmig François
  • Le Gall Arthur
  • Windsor Camille
  • Vallée Fabrice
  • Chapelle Dominique
  • Moireau Philippe
  , 2025. This proof of concept study demonstrates the capabilities of a virtually automatically generated digital twin framework for enhancing hemodynamic monitoring in critical care. By combining a deterministic cardiovascular model with patient-specific data through data assimilation techniques, the digital twin can act as a data denoiser, reconstruct physiological waveforms that are typically unavailable in critical care settings and generate clinically relevant biomarkers. Validation was performed using real data from patients under general anesthesia. The proposed framework efficient calibration and ability to follow the patient's state over time supports the possibility of real-time bedside applications. (10.1007/978-3-031-94562-5_35)
  DOI : 10.1007/978-3-031-94562-5_35
• Design of experiments based on a low fidelity model for seismic fragility curves estimation
  
  • Van Biesbroeck Antoine
  • Gauchy Clément
  • Feau Cyril
  • Garnier Josselin
  ESAIM: Proceedings and Surveys, EDP Sciences, 2025, 79, pp.96-109. Seismic fragility curves are key quantities of interest for Seismic Probabilistic Risk Assessment studies. They express the probability of failure of a mechanical structure of interest conditional to a scalar value derived from the ground motion signal coined Intensity Measure. In the literature, Bayesian approaches have emerged to enable their estimation within the difficult context of limited data availability. Yet, the log-normal modeling over which most of them are based requires the use of computationally expensive Markov chain Monte Carlo methods for providing Bayesian estimators. In this work, we propose an efficient modeling for the estimation of fragility curves in the Bayesian context, based on a low fidelity model of the structure's response to the ground motion signal and an objective prior. The analytical expression of our modeling allows fast generation of estimates. Also, the representative bias arisen by the modeling choice is handled with a judicious design of experiments methodology. Finally, our method is evaluated on a real case study, and the results highlight its efficiency and its ability to robustly overcome any bias when coupled with the design of experiments we propose. (10.1051/proc/202579096)
  DOI : 10.1051/proc/202579096
• A high-order matrix-free adaptive solver for the shallow water equations with irregular bathymetry
  
  • Arpaia Luca
  • Orlando Giuseppe
  • Ferrarin Christian
  • Bonaventura Luca
  , 2025. We present the first step in the development of an Adaptive Mesh Refinement (AMR) solver for coastal engineering applications, based on a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement both static and dynamic AMR approaches. The proposed method is automatically well-balanced, allows the use of realistic bathymetry data without any regularity assumption, and includes a consistent conservative discretization for transported chemical species. Numerical experiments on idealized benchmarks validate the proposed approach, while results obtained on realistic bathymetries and complex domains show its potential for accurate and efficient adaptive simulations of coastal flows.
• Dynamics of screened particles towards equi-spaced ground states
  
  • de Luca Lucia
  • Goldman Michael
  • Ponsiglione Marcello
  , 2025. This paper deals with the dynamics - driven by the gradient flow of negative fractional seminorms - of empirical measures towards equi-spaced ground states. Specifically, we consider periodic empirical measures $\mu$ on the real line that are screened by the Lebesgue measure, i.e., with $\mu- d x$ having zero average. To each of these measures $\mu$ we associate a (periodic) function $u$ satisfying $u'= d x - \mu$. For $s\in (0,\frac 12)$ we introduce energy functionals $\mathcal E^s(\mu)$ that can be understood as the density of the $s$-Gagliardo seminorm of $u$ per unit length. Since for $s\ge \frac 12$, the $s$-Gagliardo seminorms are infinite on functions with jumps, some regularization procedure is needed: For $s\in[\frac 12,1)$ we define $\mathcal E_\varepsilon^s(\mu):= \mathcal E^s(\mu_\varepsilon)$, where $\mu_\varepsilon$ is obtained by mollifying $\mu$ on scale $\varepsilon$. We prove that the minimizers of $\mathcal E^s$ and $\mathcal E_\varepsilon^s$ are the equi-spaced configurations of particles with lattice spacing equal to one. Then, we prove the exponential convergence of the corresponding gradient flows to the equi-spaced steady states. Finally, although for $s\in[\frac 12 ,1)$ the energy functionals $\mathcal E_\varepsilon^s$ blow up as $\varepsilon\to 0$, their gradients are uniformly bounded (with respect to $\varepsilon$), so that the corresponding trajectories converge, as $\varepsilon\to 0$, to the gradient flow solution of a suitable renormalized energy.
• Numerical strategies for the microscale simulation of Li-ion batteries
  
