Publications

2025

• Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics
  
  • Durmus Alain
  • Enfroy Aurélien
  • Moulines Éric
  • Stoltz Gabriel
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2025, 61 (1). We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method. (10.1214/23-AIHP1442)
  DOI : 10.1214/23-AIHP1442
• Annealed limit for a diffusive disordered mean-field model with random jumps
  
  • Erny Xavier
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2025, 61 (1), pp.510-532. We study a sequence of $N-$particle mean-field systems, each driven by $N$ simple point processes $Z^{N,i}$ in a random environment. Each $Z^{N,i}$ has the same intensity $(f(X^N_{t-}))_t$ and at every jump time of $Z^{N,i},$ the process $X^N$ does a jump of height $U_i/\sqrt{N}$ where the $U_i$ are disordered centered random variables attached to each particle. We prove the convergence in distribution of $X^N$ to some limit process $\bar X$ that is solution to an SDE with a random environment given by a Gaussian variable, with a convergence speed for the finite-dimensional distributions. This Gaussian variable is created by a CLT as the limit of the patial sums of the $U_i.$ To prove this result, we use a coupling for the classical CLT relying on the result of [Koml\'os, Major and Tusn\'ady (1976)], that allows to compare the conditional distributions of $X^N$ and $\bar X$ given the random environment, with the same Markovian technics as the ones used in [Erny, L\"ocherbach and Loukianova (2022)]. (10.1214/23-AIHP1432)
  DOI : 10.1214/23-AIHP1432
• Uniswap v3: impermanent loss modeling and swap fees asymptotic analysis
  
  • Echenim Mnacho
  • Gobet Emmanuel
  • Maurice Anne-Claire
  , 2023. Automated Market Makers have emerged quite recently, and Uniswap is one of the most widely used platforms (it covers 60% of the total value locked on Ethereum blockchain at the time of writing this article). This protocol is challenging from a quantitative point of view, as it allows participants to choose where they wish to concentrate liquidity. There has been an increasing number of research papers on Uniswap v3 but often, these articles use heuristics or approximations that can be far from reality: for instance, the liquidity in the pool is sometimes assumed to be constant over time, which contradicts the mechanism of the protocol. The ob- jectives of this work are fourfold: first, to revisit Uniswap v3’s principles in detail (starting from the open source code) to build an unambiguous knowledge base. Second, to analyze the Im- permanent Loss of a liquidity provider by detailing its evolution, with no assumption on the swap trades or liquidity events that occur over the time period. Third, we introduce the notion of a liquidity curve. For each curve, we can construct a payoff at a given maturity, net of fees. Conversely, we show how any concave payoff can be synthetized by an initial liquidity curve and some tokens outside the pool; this paves the way for using Uniswap v3 to create options. Fourth, we analyze the asymptotic behavior of collected fees without any simplifying hypoth- esis (like a constant liquidity), under the mild assumption that the pool price coincides with a latent price (general Ito process) every time the latter changes by γ%. The asymptotic analysis is conducted as γ → 0. The value of the collected fees then coincides with an integral of call and put prices. Our derivations are supported by graphical illustrations and experiments.
• On the entropy dissipation of systems of quadratures
  
  • Pichard Teddy
  • Laurent Frédérique
  , 2025. <div>The method of moments in kinetic theory is a popular discretization technique with respect to the kinetic velocity variable. It can be seen as a non-linear Galerkin semi-discretization in velocity. Among these methods, the quadrature-based methods exploiting the theory of orthogonal polynomials are well suited for numerical purposes. However, two important properties of the original kinetic equation are lost during such approximations: the strong hyperbolicity, corresponding to transport phenomena, and the dissipation of entropy, corresponding to the trend of the solution towards an equilibrium. These two properties are closely related through symmetrization techniques. In this work, we aim to clarify the treatment of these two properties in the derivation of quadrature-based moment systems, and to prove H-theorems, namely dissipation of entropy and equilibrium representation, after such derivations. From this study, we develop of quadrature-based closures adapted to specific entropies. These closures are of two types, either with fixed quadrature points (or velocities), namely the discrete velocity methods (DVM), and with varying ones, namely the quadrature-based method of moments (QMOM). To adapt the closures to specific entropies, the number of quadrature points is increased compared to the number of moments, resulting in augmented systems (ADVM or AQMOM) with more unknowns than equations, and the additional parameters are constrained to match the entropy requirements. Similarly, the quadrature-based entropies are of two types, either those based on a symmetrization criterion on the flux vector, which are only intended to have an entropy, or those corresponding to a quadrature formula of the kinetic entropy, which are intended to reproduce the kinetic trend towards the equilibrium. At each step of these developments, based on the considered entropy, we provide definitions for the flux vectors, adapted relaxation operators (with common conservation properties), we compute the associated entropic variables, and highlight the corresponding symmetrization property of the flux and equilibria. Finally, we provide an entropy-dissipative discretization for such moment systems.</div>
• Mathematical model linking telomeres to senescence in Saccharomyces cerevisiae reveals cell lineage versus population dynamics
  
