Publications

2025

• 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 $\{\lvert T(x)-x\rvert>0\}$.
• 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
• 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.
• 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
• 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
• 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<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<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
• 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
• 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