  • Asad Ali
  , 2025. In this thesis, we develop and analyze numerical strategies for the simulation of lithium-ion batteries (LIBs) based on their continuum description at the microscale. Our focus is on addressing the computational challenges posed by this inherently multiphysics and multiscale problem, particularly the nonlinearity at the reaction interface and the stiffness of the governing equations.In the first phase of this doctoral work, we tackle the temporal multiscale nature of LIBs by decoupling the domains into subproblems that can be solved independently. Building upon an adaptive high-order coupling strategy, we implement this approach in a Python code. The effectiveness and performance of the method are demonstrated through 1D LIB half-cell simulations. Additionally, we discuss how this promising numerical strategy can be extended to 3D LIB simulations.The multiphysics nature of the LIB model further motivates us to explore adaptive methods in both space and time to reduce computational costs. To extend our study to higher dimensions, we employ a C++ framework with a monolithic solution strategy. Specifically, we implement a multiresolution-based adaptive mesh refinement (AMR) technique using SAMURAI and an adaptive high-order implicit time integrator using PETSc, examining their performance when used together. Using this fully adaptive implementation, we conduct parametric studies to evaluate the impact of interdigitated electrodes on the performance and behavior of 2D LIB half cells.The thesis concludes by synthesizing the two numerical strategies developed to address the computational challenges of LIB microscale simulations. As a natural extension of this work, we propose a unified framework that integrates these approaches, providing a robust foundation for tackling additional complexities that may arise in future microscale LIB model developments.
• Faster Latency Constrained Service Placement in Edge Computing with Deep Reinforcement Learning
  
  • Forghieri Orso
  • Carlinet Yannick
  • Hyon Emmanuel
  • Le Pennec Erwan
  • Perrot Nancy
  , 2025. To enhance the user experience on mobile devices, Mobile Edge Computing (MEC) is a paradigm which integrates computing capabilities directly within access networks. However, designing efficient computation offloading policies in MEC systems remains a challenge. In particular, the decision on whether to process an incoming computation task locally on the mobile device or offload it to the cloud must intelligently adapt to dynamic environmental conditions. This article presents a novel approach that aims to address an edge computing optimization problem, issued from industrial cases, by modeling it as a combinatorial optimization problem combining multicommodity flow and linear latencies constraints. We develop an equivalent linear formulation of the Service Placement Problem, allowing us to use traditional Integer Linear Programming (ILP) methods that turns out to be inefficient in practice. We therefore develop a use-case-based heuristic and a Reinforcement Learning (RL) methodology to model the network configuration under orchestration actions. The latest allows us to transfer learning across pre-training of the agent and shows proof of its efficiency on a dynamic real-world instance, aiming for practical deployment conditions. This comparison reveals that RL is a robust approach that can solve large realistic instances, reaching an optimality gap smaller than 25% on average below a second of runtime for dynamic service placement.
• Optimizing the diffusion coefficient of overdamped Langevin dynamics
  
  • Lelièvre Tony
  • Pavliotis Grigorios A.
  • Robin Geneviève
  • Santet Régis
  • Stoltz Gabriel
  Mathematics of Computation, American Mathematical Society, 2025. Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient. (10.1090/mcom/4098)
  DOI : 10.1090/mcom/4098
• From energy bounds to dimensional estimates in a branched transport model for type-I superconductors
  
