Publications

2025

• Signed tropicalization of polar cones
  
  • Akian Marianne
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Sergeev Sergei
  Journal of Optimization Theory and Applications, Springer Verlag, 2025, 207 (10). We study the tropical analogue of the notion of polar of a cone, working over the semiring of tropical numbers with signs. We characterize the cones which arise as polars of sets of tropically nonnegative vectors by an invariance property with respect to a tropical analogue of Fourier-Motzkin elimination. We also relate tropical polars with images by the nonarchimedean valuation of classical polars over real closed nonarchimedean fields and show, in particular, that for semi-algebraic sets over such fields, the operation of taking the polar commutes with the operation of signed valuation (keeping track both of the nonarchimedean valuation and sign). We apply these results to characterize images by the signed valuation of classical cones of matrices, including the cones of positive semidefinite matrices, completely positive matrices, completely positive semidefinite matrices, and their polars, including the cone of co-positive matrices, showing that hierarchies of classical cones collapse under tropicalization. We finally discuss an application of these ideas to optimization with signed tropical numbers. (10.1007/s10957-025-02732-2)
  DOI : 10.1007/s10957-025-02732-2
• On Global Convergence Rates for Federated Policy Gradient under Heterogeneous Environment
  
  • Labbi Safwan
  • Mangold Paul
  • Tiapkin Daniil
  • Moulines Eric
  , 2025. Ensuring convergence of policy gradient methods in federated reinforcement learning (FRL) under environment heterogeneity remains a major challenge. In this work, we first establish that heterogeneity, perhaps counter-intuitively, can necessitate optimal policies to be non-deterministic or even time-varying, even in tabular environments. Subsequently, we prove global convergence results for federated policy gradient (FedPG) algorithms employing local updates, under a Łojasiewicz condition that holds only for each individual agent, in both entropy-regularized and non-regularized scenarios. Crucially, our theoretical analysis shows that FedPG attains linear speed-up with respect to the number of agents, a property central to efficient federated learning. Leveraging insights from our theoretical findings, we introduce b-RS-FedPG, a novel policy gradient method that employs a carefully constructed softmax-inspired parameterization coupled with an appropriate regularization scheme. We further demonstrate explicit convergence rates for b-RS-FedPG toward near-optimal stationary policies. Finally, we demonstrate that empirically both FedPG and b-RS-FedPG consistently outperform federated Q-learning on heterogeneous settings. (10.48550/arXiv.2505.23459)
  DOI : 10.48550/arXiv.2505.23459
• Large-Eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite materials
  
  • Grenouilloux Adrien
  • Bioche Kévin
  • Dellinger Nicolas
  • Letournel Roxane
  • Bechane Yacine
  • Moureau Vincent
  , 2025.
• Improving GFlowNets with Monte Carlo Tree Search
  
  • Morozov Nikita
  • Tiapkin Daniil
  • Samsonov Sergey
  • Naumov Alexey
  • Vetrov Dmitry
  , 2024. Generative Flow Networks (GFlowNets) treat sampling from distributions over compositional discrete spaces as a sequential decision-making problem, training a stochastic policy to construct objects step by step. Recent studies have revealed strong connections between GFlowNets and entropy-regularized reinforcement learning. Building on these insights, we propose to enhance planning capabilities of GFlowNets by applying Monte Carlo Tree Search (MCTS). Specifically, we show how the MENTS algorithm (Xiao et al., 2019) can be adapted for GFlowNets and used during both training and inference. Our experiments demonstrate that this approach improves the sample efficiency of GFlowNet training and the generation fidelity of pre-trained GFlowNet models. (10.48550/arXiv.2406.13655)
  DOI : 10.48550/arXiv.2406.13655
• Contribution to the study of the mean field games master equation
  
  • Meynard Charles
  , 2025. This thesis focuses on the problems of uniqueness and existence to the mean field games master equation. The first part is devoted to the study of this equation for mean field games in which the optimization problem solved by an individual player depends on the realization of a stochastic process common to all players. This process may be autonomous, but it can also depend on the evolution of the game through the distribution of players. We provide sufficient conditions ensuring the existence and uniqueness of a solution to the master equation associated with such games, first in the context of finite state space mean field games. We then extend these results to master equations defined on the space of probability measures under several notions of monotonicity. In the second part, we introduce the notion of monotone solution for so-called displacement monotone mean field games. This notion of weak solution to the master equation does not require any differentiability assumptions on the solution with respect to probability measures. Under suitable monotonicity conditions, we establish uniqueness, stability, and finally existence of such solutions.
• High-order adaptive multi-domain time integration scheme for microscale lithium-ion batteries simulations
  
