Publications

2025

• Bridging multifluid and drift-diffusion models for bounded plasmas
  
  • Gangemi G M
  • Alvarez Laguna Alejandro
  • Massot M.
  • Hillewaert K.
  • Magin T.
  Physics of Plasmas, American Institute of Physics, 2025, 32 (2), pp.023502. Fluid models represent a valid alternative to kinetic approaches in simulating low-temperature discharges: a well-designed strategy must be able to combine the ability to predict a smooth transition from the quasineutral bulk to the sheath, where a space charge is built at a reasonable computational cost. These approaches belong to two families: multifluid models, where momenta of each species are modeled separately, and drift-diffusion models, where the dynamics of particles is dependent only on the gradient of particle concentration and on the electric force. It is shown that an equivalence between the two models exists and that it corresponds to a threshold Knudsen number, in the order of the square root of the electron-to-ion mass ratio; for an argon isothermal discharge, this value is given by a neutral background pressure Pn≳1000 Pa. This equivalence allows us to derive two analytical formulas for a priori estimation of the sheath width: the first one does not need any additional hypothesis but relies only on the natural transition from the quasineutral bulk to the sheath; the second approach improves the prediction by imposing a threshold value for the charge separation. The new analytical expressions provide better estimations of the floating sheath dimension in collisions-dominated regimes when tested against two models from the literature. (10.1063/5.0240640)
  DOI : 10.1063/5.0240640
• 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
• Inverse scattering for the multipoint potentials of Bethe-Peierls-Thomas-Fermi type
  
  • Kuo Pei-Cheng
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2025, 41 (6), pp.065021. <div><p>We consider the Schrödinger equation with a multipoint potential of the Bethe-Peierls-Thomas-Fermi type. We show that such a potential in dimension d = 2 or d = 3 is uniquely determined by its scattering amplitude at a fixed positive energy. Moreover, we show that there is no non-zero potential of this type with zero scattering amplitude at a fixed positive energy and a fixed incident direction. Nevertheless, we also show that a multipoint potential of this type is not uniquely determined by its scattering amplitude at a positive energy E and a fixed incident direction. Our proofs also contribute to the theory of inverse source problem for the Helmholtz equation with multipoint source.</p></div> (10.1088/1361-6420/ade282)
  DOI : 10.1088/1361-6420/ade282
• PSWF-Radon approach to reconstruction from band-limited Hankel transform
  
  • Goncharov Fedor
  • Isaev Mikhail
  • Novikov Roman
  • Zaytsev Rodion
  Applied and Computational Harmonic Analysis, Elsevier, 2025. We give new formulas for reconstructions from band-limited Hankel transform of integer or half-integer order. Our formulas rely on the PSWF-Radon approach to super-resolution in multidimensional Fourier analysis. This approach consists of combining the theory of classical one-dimensional prolate spheroidal wave functions with the Radon transform theory. We also use the relation between Fourier and Hankel transforms and Cormack-type inversion of the Radon transform. Finally, we investigate numerically the capabilities of our approach to super-resolution for band-limited Hankel inversion in relation to varying levels of noise.
• Byzantine-Robust Gossip: Insights from a Dual Approach
  
  • Gaucher Renaud
  • Dieuleveut Aymeric
  • Hendrikx Hadrien
  , 2025. Distributed learning has many computational benefits but is vulnerable to attacks from a subset of devices transmitting incorrect information. This paper investigates Byzantine resilient algorithms in a decentralized setting, where devices communicate directly in a peer-to-peer manner within a communication network. We leverage the so-called dual approach for decentralized optimization and propose a Byzantine-robust algorithm. We provide convergence guarantees in the average consensus subcase, discuss the potential of the dual approach beyond this subcase, and re-interpret existing algorithms using the dual framework. Lastly, we experimentally show the soundness of our method.
• Taking Advantage of Multiple Scattering for Optical Reflection Tomography
  
  • Wasik Thomas
  • Barolle Victor
  • Aubry Alexandre
  • Garnier Josselin
  , 2025. <div><p>Optical Diffraction Tomography (ODT) is a powerful non-invasive imaging technique widely used in biological and medical applications. While significant progress has been made in transmission configuration, reflection ODT remains challenging due to the ill-posed nature of the inverse problem. We present a novel optimization algorithm for 3D refractive index (RI) reconstruction in reflection-mode microscopy. Our method takes advantage of the multiply-scattered waves that are reflected by uncontrolled background structures and that illuminate the foreground RI from behind. It tackles the ill-posed nature of the problem using weighted time loss, positivity constraints and Total Variation regularization. We have validated our method with data generated by detailed 2D and 3D simulations, demonstrating its performance under weak multiple scattering conditions and with simplified forward models used in the optimization routine for computational efficiency.</p><p>In addition, we highlight the need for multi-wavelength analysis and the use of regularization to ensure the reconstruction of the low spatial frequencies of the foreground RI.</p></div>
• Long time behavior of a degenerate stochastic system modeling the response of a population face to environmental impacts
  
