Publications

2026

• Surrogate-Based Strategies for Accelerated Bayesian Calibration of Computer Codes With Complete Maximum a Posteriori Estimation of Model Error
  
  • Kahol Omar
  • Le Maître Olivier
  • Congedo Pietro M
  • Denimal Goy Enora
  Journal of Mechanical Design, American Society of Mechanical Engineers, 2026, 148 (9). The calibration of a computer code is a process that reduces the uncertainty of model parameters by matching the code’s predictions to experimental observations of a quantity of interest. A more faithful representation of the global uncertainty is achieved by including a model error term, a discrepancy between the physical system and the computer code. The recently proposed complete maximum a posteriori (CMP) method is able to infer both a posterior distribution of the model parameters and a model error term, improving upon traditional frameworks. On the other hand, the CMP method relies on an optimization step which increases the cost of complex calibration problems. This article proposes a surrogate-based strategy to reduce the computational cost of the CMP method. First, we build a surrogate model of the model error’s hyperparameters using Gaussian processes. Second, we propose an iterative algorithm that builds a training set in regions of the parameter space that are more likely, reducing the overall cost of the algorithm and improving the accuracy of the surrogate. The proposed strategy is applied to four different examples, including a design problem in solid mechanics and a complex test case in fluid dynamics. The results show that the proposed strategy is able to accelerate the CMP method without losing accuracy, making it suitable for real-world applications. In an industrial application, we demonstrate a speed-up of almost 100 compared to the original CMP method. (10.1115/1.4071071)
  DOI : 10.1115/1.4071071
• On the simulation of extreme events with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2026. This article aims at investigating the use of generative methods based on neural networks to simulate extreme events. Although very popular, these methods are mainly invoked in empirical works. Therefore, providing theoretical guidelines for using such models in extreme values context is of primal importance. To this end, we propose an overview of most recent generative methods dedicated to extremes, giving some theoretical and practical tips on their tail behaviour thanks to both extreme-value and copula tools.
• Entropic Mirror Monte Carlo
  
  • Cherradi Anas
  • Janati Yazid
  • Durmus Alain
  • Le Corff Sylvain
  • Petetin Yohan
  • Stoehr Julien
  , 2026. Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.
• Numerical analysis of an optimal control approach to solve a tsunami inverse problem
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This paper concerns the reconstruction of an abrupt bottom displacement of the ocean from the measurement of the induced perturbation of the free surface, which is a severely ill-posed inverse problem. This problem is solved by using an optimal control approach, the physics being governed by a time evolution system based on a simple oceanography model. We firstly recast the problem in an abstract framework, secondly propose an implicit Euler scheme for the time discretization combined with a Finite Element method for the space discretization. The main result is an error estimate between the solution to the discrete control optimal problem and the solution to the continuous optimal problem, which is obtained by considering the discrete and continuous weak mixed formulations that characterize the optimality for these two problems. Some numerical experiments illustrate the efficiency of our approach and the consistency of our error estimate.
• An inverse tsunami problem in the time domain: a well-posedness analysis of the forward problem and an inversion strategy based on a mixed formulation of the Tikhonov regularization
  
  • Bourgeois Laurent
  • Moireau Philippe
  • Terrine Raphaël
  , 2026. This contribution concerns an inverse problem related to a tsunami in the ocean, the tsunami being caused by a submarine earthquake. Considering the very beginning of the phenomenon, a simple linear model incorporating both gravity and acoustic waves is proposed. The main objective is to develop a strategy to solve the inverse problem of retrieving the bottom displacement from the induced free surface perturbation. Such strategy is based on a mixed formulation of the Tikhonov regularization in the space/time domain, the regularization parameter being determined by using the Morozov principle by means of duality in optimization. Some numerical experiments in 2D, which rely on a tensorized finite element method, show that our strategy is effective. A secondary objective is to prove existence and uniqueness of both strong and variational solutions to the forward problem.
• A robust computational framework for the mixture-energy-consistent six-equation two-phase model with instantaneous mechanical relaxation terms
  