  • Rat Anaïs
  • Martinez Fernandez Veronica
  • Doumic Marie
  • Teixeira Maria Teresa
  • Xu Zhou
  Nature Communications, Nature Publishing Group, 2025, 16 (1), pp.1024. Telomere shortening ultimately causes replicative senescence. However, identifying the mechanisms driving replicative senescence in cell populations is challenging due to the heterogeneity of telomere lengths and the asynchrony of senescence onset. Here, we present a mathematical model of telomere shortening and replicative senescence in Saccharomyces cerevisiae which is quantitatively calibrated and validated using data of telomerase-deficient single cells. Simulations of yeast populations, where cells with varying proliferation capacities compete against each other, show that the distribution of telomere lengths of the initial population shapes population growth, especially through the distribution of cells’ shortest telomere lengths. We also quantified how factors influencing cell viability independently of telomeres can impact senescence rates. Overall, we demonstrate a temporal evolution in the composition of senescent cell populations—from a state directly linked to critically short telomeres to a state where senescence onset becomes stochastic. This population structure may promote genome instability and facilitate senescence escape. (10.1038/s41467-025-56196-z)
  DOI : 10.1038/s41467-025-56196-z
• Individual cell fate and population dynamics revealed by a mathematical model linking telomere length and replicative senescence
  
  • Rat Anaïs
  • Martinez Fernandez Veronica
  • Doumic Marie
  • Teixeira Maria Teresa
  • Xu Zhou
  Nature Communications, Nature Publishing Group, 2025, 16 (1), pp.1024. Progressive shortening of telomeres ultimately causes replicative senescence and is linked with aging and tumor suppression. Studying the intricate link between telomere shortening and senescence at the molecular level and its population-scale effects over time is challenging with current approaches but crucial for understanding behavior at the organ or tissue level. In this study, we developed a mathematical model for telomere shortening and the onset of replicative senescence using data from Saccharomyces cerevisiae without telomerase. Our model tracks individual cell states, their telomere length dynamics, and lifespan over time, revealing selection forces within a population. We discovered that both cell genealogy and global telomere length distribution are key to determine the population proliferation capacity. We also discovered that cell growth defects unrelated to telomeres also affect subsequent proliferation and may act as confounding variables in replicative senescence assays. Overall, while there is a deterministic limit for the shortest telomere length, the stochastic occurrence of non-terminal arrests drive cells into a totally different regime, which may promote genome instability and senescence escape. Our results offer a comprehensive framework for investigating the implications of telomere length on human diseases. (10.1101/2023.11.22.568287)
  DOI : 10.1101/2023.11.22.568287
• Convergence and Error Estimates of A Semi-Lagrangian scheme for the Minimum Time Problem
  
  • Akian Marianne
  • Liu Shanqing
  , 2024. We consider a semi-Lagrangian scheme for solving the minimum time problem, with a given target, and the associated eikonal type equation. We first use a discrete time deterministic optimal control problem interpretation of the time discretization scheme, and show that the discrete time value function is semiconcave under regularity assumptions on the dynamics and the boundary of target set. We establish a convergence rate of order $1$ in terms of time step based on this semiconcavity property. Then, we use a discrete time stochastic optimal control interpretation of the full discretization scheme, and we establish a convergence rate of order $1$ in terms of both time and spatial steps using certain interpolation operators, under further regularity assumptions. We extend our convergence results to problems with particular state constraints. We apply our results to analyze the convergence rate and computational complexity of the fast-marching method. We also consider the multi-level fast-marching method recently introduced by the authors.
• Small-scale interface dynamic modelling based on the geometric method of moments for a two-scale two-phase flow model with a disperse small scale
  