  • Collet Pierre
  • Ecotière Claire
  • Méléard Sylvie
  Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2025, 30 (none). We study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two different behaviors. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. That allows to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibrium of the dynamical system having the smallest diffusion coefficient. (10.1214/24-ECP650)
  DOI : 10.1214/24-ECP650
• Optimisation of space-time periodic eigenvalues
  
  • Bogosel Beniamin
  • Mazari Idriss
  • Nadin Grégoire
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2025. <div><p>The goal of this paper is to provide a qualitative analysis of the optimisation of space-time periodic principal eigenvalues. Namely, considering a fixed time horizon T and the d-dimensional torus T d , let, for any m ∈ L ∞ ((0, T ) × T d ), λ(m) be the principal eigenvalue of the operator ∂t -∆ -m endowed with (time-space) periodic boundary conditions. The main question we set out to answer is the following: how to choose m so as to minimise λ(m)? This question stems from population dynamics. We prove that in several cases it is always beneficial to rearrange m with respect to time in a symmetric way, which is the first comparison result for the rearrangement in time of parabolic equations. Furthermore, we investigate the validity (or lack thereof) of Talenti inequalities for the rearrangement in time of parabolic equations. The numerical simulations which illustrate our results were obtained by developing a framework within which it is possible to optimise criteria with respect to functions having a prescribed rearrangement (or distribution function).</p></div> (10.2422/2036-2145.202501_012)
  DOI : 10.2422/2036-2145.202501_012
• Adaptive Destruction Processes for Diffusion Samplers
  
  • Gritsaev Timofei
  • Morozov Nikita
  • Tamogashev Kirill
  • Tiapkin Daniil
  • Samsonov Sergey
  • Naumov Alexey
  • Vetrov Dmitry
  • Malkin Nikolay
  , 2025. This paper explores the challenges and benefits of a trainable destruction process in diffusion samplers -- diffusion-based generative models trained to sample an unnormalised density without access to data samples. Contrary to the majority of work that views diffusion samplers as approximations to an underlying continuous-time model, we view diffusion models as discrete-time policies trained to produce samples in very few generation steps. We propose to trade some of the elegance of the underlying theory for flexibility in the definition of the generative and destruction policies. In particular, we decouple the generation and destruction variances, enabling both transition kernels to be learned as unconstrained Gaussian densities. We show that, when the number of steps is limited, training both generation and destruction processes results in faster convergence and improved sampling quality on various benchmarks. Through a robust ablation study, we investigate the design choices necessary to facilitate stable training. Finally, we show the scalability of our approach through experiments on GAN latent space sampling for conditional image generation. (10.48550/arXiv.2506.01541)
  DOI : 10.48550/arXiv.2506.01541
• Efficient treatment of the model error in the calibration of computer codes: the Complete Maximum a Posteriori method
  
  • Kahol Omar
  • Congedo Pietro Marco
  • Le Maitre Olivier
  • Denimal Goy Enora
  International Journal for Uncertainty Quantification, Begell House Publishers, 2025. <div><p>Computer models are widely used for the prediction of complex physical phenomena. Based on observations of these physical phenomena, it is possible to calibrate the model parameters. In most cases, such computer models are mis-specified, and the calibration process must be improved by including a model error term. The model error hyperparameters are, however, rarely learned jointly with the model parameters to reduce the dimensionality of the problem. Sequential and non-sequential approaches have been introduced to estimate the hyperparameters. The former, such as the Kennedy and O'Hagan (KOH) framework, estimates the model error hyperparameters before calibrating the model parameters. The latter, such as the Full Maximum a Posteriori (FMP), introduces a functional dependence between the model parameters and the model error hyperparameters. Despite being more reliable in some cases (bimodality e.g.), the FMP method still fails to estimate correctly the posterior distribution shape. This work proposes a new methodology for treating the model error term in computer code calibration. It builds upon the KOH and FMP framework. Called the Complete Maximum a Posteriori (CMP) method, it provides a closed-form expression for the marginalization integral over the model error hyperparameters, significantly reducing the dimensionality of the calibration problem. Such expression re- lies on a set of assumptions that are more general and less stringent than the ones usually employed. The CMP method is applied to four examples of increasing complexity, from elementary to real fluid dynamics problems, including or not bimodality. Compared to the true reference solution and unlike the KOH and FMP, the CMP method correctly captures the shape of the posterior distribution, including all modes and their weights. Moreover, it provides an accurate estimate of the distribution tails</p></div> (10.1615/Int.J.UncertaintyQuantification.2025056317)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2025056317
• Crediting football players for creating dangerous actions in an unbiased way: the generation of threat (GoT) indices
  