  • Orlando Giuseppe
  • Haegeman Ward
  • Pelanti Marica
  • Massot Marc
  , 2026. We present a robust computational framework for the numerical solution of a hyperbolic 6-equation single-velocity two-phase system. The system's main interest is that, when combined with instantaneous mechanical relaxation, it recovers the solution of the 5-equation model of Kapila. Several numerical methods based on this strategy have been developed over the years. However, neither the 5- nor 6-equation model admits a complete set of jump conditions because they involve non-conservative products. Different discretizations of these terms in the 6-equation model exist. The precise impact of these discretizations on the numerical solutions of the 5-equation model, in particular for shocks, is still an open question to which this work provides new insights. We consider the phasic total energies as prognostic variables to naturally enforce discrete conservation of total energy and compare the accuracy and robustness of different discretizations for the hyperbolic operator. Namely, we discuss the construction of an HLLC approximate Riemann solver in relation to jump conditions. We then compare an HLLC wave-propagation scheme which includes the non-conservative terms, with Rusanov and HLLC solvers for the conservative part in combination with suitable approaches for the non-conservative terms. We show that some approaches for the discretization of non-conservative terms fit within the framework of path-conservative schemes for hyperbolic problems. We then analyze the use of various numerical strategies on several relevant test cases, showing both the impact of the theoretical shortcomings of the models as well as the importance of the choice of a robust framework for the global numerical strategy.
• Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES
  
  • Matalon Pierre
  • Spillane Nicole
  , 2026. The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pták, Strakoš, 1996] is that, given any convergence curve, one can build a linear system for which GMRES realizes that convergence curve. What is even more extraordinary is that the eigenvalues of the problem matrix can be chosen arbitrarily. We build upon this idea to derive novel results about weighted GMRES. We prove that for any linear system and any prescribed convergence curve, there exists a weight matrix M for which weighted GMRES (i.e. GMRES in the inner product induced by M ) realizes that convergence curve, and we characterize the form of M . Additionally, we exhibit a necessary and sufficient condition on M for the simultaneous prescription of two convergence curves, one realized by GMRES in the Euclidean inner product, and the other in the inner product induced by M . These results are then applied to infer some properties of preconditioned GMRES when the preconditioner is applied either on the left or on the right. For instance, we show that any two convergence curves are simultaneously possible for left and right preconditioned GMRES.
• Volterra clocks and their pure-jump limits: hitting times of curved boundaries
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Søjmark Andreas
  , 2026. We introduce a class of continuous Volterra processes, called Volterra clocks, and study their singular limit as the memory kernel collapses to a Dirac mass at zero. The dynamics are parametrised by a function f acting as a nonlinear time- change, generalising the Volterra square-root process and recovering it when f is affine. In the singular limit, the continuous Volterra clock converges weakly to a pure-jump process given by first passage times of a Brownian motion to curved boundaries, including affine and square-root boundaries when f is, respectively, affine or quadratic. Outside the affine setting, characteristic function methods are no longer available, and we instead identify the limit directly from the dynamics. We do this through a topological framework adapted to the time-change structure which involves Skorokhod’s M1 topology and a decorated notion of convergence. Our analysis unifies several regimes of interest for general Volterra clocks, including large-time asymptotics, fast mean reversion, and hyper-roughness. In particular, this subsumes and extends existing results in the affine setting.
• Assessing Per-Sample Membership Inference Vulnerability without Retraining
  
  • Dorseuil Valentin
  • Atif Jamal
  • Cappé Olivier
  , 2026. Recent work in the privacy literature shows that sample-targeted membership inference attacks (MIAs) significantly outperform untargeted approaches by a wide margin. Motivated by this observation, we address the following question: can the privacy vulnerability of individual training points be assessed without training shadow models? We show that per-sample exposure to MIA is governed not only by a point's loss, but also by a data-dependent geometric measure. In the linear setting, we derive a closed-form decomposition of individual black-box MIA vulnerability into a population leverage score and a residual loss term, making explicit how sample-dependent geometry translates into privacy exposure. Since the final layer of most modern architectures is linear, we extend this framework to deep networks and propose a surrogate score operating on last-layer representations that requires only a single trained model and no shadow models. Empirical evaluations across diverse datasets and architectures show that our score outperforms loss and gradient-norm baselines at identifying the highest-risk points under state-of-the-art attacks, providing a computationally efficient and theoretically grounded tool for per-sample privacy risk assessment.
• High-order adaptive discontinuous finite elements for the shallow water equations with sub-grid irregular bathymetry
  
