Publications

2025

• Concentration inequalities for semidefinite least squares based on data
  
  • Fabiani Filippo
  • Simonetto Andrea
  , 2025. We study data-driven least squares (LS) problems with semidefinite (SD) constraints and derive finite-sample guarantees on the spectrum of their optimal solutions when these constraints are relaxed. In particular, we provide a high confidence bound allowing one to solve a simpler program in place of the full SDLS problem, while ensuring that the eigenvalues of the resulting solution are $\varepsilon$-close of those enforced by the SD constraints. The developed certificate, which consistently shrinks as the number of data increases, turns out to be easy-to-compute, distribution-free, and only requires independent and identically distributed samples. Moreover, when the SDLS is used to learn an unknown quadratic function, we establish bounds on the error between a gradient descent iterate minimizing the surrogate cost obtained with no SD constraints and the true minimizer. (10.48550/arXiv.2509.13166)
  DOI : 10.48550/arXiv.2509.13166
• On the application of the T-coercivity method for the Helmholtz problem with sign-changing coefficients
  
  • Chaaban Farah
  , 2025. To solve transmission problems with sign-changing coefficients, one can apply the T-coercivity method, which imposes specific mesh conditions near the interface to ensure optimal convergence rates for the finite element approximation. This method was initially proposed and analyzed for the quasi-static case. The aim in this work is to extend it and prove its convergence for the case of non-zero frequency. Additionally, we check its convergence for a general compact perturbation and outline key ideas for a 3D polyhedral interface. Our theoretical results are validated through a numerical test.
• Spectrum of slip dynamics, scaling and statistical laws emerge from simplified model of fault and damage zone architecture
  
  • Almakari Michelle
  • Kheirdast N.
  • Villafuerte C.
  • Thomas Marion Y.
  • Dubernet P.
  • Cheng J.
  • Gupta A.
  • Romanet P.
  • Chaillat S.
  • Bhat Harsha S.
  Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2025. <div><p>Seismological and geodetic observations of a fault zone reveal a wide range of slip dynamics, scaling, and statistical laws. However, the underlying physical mechanisms remain unclear. In this study, we show that incorporating an off-fault damage zone-characterized by distributed fractures surrounding a main fault-can reproduce many key features observed in seismic and geodetic data. We model a 2D shear fault zone in which off-fault cracks follow power-law size and density distributions, and are oriented either optimally or parallel to the main fault. All fractures follow the rate-and-state friction law with parameters chosen such that each can host slip instabilities. We do not introduce spatial heterogeneities in the frictional properties of the fault. Using quasi-dynamic boundary integral simulations accelerated by hierarchical matrices, we simulate slip dynamics of this system and analyze the events produced both on and off the main fault. Despite the spatially uniform frictional properties, we observe a natural continuum from slow to fast ruptures, as observed in nature. Our simulations reproduce the Omori law, the inverse Omori law, the Gutenberg-Richter scaling, and the moment-duration scaling. We also observe seismicity localizing toward the main fault when an event is about to nucleate on the main fault. During slow slip events, off-fault seismicity migrates in a pattern resembling a fluid diffusion front, despite the absence of fluids in the model. We also show that tremors, Very Low Frequency Earthquakes (VLFEs), Low Frequency Earthquakes (LFEs), Slow Slip Events (SSEs), and earthquakes (EQs) can all emerge naturally in the ‘digital twin’ framework.</p></div>
• A global-in-time domain decomposition approach for transient acoustic-elastic interaction
  
  • Bonnet Marc
  • Chaillat Stéphanie
  • Nassor Alice
  , 2025. This work develops a global-in-time iterative domain decomposition approach for transient fluid-structure interaction problems involving acoustic scattering by elastic obstacles. The proposed method, inspired by optimized Schwarz waveform relaxation algorithms, proceeds by iteratively exchanging Robin boundary conditions, enabling the non-intrusive coupling of distinct fluid and structure solvers, and works for arbitrary transient incident acoustic fields. We prove the convergence of the proposed coupling iterations in continuous form. A BEM-FEM coupling implementation of the method is then validated against a reference analytical solution, and its efficiency, accuracy and robustness demonstrated through numerical experiments on configurations representative of potential industrial applications.
• Portage GPU d'un solveur éléments finis discontinus hybridisé pour les problèmes d'ondes en fréquence
  