  • Baouan Ali
  • Coustou Sebastien
  • Lacome Mathieu
  • Pulido Sergio
  • Rosenbaum Mathieu
  Journal of Quantitative Analysis in Sports, De Gruyter, 2025. We introduce an innovative methodology to identify football players at the origin of threatening actions in a team. In our framework, a threat is defined as entering the opposing team's danger area. We investigate the timing of threat events and ball touches of players, and capture their correlation using Hawkes processes. Our model-based approach allows us to evaluate a player's ability to create danger both directly and through interactions with teammates. We define a new index, called Generation of Threat (GoT), that measures in an unbiased way the contribution of a player to threat generation. For illustration, we present a detailed analysis of Chelsea's 2016-2017 season, with a standout performance from Eden Hazard. We are able to credit each player for his involvement in danger creation and determine the main circuits leading to threat. In the same spirit, we investigate the danger generation process of Stade Rennais in the 2021-2022 season. Furthermore, we establish a comprehensive ranking of Ligue 1 players based on their generated threat in the 2021-2022 season. Our analysis reveals surprising results, with players such as Jason Berthomier, Moses Simon and Frederic Guilbert among the top performers in the GoT rankings. We also present a ranking of Ligue 1 central defenders in terms of generation of threat and confirm the great performance of some center-back pairs, such as Nayef Aguerd and Warmed Omari.
• Signature volatility models: pricing and hedging with Fourier
  
  • Abi Jaber Eduardo
  • Gérard Louis-Amand
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2025, 16 (2). We consider a stochastic volatility model where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinitedimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study. (10.1137/24M1636952)
  DOI : 10.1137/24M1636952
• Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025 (1930-8337). <div><p>In this paper, we present a mathematical model and analysis for a new experimental method [Bureau and al., arXiv:2409.13901, 2024] for effective sound velocity estimation in medical ultrasound imaging. We perform a detailed analysis of the point spread function of a medical ultrasound imaging system when there is a mismatch between the effective sound speed in the medium and the one used in the backpropagation imaging functional. Based on this analysis, an estimator for the speed of sound error is introduced. Using recent results on stochastic homogenization of the Helmholtz equation, we provide a representation formula for the field scattered by a random multi-scale medium (whose acoustic behavior is similar to a biological tissue) in the time-harmonic regime. We then prove that statistical moments of the imaging function can be accessed from data collected with only one realization of the medium. We show that it is possible to locally extract the point spread function from an image constituted only of speckle and build an estimator for the effective sound velocity in the micro-structured medium. Some numerical illustrations are presented at the end of the paper.</p></div> (10.3934/ipi.2026001)
  DOI : 10.3934/ipi.2026001
• Partial regularity for optimal transport with $p$-cost away from fixed points
  
  • Goldman Michael
  • Koch Lukas
  Proceedings of the American Mathematical Society, American Mathematical Society, 2025. We consider maps $T$ solving the optimal transport problem with a cost $c(x-y)$ modeled on the $p$-cost. For Hölder continuous marginals, we prove a $C^{1,\alpha}$-partial regularity result for $T$ in the set $\{\lvert T(x)-x\rvert&gt;0\}$.
• Learning homogenized hyperelastic behavior for topology optimization of lattice structures
  
  • Ribeiro Nogueira Breno
  • Allaire Grégoire
  , 2025. (10.5281/zenodo.14900138)
  DOI : 10.5281/zenodo.14900138
• Finite elements for Wasserstein $W_p$ gradient flows
  
  • Cancès Clément
  • Matthes Daniel
  • Nabet Flore
  • Rott Eva-Maria
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (3), pp.1565-1600. Wasserstein $\bbW_p$ gradient flows for nonlinear integral functionals of the density yield degenerate parabolic equations involving diffusion operators of $q$-Laplacian type, with $q$ being $p$'s conjugate exponent. We propose a finite element scheme building on conformal $\mathbb{P}_1$ Lagrange elements with mass lumping and a backward Euler time discretization strategy. Our scheme preserves mass and positivity while energy decays in time. Building on the theory of gradient flows in metric spaces, we further prove convergence towards a weak solution of the PDE that satisfies the energy dissipation equality. The analytical results are illustrated by numerical simulations. (10.1051/m2an/2025035)
  DOI : 10.1051/m2an/2025035
• Volume growth of Funk geometry and the flags of polytopes
  