  • Arpaia Luca
  • Orlando Giuseppe
  • Ferrarin Christian
  • Bonaventura Luca
  , 2026. We present a discontinuous finite element method for the shallow water equations which exploits high-resolution realistic bathymetry data without any regularity assumption, also in the case of high-order discretizations. We prove a number of mathematical properties specific to the proposed method that is well-balanced, mass-conserving and positivity-preserving under a mild CFL condition also in the presence of wet-dry fronts. The method includes a consistent conservative discretization for passive tracers. We use a high-order Discontinuous Galerkin (DG) method as implemented in the deal.II library. This environment provides efficient and native parallelization techniques and automatically handles non-conforming meshes to implement adaptive strategies which are tested in a coastal environment. Idealized test cases show the robustness in presence of irregular bathymetries also with under-resolved features at the grid scale. A benchmark with realistic bathymetry and a complex domain shows the potential of the proposed discretization for adaptive simulations of coastal flows.
• Certified Per-Instance Unlearning Using Individual Sensitivity Bounds
  
  • Benarroch Hanna
  • Atif Jamal
  • Cappé Olivier
  , 2026. Certified machine unlearning can be achieved via noise injection leading to differential privacy guarantees, where noise is calibrated to worst-case sensitivity. Such conservative calibration often results in performance degradation, limiting practical applicability. In this work, we investigate an alternative approach based on adaptive per-instance noise calibration tailored to the individual contribution of each data point to the learned solution. This raises the following challenge: how can one establish formal unlearning guarantees when the mechanism depends on the specific point to be removed? To define individual data point sensitivities in noisy gradient dynamics, we consider the use of per-instance differential privacy. For ridge regression trained via Langevin dynamics, we derive high-probability per-instance sensitivity bounds, yielding certified unlearning with substantially less noise injection. We corroborate our theoretical findings through experiments in linear settings and provide further empirical evidence on the relevance of the approach in deep learning settings.
• Erratum for Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models
  
  • Giovangigli Vincent
  • Nabet Flore
  , 2023. An inaccurate assumption has been identified in Vincent Giovangigli, Yoann Le Calvez and Flore Nabet ``Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models'', J.~Math.~Fluid Dyn., Volume 25, 82, (2023),https://doi.org/10.1007/s00021-023-00825-4 and is corrected here.
• Optimisation topologique robuste avec microstructure incertaine pour la fabrication additive métallique
  
  • Masson Hugo
  • Denimal Goy Enora
  • Peigney Michael
  • Renson Ludovic
  , 2026, pp.8 p.. Ce travail propose de prendre en compte la variabilité de microstructure observée en fabrication additive métallique à travers un modèle matériau simple et incertain. Ce modèle est intégré à une optimisation topologique par densité, pour une minimisation de souplesse. Les incertitudes son propagées par quadrature de Gauss. La méthode réduit la variabilité de performance des formes optimisées, même dans le cas de propriétés matériaux réalistes. Des premiers résultats sur un cas industriel d’aube en 3D sont présentés.
• Development of a 2D test bench for characterising the dynamic mechanical response of a solid rocket propellant under rapid depressurisation
  