  • Loison Arthur
  • Pichard Teddy
  • Kokh Samuel
  • Massot Marc
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2025, 1003 (A27), pp.1--42. In this contribution, we develop a versatile formalism to derive unified two-phase models describing both the separated and disperse regimes as introduced by Loison et al. (2024). It relies on the stationary action principle and interface geometric variables. This contribution provides a novel method to derive small-scale models for the dynamics of the interface geometry. They are introduced here on a simplified case where all the scales and phases have the same velocity and that does not take into account large-scale capillary forces. The derivation tools yield a proper mathematical framework through hyperbolicity and signed entropy evolution. The formalism encompasses a hierarchy of small-scale reduced-order models based on a statistical description at a mesoscopic kinetic level and is naturally able to include the description of a disperse phase with polydispersity in size. This hierarchy includes both a cloud of spherical droplets and non-spherical droplets experiencing a dynamical behaviour through incompressible oscillations. The associated small-scale variables are moments of a number density function resulting from the geometric method of moments (GeoMOM). This method selects moments as small-scale geometric variables compatible with the structure and dynamics of the interface; they are defined independently of the flow topology and, therefore, this model allows the coupling of the two-scale flow with an inter-scale transfer. It is shown in particular that the resulting dynamics provides partial closures for the interface area density equation obtained from the averaging approach. (10.1017/jfm.2024.1200)
  DOI : 10.1017/jfm.2024.1200
• Brochette first-passage percolation
  
  • Marivain Maxime
  , 2025. We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point convergence theorem, identifying the time constant. In particular, we explore the case where the time constant vanishes and demonstrate the existence of a wide range of possible behaviours. Next, we prove a shape theorem, showing that the limiting shape is the $L^1$ diamond. Finally, we extend the analysis by proving a point-to-point convergence theorem in the setting where passage times are allowed to be infinite.
• Kernel density estimation for stationary random fields with values in a finite-dimensional Riemannian manifold
  
  • Nefzi Wiem
  • Yao Anne-Françoise
  • KHARDANI Salah
  , 2025. <div><p>This paper investigates some asymptotic properties of the kernel spatial density estimation for stationary α-mixing process on a finite-dimensional Riemannian manifold without boundary. The results extend beyond the classical independently and identically distributed (i.i.d.) data, focusing on the case where the manifold is known and extending the classical theory to random fields.</p></div>
• Nonparametric Regression on Riemannian manifold under α-Mixing process
  
  • Nefzi Wiem
  • Yao Anne-Françoise
  • KHARDANI Salah
  , 2025. <div><p>The main focus of our paper is to investigate the behavior of the kernel estimator for the regression function between a real-valued random variable Y and a random variable X, where X takes values in a Riemannian submanifold. The estimator is adapted from the article of Pelletier (2006). Additionally, we study data that adheres to the α-mixing condition, which imposes valuable constraints on the dependence structure of the observations. Specifically, we provide the rate of convergence in mean square error, enabling us to assess the precision and efficiency of the estimator.</p></div>
• Derivation of a 4-moment model for electron transport in Hall thrusters from a gyrokinetic model
  
  • Tazakkati Zoubaïr
  • Laguna Alejandro Alvarez
  • Massot Marc
  • Pichard Teddy
  , 2025. <div><p>We model the motion of a population of electrons in a strong electromagnetic field undergoing elastic electron/electron collisions. This regime is derived from a dimensional analysis of the electron confinement in Hall-effect thrusters. The electrons exhibit a very high cyclotron frequency and a E × B-drift, modelled by stiff PDEs at the mesoscopic scale. We obtain a gyrokinetic model in which the fastest oscillations of the system are filtered out by averaging the rotation of the electrons around the magnetic field lines. The model is derived in the strong electromagnetic field limit. Based on this gyrokinetic model, we then develop a 10-moment model. The averaging operation performed at the kinetic scale leads to symmetry properties that allow to reduce the 10-moment model to a 4-moment model.</p></div>
• Méthodes d'assimilation de données pour des simulations lagrangiennes
  