  • Chabib Ahmed
  • Greffe Roland
  • Geuzaine Christophe
  • Modave Axel
  , 2025. Dans ce travail, nous nous intéressons à la résolution par éléments finis de problèmes de propagation d'ondes en régime harmonique de très grande taille. L'utilisation de cartes graphiques (GPU) permet d'accélérer les calculs, mais il est difficile d'en exploiter pleinement la puissance. Nous considérons une méthode d'éléments finis discontinus de type Galerkin (DG) avec des flux amonts, hybridisée utilisant des variables de transmission définies aux faces des éléments. L'élimination des variables physiques conduit à un système linéaire adapté pour une résolution itérative et une implémentation parallèle efficace sur GPU. Après une description de la méthode, appelée CHDG, nous présentons quelques stratégies de mise en oeuvre sur GPU et nous comparons et discutons leurs performances.
• Eigenvalue falls in thin broken quantum strips
  
  • Chesnel Lucas
  • Nazarov Sergei A.
  , 2025. We are interesting in the spectrum of the Dirichlet Laplacian in thin broken strips with angle α. Playing with symmetries, this leads us to investigate spectral problems for the Laplace operator with mixed boundary conditions in thin trapezoids characterized by a parameter ε small. We give an asymptotic expansion of the first eigenvalues and corresponding eigenfunctions as ε tends to zero. The new point in this work is to study the dependence with respect to α. We show that for a small fixed ε&gt;;0, at certain particular angles α_k , k = 0, 1, . . . , that we characterize, an eigenvalue dives, i.e. moves down rapidly, below the normalized threshold π^2 /ε^2 as α&gt;0 increases. We describe the way the eigenvalue dives below π^2 /ε^2 and prove that the phenomenon is milder at α_0 = 0 than at α_k for k ≥ 1.
• Optimal Control Problem Under Signal Temporal Logic Constraints: A Robust Reformulation using Augmented Dynamics
  
  • Lai En
  • Bonalli Riccardo
  • Girard Antoine
  • Jean Frédéric
  , 2025. This work presents a novel approach for solving optimal control problems under Signal Temporal Logic (STL) constraints. The proposed method reformulates the original problem as a classical continuous-time optimal control problem by augmenting the system dynamics with auxiliary variables that encode STL satisfaction through their evolution and boundary conditions. Introducing a robustness parameter, we also establish the convergence of the reformulated problem to the original one as this parameter tends to zero. Numerical simulations are realized to demonstrate the feasibility of our method, highlighting its potential for practical applications.
• McKean-Vlasov equations with singular coefficients - a review of recent results
  
  • Bondi Luca
  • Issoglio Elena
  • Russo Francesco
  , 2025. This paper focuses on recent works on McKean-Vlasov stochastic differential equations (SDEs) involving singular coefficients. After recalling the classical framework, we review existing recent literature depending on the type of singularities of the coefficients: on the one hand they satisfy some integrability and measurability conditions only, while on the other hand the drift is allowed to be a generalised function. Different types of dependencies on the law of the unknown and different noises will also be considered. McKean-Vlasov SDEs are closely related to non-linear Fokker-Planck equations that are satisfied by the law (or its density) of the unknown. These connections are often established also in this singular setting and will be reviewed here. Important tools for dealing with singular coefficients are also included in the paper, such as Figalli-Trevisan superposition principle, Zvonkin transformation, Markov marginal uniqueness, and stochastic sewing lemma.
• An entropy penalized approach for stochastic control problems. Complete version
  