  • Distelzwey Léo
  • Levard Quentin
  • François Laurent
  • Voreux Olivier
  , 2026. The ignition of a solid rocket propellant is a critical moment during which the material is subjected to a rapid and large variation in chamber pressure. This occurs especially when the operculum sealing the throat is blown off as the pressure rises. Depending on the internal geometry, the ensuing pressure drop can sometimes cause rapid deformation of the propellant. In this paper we present the development of an experimental setup dedicated to the study of the 2D dynamic response of a solid propellant to propagating pressure waves. The experiment is first intended to assess whether the propellant’s response can be measured using various diagnostic techniques. The rectangular (parallelepiped‑shaped) chamber is slowly filled with air until the operculum bursts, causing the rapid depressurisation of the chamber. The resulting pressure wave travels through the chamber, impinging a specific shape of the propellant load, and induces oscillation as the wave reflects off the walls. The results show that the propellant obstacle, made of a conventional AP/HTPB research composition, oscillates in response to the pressure excitation as well as its internal mechanical forces. The test campaign is carried out at various pressure levels by adjusting the operculum thickness and involves complementary measurement techniques, such as pressure sensors that allow us to track the wave travelling through the chamber, while strain gauges measure the deformation directly on the propellant obstacle. Specific optical diagnostics (digital image correlation and point-tracking) are performed through side‑mounted windows. The setup and the measurement methods give very promising results for the characterisation of the dynamic behaviour of solid propellants during ignition, offering useful data for the development of mechanical models of the solid propellant. (10.60711/SPC2026.20260603.225861734990091247)
  DOI : 10.60711/SPC2026.20260603.225861734990091247
• Weak solutions of stochastic volterra equations in convex domains with general kernels
  
  • Abi Jaber Eduardo
  • Alfonsi Aurélien
  • Szulda Guillaume
  , 2026. 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
• POT Python Optimal Transport
  
  • Flamary Rémi
  • Vincent-Cuaz Cédric
  • Courty Nicolas
  • Gramfort Alexandre
  • Kachaiev Oleksii
  • Quang Tran Huy
  • David Laurène
  • Bonet Clément
  • Cassereau Nathan
  • Gnassounou Theo
  • Tanguy Eloi
  • Delon Julie
  • Collas Antoine
  • Mazelet Sonia
  • Chapel Laetitia
  • Kerdoncuff Tanguy
  • Yu Xizheng
  • Feickert Matthew
  • Krzakala Paul
  • Liu Tianlin
  • Fernandes Montesuma Eduardo
  , 2026. (10.5281/ZENODO.17161062)
  DOI : 10.5281/ZENODO.17161062
• Geometry of Relaxed Fair Regression: A Unified Framework for Aware and Unaware Settings
  
  • Generali Lince Marie
  • Divol Vincent
  • Flamary Rémi
  • Gaucher Solenne
  • Loiseau Patrick
  , 2026. Fairness-accuracy trade-offs are a central concern in the deployment of fairness-aware machine learning methods. When sensitive attributes are unavailable at inference time–the so called unawareness setting, principled methods for obtaining accurate predictions under relaxed fairness constraints are largely missing. In this work, we address this gap by formulating regression under a demographic parity penalty as an optimal transport problem. Our framework unifies both the \emph{aware} and \emph{unaware} settings and characterizes optimal prediction functions via optimal transport maps, under both squared Wasserstein-2 and Total Variation penalties. These results reveal that the choice of penalty reflects fundamentally different fairness philosophies: the Wasserstein penalty induces a smooth, population-wide compromise, while Total Variation enforces exact parity for a subset of individuals. Building on these theoretical characterizations, we propose an algorithm that is simple to implement, computationally efficient, and consistently matches or outperforms state-of-the-art baselines on real-world benchmarks.
• A class of optimal virtual fields for inverse problems in elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2026, 354 (G1), pp.417-449. This work addresses the identification of nonhomogeneous constitutive parameters from full-field measurements in both linear and nonlinear elasticity, considering incompressible as well as compressible materials. The inverse identification procedure relies on the Virtual Fields Method (VFM), which is based on the principle of virtual work with specifically chosen virtual fields. We propose an optimal class of virtual fields, designed to optimize the reconstruction stability with respect to measurement noise. A series of numerical experiments illustrate the effectiveness of the proposed approach. The method exhibits moderate sensitivity to measurement noise and remains robust even when the boundary conditions are only partially known. (10.5802/crmeca.361)
  DOI : 10.5802/crmeca.361
• Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing
  
  • Bonet Clément
  • Cazelles Elsa
  • Drumetz Lucas
  • Courty Nicolas
  , 2026. 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
• A characterization of Generalized functions of Bounded Deformation
  