  • Asad Ali
  • de Loubens Romain
  • François Laurent
  • Massot Marc
  SMAI Journal of Computational Mathematics, Société de Mathématiques Appliquées et Industrielles (SMAI), 2025, 11, pp.369-404. We investigate the modeling and simulation of ionic transport and charge conservation in lithium-ion batteries (LIBs) at the microscale. It is a multiphysics problem that involves a wide range of time scales. The associated computational challenges motivate the investigation of numerical techniques that can decouple the time integration of the governing equations in the liquid electrolyte and the solid phase (active materials and current collectors). First, it is shown that semi-discretization in space of the non-dimensionalized governing equations leads to a system of index-1 semi-explicit differential algebraic equations (DAEs). Then, a new generation of strategies for multi-domain integration is presented, enabling high-order adaptive coupling of both domains in time, with efficient and potentially different domain integrators. They reach a high level of flexibility for real applications, beyond the limitations of multirate methods. A simple 1D LIB half-cell code is implemented as a demonstrator of the new strategy for the simulation of different modes of cell operation. The integration of the decoupled subsystems is performed with high-order accurate implicit nonlinear solvers. The accuracy of the space discretization is assessed by comparing the numerical results to the analytical solutions. Then, temporal convergence studies demonstrate the accuracy of the new multi-domain coupling approach. Finally, the accuracy and computational efficiency of the adaptive coupling strategy are discussed in the light of the conditioning of the decoupled subproblems compared to the one of the fully-coupled problem. This new approach will constitute a key ingredient for the high-fidelity 3D LIB simulations based on actual electrode microstructures. (10.5802/smai-jcm.128)
  DOI : 10.5802/smai-jcm.128
• samurai/ponio: two complementary software libraries for the efficient implementation of adaptive space/time numerical schemes
  
  • Dubois Sébastien
  • Gouarin Loïc
  • Hoffmann Alexandre
  • Massot Josselin
  • Massot Marc
  • Matalon Pierre
  • Series Laurent
  , 2025. We present two complementary open-source C ++ libraries for the numerical simulation of multiscale physical systems. samurai provides an efficient adaptive Cartesian meshing framework, while ponio offers a wide range of time-integration schemes for ordinary differential equations. Together, they enable the rapid implementation and testing of high-performance, adaptive space-time numerical methods for PDEs. Both libraries emphasize modularity, data locality, and modern C ++ design. Their capabilities are illustrated through applications developed during the Modeling Summer Visit 2025, including two-phase flows, rarefied gases, and reactive flows for aerospace applications relevant to NASA.
• Longest increasing subsequences for distributions with atoms, and an inhomogeneous Hammersley process
  
  • Basdevant Anne-Laure
  • Gerin Lucas
  • Marivain Maxime
  , 2025. A famous result by Hammersley and Versik-Kerov states that the length $L_n$ of the longest increasing subsequence among $n$ iid continuous random variables grows like $2\sqrt{n}$. We investigate here the asymptotic behavior of $L_n$ for distributions with atoms. For purely discrete random variables, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt{n})$, depending on the tail behavior of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.
• Exponentially Fading Memory Signature
  
  • Abi Jaber Eduardo
  • Sotnikov Dimitri
  , 2025. We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for timeseries analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation.
• Stochastic and deterministic approaches for studying telomere shortening and other cell dynamics
  
  • Olayé Jules
  , 2025. Telomeres are non-coding regions situated at the ends of chromosomes of eukaryotic cells, whose lengths vary during cell divisions through shortening and/or elongation. The shortening of telomeres is one of the main factors leading to replicative senescence, an irreversible state in which cells stop to divide. The study of telomeres has gained interest in recent years, notably due to their link with the emergence of cancer cells and with the aging of individuals. These studies have led to the development of mathematical models, allowing the description of the experiments conducted by biologists. Using these models often relies on assuming that a stationary profile of telomere lengths exists, or doing model approximations, for which theoretical results are not well-established. The main objective of this thesis is to provide theoretical results, through two axes of study.The first axis is the analysis of the long-time behaviour of the telomere lengths density in a telomere shortening model with elongation. The difficulties that we need to manage are the non-compact nature of the operator we study, and the semi-Markovian aspect of our model. The second axis is the study of an inverse problem, in which we aim to estimate an initial distribution of telomere lengths from measurements of senescence times. The main argument in solving this problem is to approximate our models by transport and transport-diffusion equations, on which estimators are easier to construct.In parallel with these works related to telomeres, two additional studies have been conducted in this thesis, still related to cell dynamics. In the first one, we propose a model for the neurogenesis phenomenon based on compound Poisson processes. In the second one, we perform a theoretical and numerical study of a method for estimating the division times distribution of bacteria, developed by a biologist during his thesis.
• A quantitative comparison of high-order asymptotic-preserving and asymptotically-accurate IMEX methods for the Euler equations with non-ideal gases
  