  • Duvillard Marius
  , 2025. Cette thèse porte sur le développement de méthodes d'assimilation de données pour les simulations lagrangiennes basées sur une discrétisation particulaire, avec des applications pour la simulation en mécanique des fluides. Nous étudions des situations où un ensemble de simulations et des observations à des temps discrets sont utilisés sont pour corriger l'estimation de l'état du système. Dans ce contexte, la procédure de mise à jour de la discrétisation particulaire à partir des observations disponibles constitue une problématique centrale.Dans un premier temps, nous adaptons le filtre de Kalman d'ensemble pour corriger les champs en modifiant uniquement les intensités des particules de la discrétisation. Les positions des particules restent alors inchangées ou sont régénérées sur une grille régulière, conduisant à deux méthodes distinctes.Ensuite, nous présentons une approche variationnelle d'ensemble pour corriger les positions des particules. Nous montrons que cette approche peut être combinée avec les premiers filtres pour corriger séquentiellement les positions et les intensités. Nous évaluons ces différentes méthodes sur des applications en dynamique des fluides incompressibles discrétisées par des méthodes de vortex, et nous analysons l'efficacité des filtres sur des problèmes d'advection où l'erreur de position peut être importante. (10.70675/f6d82b19z54bcz49cdzb0b9zd6b5fac44a40)
  DOI : 10.70675/f6d82b19z54bcz49cdzb0b9zd6b5fac44a40
• A stochastic algorithm for deterministic multistage optimization problems
  
  • Akian Marianne
  • Chancelier Jean-Philippe
  • Tran Benoît
  Annals of Operations Research, Springer Verlag, 2025, 345, pp.1-38. Several attempt to dampen the curse of dimensionnality problem of the Dynamic Programming approach for solving multistage optimization problems have been investigated. One popular way to address this issue is the Stochastic Dual Dynamic Programming method (SDDP) introduced by Perreira and Pinto in 1991 for Markov Decision Processes.Assuming that the value function is convex (for a minimization problem), one builds a non-decreasing sequence of lower (or outer) convex approximations of the value function. Those convex approximations are constructed as a supremum of affine cuts. On continuous time deterministic optimal control problems, assuming that the value function is semiconvex, Zheng Qu, inspired by the work of McEneaney, introduced in 2013 a stochastic max-plus scheme that builds upper (or inner) non-increasing approximations of the value function. In this note, we build a common framework for both the SDDP and a discrete time version of Zheng Qu's algorithm to solve deterministic multistage optimization problems. Our algorithm generates monotone approximations of the value functions as a pointwise supremum, or infimum, of basic (affine or quadratic for example) functions which are randomly selected. We give sufficient conditions on the way basic functions are selected in order to ensure almost sure convergence of the approximations to the value function on a set of interest. (10.1007/s10479-024-06153-8)
  DOI : 10.1007/s10479-024-06153-8
• Refined Analysis of Federated Averaging's Bias and Federated Richardson-Romberg Extrapolation
  
  • Mangold Paul
  • Durmus Alain
  • Dieuleveut Aymeric
  • Samsonov Sergey
  • Moulines Eric
  , 2025. In this paper, we present a novel analysis of FedAvg with constant step size, relying on the Markov property of the underlying process. We demonstrate that the global iterates of the algorithm converge to a stationary distribution and analyze its resulting bias and variance relative to the problem's solution. We provide a first-order bias expansion in both homogeneous and heterogeneous settings. Interestingly, this bias decomposes into two distinct components: one that depends solely on stochastic gradient noise and another on client heterogeneity. Finally, we introduce a new algorithm based on the Richardson-Romberg extrapolation technique to mitigate this bias.
• Heath-Jarrow-Morton meet lifted Heston in energy markets for joint historical and implied calibration
  
  • Abi Jaber Eduardo
  • Bruneau Soukaïna
  • de Carvalho Nathan
  • Sotnikov Dimitri
  • Tur Laurent
  , 2025. In energy markets, joint historical and implied calibration is of paramount importance for practitioners, yet notoriously challenging due to the need to align historical correlations of futures contracts with implied volatility smiles from the option market. We address this crucial problem with a multiplicative multi-factor Heath-Jarrow-Morton (HJM) model for forward curves, combined with a stochastic volatility factor coming from the lifted Heston model. We develop a sequential fast calibration procedure leveraging the Kemna-Vorst approximation of futures contracts: (i) historical correlations and the Variance Swap (VS) volatility term structure are captured through Level, Slope, and Curvature factors, (ii) the VS volatility term structure can then be corrected for a perfect match via a fixed-point algorithm, (iii) implied volatility smiles are calibrated using Fourier-based techniques. The main advantage of the proposed calibration framework is the decoupling of the calibration steps: each step tackles a simpler calibration subproblem and guaranties that the previously optimized parameters remain unchanged. Our model displays remarkable joint historical and implied calibration fits on the German power market and enables realistic interpolation within the implied volatility hypercube.
• Wavelet-Based Multiscale Flow For Realistic Image Deformation in the Large Diffeomorphic Deformation Model Framework
  