  • De Philippis Guido
  • Goldman Michael
  • Berardo Ruffini
  , 2025. We consider a branched transport type problem which describes the magnetic flux through type-I superconductors in a regime of very weak applied fields. At the boundary of the sample, deviation of the magnetization from being uniform is penalized through a negative Sobolev norm. It was conjectured by S. Conti, F. Otto and S. Serfaty that as a result, the trace of the magnetization on the boundary should be a measure of Hausdorff dimension 8/5. We prove that this conjecture is equivalent to the proof of local energy bounds with an optimal exponent. We then obtain local bounds which are however not optimal. These yield improved lower bounds on the dimension of the irrigated measure but unfortunately does not improve on the trivial upper bound. In order to illustrate the dependence of this dimension on the choice of penalization, we consider in the last part of the paper a toy model where the boundary energy is given by a Wasserstein distance to Lebesgue. In this case minimizers are finite graphs and thus the trace is atomic.
• Asymptotic Analysis of a bi-monomeric nonlinear Becker-Döring system
  
  • Doumic Marie
  • Fellner Klemens
  • Mezache Mathieu
  • Velázquez Juan J L
  Nonlinearity, IOP Publishing, 2025. To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-D\"oring system has been proposed in~(Doumic, Fellner, Mezache, Rezaei, J. of Theor. Biol., 2019). When all reaction rates are constant, the equations are the following: \begin{align*} \frac{dv}{dt} & =-vw+v\sum_{j=2}^{\infty}c_{j}, \qquad \frac{dw}{dt} =vw-w\sum_{j=1}^{\infty}c_{j}, \\ \frac{dc_{j}}{dt} & =J_{j-1}-J_{j}\ \ ,\ \ j\geq1\ \ ,\ \ \ J_{j}=wc_{j}-vc_{j+1}\ \ ,\ \ j\geq1\ \ ,\ J_{0}=0, \end{align*} where $v$ and $w$ are two distinct unit species, and $c_i$ represents the concentration of clusters containing $i$ units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system. (10.1088/1361-6544/adc3e5)
  DOI : 10.1088/1361-6544/adc3e5
• Model Updating of Rotating Wind Turbines Using Operational Modal Analysis and Floquet Mode Decomposition
  
  • Delette Nina
  • Denimal Goy Enora
  • Pfister Jean-Lou
  • El Amri Reda
  • Mevel Laurent
  , 2025, pp.1-7. The structural complexity of modern wind turbines, combined with numerous uncertain or unknown parameters, presents significant challenges for accurate predictive modeling. Model updating, which refines numerical model parameters using measurement data, offers a means to mitigate these discrepancies. While extensively applied to stationary structures, its extension to rotating wind turbines remains limited, as their time-periodic dynamics violate key assumptions underlying conventional methods. This study develops a numerical framework for model updating of rotating wind turbines based on an equivalent Linear Time-Invariant (LTI) approximation, derived through a Fourier decomposition of the system’s Floquet modes. A simplified 5 Degrees of Freedom (DoF) turbine model is employed to evaluate the effectiveness of a deterministic model updating strategy leveraging this approximation. Synthetic vibration data, generated from the model using a predefined parameter set, serve as reference measurements for assessing parameter recovery accuracy. Modal features extracted via Operational Modal Analysis (OMA) are used to construct the cost function that quantifies discrepancies between predicted and observed modes. The results underscore the potential of equivalent LTI representations in facilitating model updating for rotating systems, as they effectively capture the modal characteristics identified via OMA. This study establishes a foundation for extending this methodology to more complex, industrial-scale wind turbine models, provided that the computational cost of model evaluation remains manageable.
• The equilibrium price of bubble assets
  