  • Orlando Giuseppe
  • Boscarino Sebastiano
  • Russo Giovanni
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2025, 442, pp.118037. We present a quantitative comparison between two different Implicit-Explicit Runge-Kutta (IMEX-RK) approaches for the Euler equations of gas dynamics, specifically tailored for the low Mach limit. In this regime, a classical IMEX-RK approach involves an implicit coupling between the momentum and energy balance so as to avoid the acoustic CFL restriction, while the density can be treated in a fully explicit fashion. This approach leads to a mildly nonlinear equation for the pressure, which can be solved according to a fixed point procedure. An alternative strategy consists of employing a semi-implicit temporal integrator based on IMEX-RK methods (SI-IMEX-RK). The stiff dependence is carefully analyzed, so as to avoid the solution of a nonlinear equation for the pressure also for equations of state (EOS) of non-ideal gases. The spatial discretization is based on a Discontinuous Galerkin (DG) method, which naturally allows high-order accuracy. The asymptotic-preserving (AP) and the asymptotically-accurate (AA) properties of the two approaches are assessed on a number of classical benchmarks for ideal gases and on their extension to non-ideal gases. (10.1016/j.cma.2025.118037)
  DOI : 10.1016/j.cma.2025.118037
• Computer-assisted proofs for differential equations and dynamical systems
  
  • Breden Maxime
  , 2025.
• On random 3/2-stable maps
  
  • Kammerer Emmanuel
  , 2025. This thesis focuses on the asymptotic behaviour of random planar maps and random trees. In a first part, we study the geometry of large random planar maps with high degree vertices, also called stable maps, with a particular interest in the critical case: the case of 3/2-stable maps. We prove that these maps appear naturally in the study of random planar maps coupled with a loop O(n) model, a classical model of statistical mechanics, in the case n=2. We obtain the asymptotic behaviour of the distances between two uniform random vertices and we prove that the diameter is of the same order. A consequence of these results is that 3/2-stable maps do not satisfy scaling limits in the usual sense of Gromov-Hausdorff. However, we get the scaling limit of the distances between high degree vertices and the root. This scaling limit corresponds to a distance between the loops of a conformal loop ensemble and the boundary of the disc, measured using an independent Liouville quantum gravity, which is the Lamperti transform of the "quantum distance" to the boundary introduced by Aru, Holden, Powell and Sun. We then study the geodesics to the root for the first passage percolation distance on stable maps. We construct the scaling limit of the geodesics by means of a coalescing flow of pure jump diffusions.In a second part, we introduce a generalisation of the model of random recursive trees by adding a freezing mechanism. At some steps, a uniform random vertex is frozen and new vertices can no longer be attached to this vertex. The infection tree in an SIR model on the complete graph falls within this framework. We obtain local limits of these trees and scaling limits of the distances between the root and a typical vertex, of the distance between two uniform typical vertices and of the total height of the tree obtained after n steps. In the sub-linear case where the number of non-frozen vertices evolves as a given power of n, we identify a phase transition. Lastly, we consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalised converge.
• Some mathematical models for flagellar activation mechanisms
  
  • Alouges François
  • Anello Irene
  • Desimone Antonio
  • Lefebvre-Lepot Aline
  • Levillain Jessie
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2025, 35 (11), pp.2395-2424. This paper focuses on studying a model for molecular motors responsible for the bending of the axoneme in the flagella of microorganisms. The model is a coupled system of partial differential equations inspired by Jülicher et al. or Camalet, incorporating two rows of molecular motors between microtubules filaments. Existence and uniqueness of a solution is proved, together with the presence of a supercritical Hopf bifurcation. Additionally, numerical simulations are provided to illustrate the theoretical results. A brief study on the generalization to N-rows is also included. (10.1142/S0218202525500423)
  DOI : 10.1142/S0218202525500423
• Effects of a mutation on the fitness of a bacterium : theoretical estimation and numerical implementation
  