  • Gaudfernau Fleur
  • Blondiaux Eléonore
  • Allassonnière Stéphanie
  • Le Pennec Erwan
  Journal of Mathematical Imaging and Vision, Springer Verlag, 2025, 67 (2), pp.10. Estimating accurate high-dimensional transformations remains very challenging, especially in a clinical setting. In this paper, we introduce a multiscale parameterization of deformations to enhance registration and atlas estimation in the Large Deformation Diffeomorphic Metric Mapping framework. Using the Haar wavelet transform, a multiscale representation of the initial velocity fields is computed to optimize transformations in a coarse-to-fine fashion. This additional layer of spatial regularization does not modify the underlying model of deformations. As such, it preserves the original kernel Hilbert space structure of the velocity fields, enabling the algorithm to perform efficient gradient descent. Numerical experiments on several datasets, including abnormal fetal brain images, show that compared to the original algorithm, the coarse-to-fine strategy reaches higher performance and yields template images that preserve important details while avoiding unrealistic features. This highly versatile strategy can easily be applied to other mathematical frameworks for almost no additional computational cost. (10.1007/s10851-024-01219-5)
  DOI : 10.1007/s10851-024-01219-5
• Surface Waves in Randomly Perturbed Discrete Models
  
  • Garnier Josselin
  • Sharma Basant Lal
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2025, 23 (1), pp.158-186. (10.1137/24M165510X)
  DOI : 10.1137/24M165510X
• An exterior optimal transport problem
  
  • Candau-Tilh Jules
  • Goldman Michael
  • Merlet Benoît
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2025, 64 (2), pp.45. This paper deals with a variant of the optimal transportation problem. Given f ∈ L 1 (R d , [0, 1]) and a cost function c ∈ C(R d × R d) of the form c(x, y) = k(y − x), we minimise ∫ c dγ among transport plans γ whose first marginal is f and whose second marginal is not prescribed but constrained to be smaller than 1 − f. Denoting by Υ(f) the infimum of this problem, we then consider the maximisation problem sup{Υ(f) : ∫ f = m} where m &gt; 0 is given. We prove that maximisers exist under general assumptions on k, and that for k radial, increasing and coercive these maximisers are the characteristic functions of the balls of volume m. (10.1007/s00526-024-02900-8)
  DOI : 10.1007/s00526-024-02900-8
• Polynomial approximations in a generalized Nyman–Beurling criterion
  
  • Alouges François
  • Darses Sébastien
  • Hillion Erwan
  , 2023, pp.767 - 785. The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance. (10.5802/jtnb.1227)
  DOI : 10.5802/jtnb.1227
• Bayesian calibration for prediction in a multi-output transposition context
  
  • Sire Charlie
  • Garnier Josselin
  • Kerleguer Baptiste
  • Durantin Cédric
  • Defaux Gilles
  • Perrin Guillaume
  International Journal for Uncertainty Quantification, Begell House Publishers, 2025, 15 (6), pp.37-59. Numerical simulations are widely used to predict the behavior of physical systems, with Bayesian approaches being particularly well suited for this purpose. However, experimental observations are necessary to calibrate certain simulator parameters for the prediction. In this work, we use a multi-output simulator to predict all its outputs, including those that have never been experimentally observed. This situation is referred to as the transposition context. To accurately quantify the discrepancy between model outputs and real data in this context, conventional methods cannot be applied, and the Bayesian calibration must be augmented by incorporating a joint model error across all outputs. To achieve this, the proposed method is to consider additional numerical input parameters within a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. This approach is applied on a computer code with three outputs that models the Taylor cylinder impact test with a small number of observations. The outputs are considered as the observed variables one at a time, to work with three different transposition situations. The proposed method is compared with other approaches that embed model errors to demonstrate the significance of the hierarchical formulation. (10.1615/Int.J.UncertaintyQuantification.2025056586)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2025056586
• Collision-induced mass loss and mass gain on an extremely massive star
  