  • Bertucci Charles
  • Lasry Jean-Michel
  • Lions Pierre Louis
  , 2025. Considering a simple economy, we derive a new Hamilton-Jacobi equation which is satisfied by the value of a "bubble" asset. We then show, by providing a rigorous mathematical analysis of this equation, that a unique non-zero stable solution exists under certain assumptions. The economic interpretation of this result is that, if the bubble asset can produce more stable returns than fiat money, agents protect themselves from hazardous situations through the bubble asset, thus forming a bubble's consensus value. Our mathematical analysis uses different ideas coming from the study of semi-linear elliptic equations.
• Robust topology optimization accounting for uncertain microstructural changes
  
  • Masson Hugo
  • Peigney Michaël
  • Denimal Goy Enora
  , 2025.
• Mean Mesh Adaptation for Efficient CFD Simulations with Operating Conditions Variability
  
  • Dornier Hugo
  • Le Maitre Olivier
  • Congedo Pietro Marco
  • Salah El Din Itham
  • Marty Julien
  • Bourasseau Sébastien
  Computers and Fluids, Elsevier, 2025, 298, pp.106666. When numerically solving partial differential equations, for a given problem and operating condition producing a steady-state, mesh adaptation has proven its efficiency to automatically build a discretization achieving a prescribed error level at low cost. However, with continuously varying operating conditions, such as those encountered in uncertainty quantification, adapting a mesh for each condition and controlling the error level becomes complex and computationally expensive. To enable more effective error and cost control, this work proposes a novel approach to mesh adaptation. The method consists in building a single adapted mesh that aims to minimize the average error for a continuous set of operating conditions. In the proposed implementation, this single mesh is built iteratively, informed by an estimate of the local average interpolation error. The proposed method leverages the iterative nature of mesh adaptation by re-sampling Monte Carlo quadratures to obtain accurate average error estimates over a reduced set of sample conditions, ensuring a low computational cost. This approach is especially effective for localized flow features whose positions change only slightly with operating conditions, such as moving shocks in supersonic flows, as the refinement is confined to smaller areas of the computational domain. The study focuses on evaluating the method’s average error convergence, robustness, and computational cost in comparison to state-of-the-art adaptation techniques. Additionally, the sensitivity of the approach to the choice and size of the quadrature, as well as to the error estimation method, is assessed. For this purpose, the methodology is applied to a one-dimensional variable-step solution of the Burgers equation and a two-dimensional Euler scramjet flow with a variable inlet Mach number. The results show that Mean Mesh Adaptation (MMA) achieves error convergence comparable to specific mesh adaptation while reducing the evaluation cost by up to a factor of five (in the scramjet case). This efficiency gain stems from the reduced dependence on the number of sampled conditions, thanks to robust Monte Carlo re-sampling, as well as the shared computational expense of mesh construction across multiple evaluations. Therefore, the proposed method enables computational efficiency while maintaining error control across varying operating conditions within a prescribed parameter variation range. (10.1016/j.compfluid.2025.106666)
  DOI : 10.1016/j.compfluid.2025.106666
• Asymptotic-preserving IMEX schemes for the Euler equations of non-ideal gases
  
  • Orlando Giuseppe
  • Bonaventura Luca
  Journal of Computational Physics, Elsevier, 2025, 529, pp.113889. We analyze schemes based on a general Implicit-Explicit (IMEX) time discretization for the compressible Euler equations of gas dynamics, showing that they are asymptotic-preserving (AP) in the low Mach number limit. The analysis is carried out for a general equation of state (EOS). We consider both a single asymptotic length scale and two length scales. We then show that, when coupling these time discretizations with a Discontinuous Galerkin (DG) space discretization with appropriate fluxes, a numerical method effective for a wide range of Mach numbers is obtained. A number of benchmarks for ideal gases and their non-trivial extension to non-ideal EOS validate the performed analysis. (10.1016/j.jcp.2025.113889)
  DOI : 10.1016/j.jcp.2025.113889
• On reconstruction from imaginary part for radiation solutions in two dimensions
  