  • Faifman Dmitry
  • Vernicos Constantin
  • Walsh Cormac
  Geometry and Topology, Mathematical Sciences Publishers, 2025, 29 (7), pp.3773-3811. We consider the Holmes-Thompson volume of balls in the Funk geometry on the interior of a convex domain. We conjecture that for a fixed radius, this volume is minimized when the domain is a simplex and the ball is centered at the barycenter, or in the centrally-symmetric case, when the domain is a Hanner polytope. This interpolates between Mahler's conjecture and Kalai's flag conjecture. We verify this conjecture for unconditional domains. For polytopal Funk geometries, we study the asymptotics of the volume of balls of large radius, and compute the two highest-order terms. The highest depends only on the combinatorics, namely on the number of flags. The second highest depends also on the geometry, and thus serves as a geometric analogue of the centro-affine area for polytopes. We then show that for any polytope, the second highest coefficient is minimized by a unique choice of center point, extending the notion of Santaló point. Finally, we show that, in dimension two, this coefficient, with respect to the minimal center point, is uniquely maximized by affine images of the regular polygon. (10.2140/gt.2025.29.3773)
  DOI : 10.2140/gt.2025.29.3773
• A holographic uniqueness theorem for the two-dimensional Helmholtz equation
  
  • Nair Arjun
  • Novikov Roman
  The Journal of Geometric Analysis, Springer, 2025, 35 (4), pp.123. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb R^2$. We consider a straight line in this region, such that the direction of propagation of the plane wave is not parallel to this line. We show that the radiation solution in the exterior region is uniquely determined by the intensity of the total solution on an interval of this line. In particular, this result solves one of the old mathematical questions of holography in its two-dimensional setting. Our proofs also contribute to the theory of the Karp expansion of radiation solutions in two dimensions. (10.1007/s12220-025-01949-x)
  DOI : 10.1007/s12220-025-01949-x
• Objective assessment of cardiac function using patient-specific biophysical modeling based on cardiovascular MRI combined with catheterization
  
  • Gusseva Maria
  • Castellanos Daniel Alexander
  • Veeram Reddy Surendranath
  • Hussain Tarique
  • Chapelle Dominique
  • Chabiniok Radomír
  AJP - Heart and Circulatory Physiology, American Physiological Society, 2025, 329 (5), pp.H118-H1191. Synthesizing multi-modality data, such as cardiovascular magnetic resonance imaging (MRI) combined with catheterization, into a single framework is challenging. Different acquisition systems are subjected to different measurement errors. Coupling clinical data with biomechanical models can assist in clinical data processing (e.g., model-based filtering of measurement noise) and quantify myocardial mechanics via metrics not readily available in the data, such as myocardial contractility. In this work we use a biomechanical modeling with the aim 1) to quantitatively compare model- and data-derived signals, and 2) to explore the potential of model-derived myocardial contractility and distal resistance of the circulation (Rd) to robustly quantify cardiovascular physiology. We used 51 ventricular catheterization pressure and cine MRI volume datasets from patients with single-ventricle physiology and left and right ventricles of patients with repaired tetralogy of Fallot. Ventricular time-varying elastance (TVE) metrics and linear regression were used to quantify the relationship between the maximum value of TVE (Emax) and maximum time derivative of ventricular pressure (max(dP/dt)) in data- and model-derived pressure and volume signals at p&lt;0.05. Pearson’s correlations were used to compare model-derived contractility and data-derived Emax and max(dP/dt), and model-derived Rd and data-derived vascular resistance. All data and model-derived linear regressions were significant (p&lt;0.05). Model-derived max(dP/dt) vs. data-derived Emax produced higher R2 than data-derived max(dP/dt) vs. data-derived Emax. Correlations demonstrated significant relationships between most data- and model-derived metrics. This work revealed the clinical value of biomechanical modeling to assist in clinical data processing by providing high-quality pressure and volume signals, and to quantify cardiovascular pathophysiology. (10.1152/ajpheart.00232.2025)
  DOI : 10.1152/ajpheart.00232.2025
• Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing
  
  • Bonet Clément
  • Cazelles Elsa
  • Drumetz Lucas
  • Courty Nicolas
  , 2025. The Busemann function has recently found much interest in a variety of geometric machine learning problems, as it naturally defines projections onto geodesic rays of Riemannian manifolds and generalizes the notion of hyperplanes. As several sources of data can be conveniently modeled as probability distributions, it is natural to study this function in the Wasserstein space, which carries a rich formal Riemannian structure induced by Optimal Transport metrics. In this work, we investigate the existence and computation of Busemann functions in Wasserstein space, which admits geodesic rays. We establish closed-form expressions in two important cases: one-dimensional distributions and Gaussian measures. These results enable explicit projection schemes for probability distributions on $\mathbb{R}$, which in turn allow us to define novel Sliced-Wasserstein distances over Gaussian mixtures and labeled datasets. We demonstrate the efficiency of those original schemes on synthetic datasets as well as transfer learning problems. (10.48550/arXiv.2510.04579)
  DOI : 10.48550/arXiv.2510.04579
• 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
• 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