  • Ramírez-Galeano Laura
  • Charbonnel Corinne
  • Fragos Tassos
  • Tazakkati Zoubaïr
  • Roman-Garza Jaime
  • Gieles Mark
  Astronomy & Astrophysics - A&A, EDP Sciences, 2025, 699, pp.A223. Aims: The objective of this study is to analytically explore mass loss and gain induced by stellar collisions on a gas-accreting extremely massive star (aEMS, 10 3 M/M 10 4 ). We also consider its contribution to the mass budget in the context of forming multiple stellar populations (MPs) in a typical protoglobular cluster (GC). Methods: We used MESA to build a series of aEMS models up to 2 × 10 4 M for three [Fe/H] values, covering the metallicity range of Galactic GCs, with different treatments of super-adiabatic convection. We set analytical prescriptions to quantify collision-induced mass loss when a star spirals and deposits energy into the envelope of the aEMS. We used a Monte Carlo approach to simulate the effects of multiple collisions on an aEMS of initial mass of 10 3 M in a static proto-GC, accounting for mass loss and gain from collisions, gas accretion, and stellar winds. Results: We show that assumptions on super-adiabaticity in radiation-dominated layers significantly impact the aEMSs properties and their collision responses: extended stars tend to lose mass, while compact ones are more likely to gain it. Our MC simulations predict the total mass lost and gained, along with the corresponding timescales and contributions from stellar winds and collisions. The results are influenced by the structural characteristics of the aEMS and by the gas accretion rate during the collision phase. Under certain conditions, the EMS exhibits a "conveyor belt" behavior, processing up to 10 5.5 M of material in ∼5 Myrs. Conclusions: This study provides theoretical predictions supporting aEMSs as contributors to the abundance anomalies observed in GCs. It emphasizes the importance of including collision dynamics and mass transfer in aEMS formation and evolution models in dense stellar environments. We provide a grid of predictions for stellar M-R-[Fe/H]-structure relations and collision-induced mass loss and gain, suitable for inclusion in hydro and N-body simulations of star clusters. (10.1051/0004-6361/202453462)
  DOI : 10.1051/0004-6361/202453462
• Provable non-accelerations of the heavy-ball method
  
  • Goujaud Baptiste
  • Taylor Adrien
  • Dieuleveut Aymeric
  Mathematical Programming, Springer Verlag, 2025. In this work, we show that the heavy-ball ($\HB$) method provably does not reach an accelerated convergence rate on smooth strongly convex problems. More specifically, we show that for any condition number and any choice of algorithmic parameters, either the worst-case convergence rate of $\HB$ on the class of $L$-smooth and $μ$-strongly convex \textit{quadratic} functions is not accelerated (that is, slower than $1 - \mathcal{O}(κ)$), or there exists an $L$-smooth $μ$-strongly convex function and an initialization such that the method does not converge. To the best of our knowledge, this result closes a simple yet open question on one of the most used and iconic first-order optimization technique. Our approach builds on finding functions for which $\HB$ fails to converge and instead cycles over finitely many iterates. We analytically describe all parametrizations of $\HB$ that exhibit this cycling behavior on a particular cycle shape, whose choice is supported by a systematic and constructive approach to the study of cycling behaviors of first-order methods. We show the robustness of our results to perturbations of the cycle, and extend them to class of functions that also satisfy higher-order regularity conditions. (10.1007/s10107-025-02269-2)
  DOI : 10.1007/s10107-025-02269-2
• Understanding the worst-kept secret of high-frequency trading
  
  • Pulido Sergio
  • Rosenbaum Mathieu
  • Sfendourakis Emmanouil
  Finance and Stochastics, Springer Verlag (Germany), 2025. Volume imbalance in a limit order book is often considered as a reliable indicator for predicting future price moves. In this work, we seek to analyse the nuances of the relationship between prices and volume imbalance. To this end, we study a market-making problem which allows us to view the imbalance as an optimal response to price moves. In our model, there is an underlying efficient price driving the mid-price, which follows the model with uncertainty zones. A single market maker knows the underlying efficient price and consequently the probability of a mid-price jump in the future. She controls the volumes she quotes at the best bid and ask prices. Solving her optimization problem allows us to understand endogenously the price-imbalance connection and to confirm in particular that it is optimal to quote a predictive imbalance. Our model can also be used by a platform to select a suitable tick size, which is known to be a crucial topic in financial regulation. The value function of the market maker's control problem can be viewed as a family of functions, indexed by the level of the market maker's inventory, solving a coupled system of PDEs. We show existence and uniqueness of classical solutions to this coupled system of equations. In the case of a continuous inventory, we also prove uniqueness of the market maker's optimal control policy.
• Inexact subgradient methods for semialgebraic functions
  
  • Bolte Jérôme
  • Le Tam
  • Moulines Éric
  • Pauwels Edouard
  Mathematical Programming, Series A, Springer, 2025. 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. (10.1007/s10107-025-02245-w)
  DOI : 10.1007/s10107-025-02245-w