  • Bourdais Thibaut
  • Oudjane Nadia
  • Russo Francesco
  , 2023. In this paper, we propose an original approach to stochastic control problems. We consider a weak formulation that is written as an optimization (minimization) problem on the space of probability measures. We then introduce a penalized version of this problem obtained by splitting the minimization variables and penalizing the discrepancy between the two variables via an entropy term. We show that the penalized problem provides a good approximation of the original problem when the weight of the entropy penalization term is large enough. Moreover, the penalized problem has the advantage of giving rise to two optimization subproblems that are easy to solve in each of the two optimization variables when the other is fixed. We take advantage of this property to propose an alternating optimization procedure that converges to the infimum of the penalized problem with a rate $O(1/k)$, where $k$ is the number of iterations. The relevance of this approach is illustrated by solving a high-dimensional stochastic control problem aimed at controlling consumption in electrical systems.
• Isolated Rotor Blade Shape Sensitivity for Aeroacoustic Optimization Using a Discrete Adjoint Framework
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Salah El Din Itham
  • Bonnet Marc
  , 2025. A discrete adjoint framework is developed to optimize rotor self-noise from steady fluid simulations in the rotating frame. To this end, a simplified expression of the off-body frequency-domain Ffowcs-Williams and Hawkings (FW-H) equation is derived for far-field observers, following the model of Hanson and Parzych (1993) originally written for on-body surfaces. The latter is implemented and compared against the results given by an established time-domain FW-H solver. Far-field acoustic pressure sensitivities are derived analytically and validated by comparison with second-order accurate finite differences. The sensitivities of any objective function expressed in terms of the acoustic pressure can therefore be reconstructed. Then the discrete adjoint of a Reynolds-averaged Navier-Stokes solver provides the objective function gradients with respect to the blade shape parameters. The complete workflow is validated against finite difference evaluations on an isolated open rotor in cruise conditions. (10.2514/6.2025-3367)
  DOI : 10.2514/6.2025-3367
• Convergence rates of curved boundary element methods for the 3D Laplace and Helmholtz equations
  
  • Faria Luiz
  • Marchand Pierre
  • Montanelli Hadrien
  , 2025. We establish improved convergence rates for curved boundary element methods applied to the three-dimensional (3D) Laplace and Helmholtz equations with smooth geometry and data. Our analysis relies on a precise analysis of the consistency errors introduced by the perturbed bilinear and sesquilinear forms. We illustrate our results with numerical experiments in 3D based on basis functions and curved triangular elements up to order four.
• Attraction of the core and the cohesion flow
  
  • Laplace Mermoud Dylan
  Theory and Decision, Springer Verlag, 2025, 99 (1-2), pp.377-392. We adopt a continuous-time dynamical system approach to study the evolution of the state of a game driven by the willingness to reduce the total dissatisfaction of the coalitions about their payment. Inspired by the work of Grabisch and Sudhölter about core stability, we define a vector field on the set of preimputations from which is defined, for any preimputation, a cohesion curve describing the evolution of the state. We prove that for each preimputation, there exists a unique cohesion curve. Subsequently, we show that, for the cohesion flow of a balanced game, the core is the unique minimal attractor of the flow, the realm of which is the whole preimputation set. These results improve our understanding of the ubiquity of the core in the study of cooperative games with transferable utility. (10.1007/s11238-025-10060-0)
  DOI : 10.1007/s11238-025-10060-0
• The algebraic structures of social organizations: the operad of cooperative games
  
  • Laplace Mermoud Dylan
  • Roca I Lucio Victor
  , 2025. <div><p>The main goal of this paper is to settle a conceptual framework for cooperative game theory in which the notion of composition/aggregation of games is the defining structure. This is done via the mathematical theory of algebraic operads: we start by endowing the collection of all cooperative games with any number of players with an operad structure, and we show that it generalises all the previous notions of sums, products and compositions of games considered by Owen, Shapley, von Neumann and Morgenstern, and many others. Furthermore, we explicitly compute this operad in terms of generators and relations, showing that the Möbius transform map induces a canonical isomorphism between the operad of cooperative games and the operad that encodes commutative triassociative algebras. In other words, we prove that any cooperative game is a linear combination of iterated compositions of the 2-player bargaining game and the 2-player dictator games. We show that many interesting classes of games (simple, balanced, capacities a.k.a fuzzy measures and convex functions, totally monotone, etc) are stable under compositions, and thus form suboperads. In the convex case, this gives by the submodularity theorem a new operad structure on the family of all generalized permutahedra. Finally, we focus on how solution concepts in cooperative game theory behave under composition: we study the core of a composite and describe it in terms of the core of its components, and we give explicit formulas for the Shapley value and the Banzhaf index of a compound game.</p></div>
• On the Formation of Steady Coalitions
  