  • Nair Arjun
  • Novikov Roman
  , 2025. We consider a radiation solution ψ for the Helmholtz equation in an exterior region in R^2 . We show that ψ in the exterior region is uniquely determined by its imaginary part Im(ψ) on an interval of a line L lying in the exterior region. This result has holographic prototype in the recent work Nair, Novikov (2025, J. Geom. Anal. 35, 123). Some other curves for measurements instead of the lines L are also considered. Applications to the Gelfand-Krein-Levitan inverse problem (from boundary values of the spectral measure in R^2) and to passive imaging are also indicated.
• Comparison Between Effective and Individual Growth Rates in a Heterogeneous Population
  
  • Doumic Marie
  • Rat Anaïs
  • Tournus Magali
  , 2025. Is there an advantage to heterogeneity in a population where individuals grow and divide by fission? This is a broad question, to which there is no easy universal answer. This article aims to provide a quantitative answer in the specific context of growth rate heterogeneity by comparing the fitness of homogeneous versus heterogeneous populations. We focus on size-structured populations, where the growth rate of each individual is set at birth by heredity and/or random mutations. The fitness (or Malthus parameter, or effective fitness) of such heterogeneous population is defined by its long-term behaviour, and we introduce the effective growth rate as the individual growth rate in the homogeneous population with the same fitness. We derive analytical formulae linking effective and individual growth rates in two paradigmatic cases: first, constant growth and division rates, second, linear growth rates and uniform fragmentation. Surprisingly, these two cases yield similar expressions. Then, by comparing the fitness and the effective growth rates of populations with different degrees of heterogeneity or different laws of heredity/mutation to those of average homogeneous populations, we quantitatively investigate the combined influence of heredity and heterogeneity, and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity.
• Avancées dans les Modèles Génératifs : Méthodologies et Applications à la Cardiologie
  
  • Bedin Lisa
  , 2025. This thesis investigates the application of generative models, particularly diffusion models, for the analysis and generation of electrocardiogram (ECG) signals. ECGs are essential diagnostic tools in cardiology, enabling rapid and non-invasive assessment of the heart's electrical activity. However, their interpretation is often complicated by noise, artifacts, and the need to reconstruct complete signals from partial observations.The research presented here focuses on developing advanced methods to address these challenges. We introduce BeatDiff, a lightweight diffusion model for generating 12-lead heartbeat morphologies, and Midpoint-Guidance Posterior Sampling (MGPS), a novel algorithm for high-dimensional inverse problems, which enhances the robustness and accuracy of ECG reconstructions. Additionally, we apply these methods to generate realistic ECGs with RhythmDiff, a diffusion model designed for complete and high-quality ECG signals.The contributions of this thesis include the development of efficient models for generating heartbeat morphologies, the introduction of new techniques for inverse problems, and the application of these methods to generate realistic ECGs. These advancements have the potential to improve cardiac diagnostics and develop more accessible and effective cardiac monitoring technologies, with significant implications for clinical practice and public health.
• Inexact subgradient methods for semialgebraic functions
  
  • Bolte Jérôme
  • Le Tam
  • Moulines Éric
  • Pauwels Edouard
  , 2024, pp.25 p.. Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(\epsilon^\rho)$ distance, where $\epsilon$ denotes the magnitude of subgradient evaluation errors, and $\rho$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $\epsilon^\rho$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits.
• On discrete X-ray transform
  
  • Novikov Roman
  • Sharma Basant Lal
  , 2025. We consider a discrete version of X-ray transform going back, in particular, to Strichartz (1982). We suggest non-overdetermined reconstruction for this discrete transform. Extensions to weighted (attenuated) analogues are given. Connections to the continuous case are presented.
• Convergence and Wave Propagation for a System of Branching Rank-Based Interacting Brownian Particles
  
  • Demircigil Mete
  • Tomasevic Milica
  , 2025. In this work we study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only K ≥ 1 particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity χ > 0. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient. Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as K → ∞ by weighting the individuals by 1/K. Then, on the microscopic level when K is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter χ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we categorize the traveling fronts as pushed or pulled according to the critical parameter χ.
• Volatility models in practice: Rough, Path-dependent or Markovian?
  