  • Garnier Guillaume
  , 2025. This thesis presents a mathematical study of genetic mutations and their effects on the fitness of individuals, a central concept in evolutionary biology. A mutation is a spontaneous or induced alteration of DNA, which can be deleterious, neutral, or beneficial. These mutations affect the fitness of individuals, i.e., their ability to survive and reproduce. The main question addressed is the estimation of the Density of Fitness Effects (DFE), which describes the distribution of the effects of mutations on fitness. Understanding the shape of the DFE is essential for predicting population evolution, genetic diversity, and the consequences of conservation programs. The starting point of this work is an experimental protocol developed by Lydia Robert et al. (2018), which allows real-time observation of mutation events in E. coli. These new data open up opportunities for improved estimation of the DFE, but also raise statistical challenges due to experimental noise .In the first part of this manuscript, we consider a stochastic model from [77] that describes the evolution of a bacterial population under mutation pressure. We develop a nonparametric estimation method based on Fourier estimators to recover the DFE, and establish convergence results for our estimator. The second part introduces two deterministic models for the evolution of fitness in a population structured by growth rate. These models allow us to study the asymptotic behavior of the population and provide mathematical insight into the dynamics of mutation accumulation observed in experiments. Finally, the third part applies various statistical methods to the experimental data, aiming to reconstruct the DFE from its empirical moments and to assess whether it is unimodal or multimodal. This work builds a bridge between experimental biology and the mathematical modeling of mutation effects.
• Robust calibration of an air-carbon ablation model employing Plasmatron and molecular beam data
  
  • Piro Vittorio
  • Capriati Michele
  • Bariselli Federico
  • Congedo Pietro Marco
  • Magin Thierry E.
  , 2025. Understanding and accurately predicting gas-surface interaction phenomena during a vehicle's re-entry phase is critical for the design of thermal protection materials. In this work, we aim to improve the robustness of an air-carbon ablation model by calibrating its reaction rates using both low-pressure molecular beam data and high-pressure Plasmatron data, as well as numerical models, following a Bayesian approach. Firstly, the subset of nitrogen chemical reactions is inferred, resulting in a finite-rate nitrogen model successfully calibrated and validated at both low-and highpressure. A surrogate-based Bayesian framework is employed, comparing artificial neural network and Kriging approaches to mimic the nitrogen model predictions at high-pressure and accelerate posterior sampling. Then, for the first time, atomic oxygen reaction rates are also included in the calibration, along with low-pressure oxygen observations. Since the experimental data used for calibration do not include high-pressure conditions, the oxygen reaction rates for accuracy at high pressure could not be fully characterised. Nevertheless, the resulting models demonstrated superior performance compared to the state-of-the-art model under low-pressure conditions.
• Bayesian model updating of rotating wind turbines
  
  • Delette Nina
  • Pfister Jean-Lou
  • Denimal Goy Enora
  • El Amri Mohamed Reda
  • Mevel Laurent
  , 2025, pp.1-3.
• Energy estimate of a Discrete Duality Finite Volume scheme for a phase-field model with surfactants
  
  • Castellano Margherita
  • Goudenège Ludovic
  • Nabet Flore
  , 2025. In this work we establish a discrete energy estimate for a phase-field model describing a ternary mixture of two immiscible fluids (air and water) and surfactants, firstly introduced by [3]. We mainly rely on the phase-field formulation presented in [5], where the authors include a stabilization term to the model analyzed in [4]. The stabilization term ensures that the free energy remains bounded from below, which is essential for deriving the discrete energy estimate. This will later guarantee the existence of an approximate solution and the convergence of the numerical scheme (which will be covered in a forthcoming paper). We propose a spatial discretization using the Discrete Duality Finite Volume (DDFV) method, which is particularly wellsuited for the simulation of non-linear diffusive problems. This method can handle very general (possibly non-conforming or anisotropic) meshes, while preserving key structural properties of the continuous problem at the discrete level.
• A sequential variational Bayesian approach to Gaussian process quantile regression for optimization
  