  • Chambolle Antonin
  • Crismale Vito
  Journal of Functional Analysis, Elsevier, 2026, 290 (9), pp.111391. <div><p>We show that Dal Maso's GBD space, introduced for tackling crack growth in linearized elasticity, can be defined by simple conditions in a finite number of directions of slicing.</p></div> (10.1016/j.jfa.2026.111391)
  DOI : 10.1016/j.jfa.2026.111391
• Regularity of solutions of the Dyson equation and applications
  
  • Pesce Valentin
  , 2026. The goal of this short paper is to investigate the regularity of the solutions of the Dyson equation. In the work of Bertucci and al. [3, 4, 5], a new notion of solutions for the Dyson equation has been introduced using the viscosity solutions theory and they proved a regularization of solutions in L^{∞} . According to the work of Biane [6] we should expect a regularization in C^{1/3} of solutions (and not better). In the spirit of the works of Bertucci and al., we shall prove using PDE methods that for almost all time t \ge 0 the solution is as expected in C^{1/3} . Our approach allows us to extend this result in addition to a drift term in the Dyson equation for which the semi explicit solutions is not necessarily known. We shall also give an application of this result, proving that the solutions of the periodic Dyson equation converge in long-time toward the uniform distribution on the circle in L^{∞} norm which was an open question in [5].
• Numerical approximation of ergodic BSDEs using non linear Feynman-Kac formulas
  
  • Gobet Emmanuel
  • Richou Adrien
  • Szpruch Lukasz
  Stochastic Processes and their Applications, Elsevier, 2026, 195, pp.104871. In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantees the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions. (10.1016/j.spa.2026.104871)
  DOI : 10.1016/j.spa.2026.104871
• On phase retrieval for continuous and discrete Fourier transforms
  
  • Novikov Roman
  • Xu Tianli
  , 2026. We continue studies on phase retrieval for continuous and discrete Fourier transforms in multidimensions. Using finite difference operators, we give a large class of unexpected examples of non-uniqueness for this problem, including examples with the sparsity condition. A prototype of this construction in the continuous case is given in the work Novikov, Xu (JFAA, 2026), using linear differential operators. The construction of the present work also yields a large class of non-trivial Pauli partners, i.e., different functions with the same intensities in both configuration and Fourier domains. Besides, our construction yields examples that solve an old open question in phase retrieval with background information arising in many areas including Fourier holography.
• A Rank-Based Reward between a Principal and a Field of Agents: Application to Energy Savings
  
  • Alasseur Clémence
  • Bayraktar Erhan
  • Dumitrescu Roxana
  • Jacquet Quentin
  , 2026. In this paper, we consider the problem of a Principal aiming at designing a reward function for a population of heterogeneous agents. We construct an incentive based on the ranking of the agents, so that a competition among the latter is initiated. We place ourselves in the limit setting of mean-field type interactions and prove the existence and uniqueness of the equilibrium distribution for a given reward, for which we can find an explicit representation. Focusing first on the homogeneous setting, we characterize the optimal reward function using a convex reformulation of the problem and provide an interpretation of its behaviour. We then show that this characterization still holds for a sub-class of heterogeneous populations. For the general case, we propose a convergent numerical method which fully exploits the characterization of the mean-field equilibrium. We develop a case study related to the French market of Energy Saving Certificates based on the use of realistic data, which shows that the ranking system allows to achieve the sobriety target imposed by the regulation.
• Malliavin calculus for signatures with applications to finance
  
  • Abi Jaber Eduardo
  • Rey Clément
  • Sotnikov Dimitri
  , 2026. Malliavin calculus is a powerful and general framework for the analysis of square-integrable random variables, but it often suffers from a lack of tractability and explicit representations. To address this limitation, we focus on a subclass of random variables given by finite linear combinations of time-extended Brownian motion signatures. The class remains rich due to the universal approximation properties of signatures. Leveraging the algebraic structure of signatures, we first derive explicit formulas for the Malliavin derivative of signatures of continuous Itô processes. As a consequence, we obtain closed-form expressions for the Clark-Ocone representation, the Ornstein-Uhlenbeck semigroup and its generator, as well as the integration-by-parts formula within the class of Brownian signature variables. These results provide purely algebraic formulations of the classical operators of Malliavin calculus. As an application, we compute Greeks for general path-dependent options under signature volatility models, and numerically compare different choices of Malliavin weights.