  • Abi Jaber Eduardo
  • Li Shaun Xiaoyuan
  Mathematical Finance, Wiley, 2025, 35 (4), pp.796–817. An extensive empirical study of the class of Volterra Bergomi models using SPX options data between 2011 and 2022 reveals the following fact-check on two fundamental claims echoed in the rough volatility literature: Do rough volatility models with Hurst index H ∈ (0, 1/2) really capture well SPX implied volatility surface with very few parameters? No, rough volatility models are inconsistent with the global shape of SPX smiles. They suffer from severe structural limitations imposed by the roughness component, with the Hurst parameter H ∈ (0, 1/2) controlling the smile in a poor way. In particular, the SPX at-the-money skew is incompatible with the power-law shape generated by rough volatility models. The skew of rough volatility models increases too fast on the short end, and decays too slow on the longer end where "negative" H is sometimes needed. Do rough volatility models really outperform consistently their classical Markovian counterparts? No, for short maturities they underperform their one-factor Markovian counterpart with the same number of parameters. For longer maturities, they do not systematically outperform the one-factor model and significantly underperform when compared to an under-parametrized two-factor Markovian model with only one additional calibratable parameter. On the positive side: our study identifies a (non-rough) path-dependent Bergomi model and an under-parametrized two-factor Markovian Bergomi model that consistently outperform their rough counterpart in capturing SPX smiles between one week and three years with only 3 to 4 calibratable parameters.
• Non-convex functionals penalizing simultaneous oscillations along two independent directions: structure of the defect measure
  
  • Goldman Michael
  • Merlet Benoît
  , 2025. We continue the analysis of a family of energies penalizing oscillations in oblique directions: they apply to functions $u(x_1,x_2)$ with $x_l\in\mathbb{R}^{n_l}$ and vanish when $u(x)$ is of the form $u_1(x_1)$ or $u_2(x_2)$. We mainly study the rectifiability properties of the defect measure $\nabla_1\nabla_2u$ of functions with finite energy. The energies depend on a parameter $\theta\in(0,1]$ and the set of functions with finite energy grows with $\theta$. For $\theta<1$ we prove that the defect measure is $(n_1-1,n_2-1)$-tensor rectifiable in $\Omega_1\times\Omega_2$. We first get the result for $n_1=n_2=1$ and deduce the general case through slicing using White's rectifiability criterion. When $\theta=1$ the situation is less clear as measures of arbitrary dimensions from zero to $n_1+n_2-1$ are possible. We show however, in the case $n_1=n_2=1$ and for Lipschitz continuous functions, that the defect measures are $1\,$-rectifiable. This case bears strong analogies with the study of entropic solutions of the eikonal equation.
• Stationary regimes of piecewise linear dynamical systems with priorities
  
  • Allamigeon Xavier
  • Capetillo Pascal
  • Gaubert Stéphane
  , 2025, pp.1-11. Dynamical systems governed by priority rules appear in the modeling of emergency organizations and road traffic. These systems can be modeled by piecewise linear time-delay dynamics, specifically using Petri nets with priority rules. A central question is to show the existence of stationary regimes (i.e., steady state solutions) -- taking the form of invariant half-lines -- from which essential performance indicators like the throughput and congestion phases can be derived. Our primary result proves the existence of stationary solutions under structural conditions involving the spectrum of the linear parts within the piecewise linear dynamics. This extends to a broader class of systems a fundamental theorem of Kohlberg (1980) dealing with nonexpansive dynamics. The proof of our result relies on topological degree theory and the notion of ``Blackwell optimality'' from the theory of Markov decision processes. Finally, we validate our findings by demonstrating that these structural conditions hold for a wide range of dynamics, especially those stemming from Petri nets with priority rules. This is illustrated on real-world examples from road traffic management and emergency call center operations. (10.1145/3716863.3718053)
  DOI : 10.1145/3716863.3718053