  • Nicolas Hugo
  • Le Maître Olivier
  • Congedo Pietro Marco
  , 2025. Quantile regression [1] extends the classical least-squares regression to the estimation of the conditional quantiles of a random variable. In the frequentist approach, one casts the quantile regression into the problem of minimizing a loss function, possibly completed with regularization terms. The Bayesian counterpart, first proposed in [2], formulates the problem as a posterior inference over a function space. Bayesian inference can rely on Markov chain Monte Carlo (MCMC) methods to sample the posterior distribution. Variation Bayesian inference techniques alleviate some of the computational burdens of MCMC by introducing latent variables and providing an analytical approximation of their posterior distribution.
• On measure-valued solutions for a structured population model with transfers
  
  • Magal Pierre
  • Raoul Gaël
  , 2025, pp.76. We consider a transfer operator where two interacting cells carrying non-negative traits transfer a random fraction of their trait to each other. These transfers can lead to population having singular distributions in trait. We extend the definition of the transfer operator to non-negative measures with a finite second moment, and we discuss the regularity of the fixed distributions of that transfer operator. Finally, we consider a dynamic transfer model where an initial population distribution is affected by a transfer operator: we prove the existence and uniqueness of mild measure-valued solutions for that Cauchy problem.
• Long-time behaviors of dynamics with mean field interactions
  
  • Wang Songbo
  , 2025. This thesis is devoted to the study of the long-time behaviors of dynamics with mean field interactions and their associated particle systems. For most cases treated in the thesis, the structural condition for the long-time behaviors is the flat convexity of the mean field energy functional, which is different from the displacement convexity studied in the classical works of optimal transport and gradient flow. The thesis is comprised of three parts. In the first part, we study the overdamped and underdamped mean field Langevin dynamics, which are gradient dynamics associated to a mean field free energy functional, and show their time-uniform propagation of chaos properties by exploiting their gradient structures and a uniform logarithmic Sobolev inequality. In the second part, we first develop some technical results on logarithmic Sobolev inequalities and apply them to get the time-uniform propagation of chaos for various McKean-Vlasov diffusions. Specifically, for the 2D viscous vortex model, we develop strong regularity bounds on its mean field limit on the whole space and show its propagation of chaos by the Jabin-Wang method; we also study its size of chaos problem using the entropy approach of Lacker and obtain time-uniform sharp bounds in the high viscosity regime. In the last part of the thesis, we explore alternative mean field dynamics that originate from convex optimization problems. For the entropy-regularized optimization, we study a fictitious self-play dynamics and a self-interacting diffusion and show their long-time convergences to the solution of the optimization problem. We also consider a non-linear Schrödinger semigroup, which is a gradient flow for the optimization problem regularized by Fisher information, and show its exponential convergence under a uniform spectral gap condition.
• Open-Canopy: Towards Very High Resolution Forest Monitoring
  
  • Fogel Fajwel
  • Perron Yohann
  • Besic Nikola
  • Saint-André Laurent
  • Pellissier-Tanon Agnès
  • Schwartz Martin
  • Boudras Thomas
  • Fayad Ibrahim
  • d'Aspremont Alexandre
  • Landrieu Loic
  • Ciais Philippe
  , 2025. Estimating canopy height and its changes at meter resolution from satellite imagery remains a challenging computer vision task with critical environmental applications. However, the lack of open-access datasets at this resolution hinders the reproducibility and evaluation of models. We introduce Open-Canopy, the first open-access, country-scale benchmark for very high-resolution (1.5 m) canopy height estimation, covering over 87,000 km² across France with 1.5 m panchromatic resolution satellite imagery and aerial LiDAR data. Additionally, we present Open-Canopy-$\Delta$, a benchmark for canopy height reduction detection between images from different years at tree level---a difficult task for current computer vision models. We evaluate state-of-the-art architectures on these benchmarks, highlighting significant challenges and opportunities for improvement. Our datasets and code are publicly available at \url{https://github.com/fajwel/Open-Canopy}. (10.1109/CVPR52734.2025.00138)
  DOI : 10.1109/CVPR52734.2025.00138
• Wave turbulence, thermalization and multimode locking in optical fibers
  
  • Ferraro M.
  • Baudin K.
  • Gervaziev M.
  • Fusaro A.
  • Picozzi Antonio
  • Garnier J.
  • Millot G.
  • Kharenko D.
  • Podivilov E.
  • Babin S.
  • Mangini F.
  • Wabnitz S.
  Physica D: Nonlinear Phenomena, Elsevier, 2025, 481, pp.134758. We present a comprehensive overview of recent advances in theory and experiments on complex light propagation phenomena in nonlinear multimode fibers. On the basis of the wave turbulence theory, we derive kinetic equations describing the out-of-equilibrium process of optical thermalization toward the Rayleigh-Jeans (RJ) equilibrium distribution. Our theory is applied to explain the effect of beam self-cleaning (BSC) in graded-index (GRIN) fibers, whereby a speckled beam transforms into a bell-shaped beam at the fiber output as the input peak power grows larger. Although the output beam is typically dominated by the fundamental mode of the fiber, higher-order modes (HOMs) cannot be fully depleted, as described by the turbulence cascades associated to the conserved quantities. We theoretically explore the role of random refractive index fluctuations along the fiber core, and show how these imperfections may turn out to assist the observation of BSC in a practical experimental setting. This conclusion is supported by the derivation of wave turbulence kinetic equations that account for the presence of a time-dependent disorder (random mode coupling). The kinetic theory reveals that a weak disorder accelerates the rate of RJ thermalization and beam cleaning condensation. On the other hand, although strong disorder is expected to suppress wave condensation, the kinetic equation reveals that an out-of-equilibrium process of condensation and RJ thermalization can occur in a regime where disorder predominates over nonlinearity. In general, the kinetic equations are validated by numerical simulations of the generalized nonlinear Schrodinger equation. We outline a series of recent experiments, which permit to confirm the statistical mechanics approach for describing beam propagation and thermalization. For example, we highlight the demonstration of entropy growth, and point out that there are inherent limits to peak-power scaling in multimode fiber lasers. We conclude by pointing out the experimental observation that BSC is accompanied by an effect of modal phase-locking. From the one hand this explains the observed preservation of the spatial coherence of the beam, but also it points to the need of extending current descriptions in future research.
• Mesh Adaptation Strategies for CFD Simulations Over a Set of Operating Conditions
  
  • Dornier Hugo
  • Le Maître Olivier P
  • Congedo Pietro Marco
  • Salah El Din Itham
  • Bourasseau Sébastien
  • Marty Julien
  , 2025. In computational fluid dynamics, evaluating accurately quantities of interest (global or local) requires capturing complex local phenomena and interactions, such as shocks and flow separations, while controlling the global error. The latter depends highly on the discretization of the computational domain, hence the mesh. In general, the location of the flow structures within the domain is sensitive to boundary and flow conditions. Proposing an a priori mesh with a discretization effort that concentrates on the demanding parts of the domain is thus usually impossible. Adaptive Mesh Refinement (AMR, [1,2,3,4]) is a method developed to iteratively adjust the local mesh resolution to the computed flow structures and construct meshes that achieve a prescribed accuracy for limited discretization and computational cost. This work concerns the problem of mesh adaptation when the operating conditions are variable and follow a prescribed continuous distribution. For instance, variability of the flow conditions appears in uncertainty quantification, operating domain analysis, and robust optimization. These analyses typically require many simulations for different conditions, making the cost-accuracy trade-off even more crucial for these problems. Several mesh adaption methods have been proposed for variable conditions ([5,6,7,8]), but they typically focus primarily on error control without simultaneously optimizing for cost. In this context, we propose two original methodologies. The first one, called Mean Mesh adaptation (MMA, [9,10]), builds a unique adapted mesh to minimize the average error over the continuous operating conditions for a given discretization effort. A key ingredient of MMA is using a small sample set of conditions to estimate the local average error at each iteration of the AMR process. The second method, Error-based Mesh Selection (EMS), tackles the optimal element selection within a library of adapted meshes to achieve the smallest possible error for any given flow conditions. The library consists of meshes independently adapted for different conditions in an offline stage, for cost efficiency, the selection uses a priori error estimations requiring no additional simulation. We used analytical and full CFD supersonic simulations [11,12] to analyze the proposed methods. We show that MMA is robust and accurately approximates the optimal mesh minimizing the average error for a limited construction cost (see figures 1 and 2). Similarly, EMS provides a robust approximation of the optimal selection with limited cost overheads (see figure 3). The EMS method is suitable to be extended for progressive library enrichment and a-posteriori correction of the error estimates.
• Weak solutions of stochastic volterra equations in convex domains with general kernels
  
  • Abi Jaber Eduardo
  • Alfonsi Aurélien
  • Szulda Guillaume
  , 2025. We establish new weak existence results for d-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional nonconvolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in Alfonsi (2025) to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a nonconvolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result. (10.48550/arXiv.2506.04911)
  DOI : 10.48550/arXiv.2506.04911