  • Laplace Mermoud Dylan
  , 2024. This paper studies the formation of the grand coalition of a cooperative game by investigating its possible internal dynamics. Each coalition is capable of forcing all players to reconsider the current state of the game when it does not provide sufficient payoff. Different coalitions may ask for contradictory evolutions, leading to the impossibility of the grand coalition forming. In this paper, we give a characterization of the impossibility, for a given state, of finding a new state dominating the previous one such that each aggrieved coalition has a satisfactory payoff. To do so, we develop new polyhedral tools related to a new family of polyhedra, appearing in numerous situations in cooperative game theory. (10.48550/arXiv.2410.05087)
  DOI : 10.48550/arXiv.2410.05087
• Projection onto the core: An optimal reallocation to correct market failure
  
  • Laplace Mermoud Dylan
  , 2024. This paper provides formulae and algorithms to compute the projection onto the core of a preimputation outside it. The core of a game is described using an exponential number of linear constraints, and we cannot know beforehand which are redundant or defining the polytope. We apply these new results to market games, a class of games in which every game has a nonempty core. Given an initial state of the game represented by a preimputation, it is not guaranteed that the state of the game evolves toward the core following the dynamics induced by the domination relations. Our results identify and compute the most efficient side payment that acts on a given state of the game and yields its closest core allocation. Using this side payment, we propose a way to evaluate the failure of a market to reach a state of the economy belonging to the core, and we propose a new solution concept consisting of preimputations that minimizes this failure. (10.48550/arXiv.2411.11810)
  DOI : 10.48550/arXiv.2411.11810
• Nonlocal vector calculus on the sphere
  
  • Montanelli Hadrien
  • Slevinsky Richard Mikael
  • Du Qiang
  , 2025. (10.48550/arXiv.2505.12372)
  DOI : 10.48550/arXiv.2505.12372
• Two-phase Trajectory Planning Method for Robust Planetary Landing in a Sensor-equipped Area
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  , 2025, pp.1296-1301. <div><p>This article addresses the planetary landing problem by considering uncertainties and leveraging the presence of a detection area where precise measurements are available. The flight consists of two distinct phases: the first phase, subject to a high level of uncertainties, and the second phase, during which the vehicle is feedback controlled to ensure precise landing. We propose a method to compute the optimal control for the initial phase, aiming to minimize fuel consumption for the entire trajectory while satisfying a probabilistic constraint that ensures the vehicle reaches the detection zone with a specified threshold.</p></div> (10.23919/ECC65951.2025.11186925)
  DOI : 10.23919/ECC65951.2025.11186925
• A few techniques to achieve invisibility in waveguides
  
  • Chesnel Lucas
  , 2025, pp.68. The aim of this lecture is to consider a concrete problem, namely the identification of situations of invisibility in waveguides, to present techniques and tools that may be useful in various fields of applied mathematics. To be more specific, we will be interested in the propagation of acoustic waves in guides which are unbounded in one direction. In general, the diffraction of an incident field in such a structure in presence of an obstacle generates a reflection and a transmission characterized by some scattering coefficients. Our goal will be to play with the geometry, the frequency and/or the index material to control these scattering coefficients. We will explain how to: - develop a continuation method based on the use of shape derivatives to construct invisible defects; - exploit complex resonances located closed to the real axis to hid obstacles; - construct a non self-adjoint operator whose eigenvalues coincide with frequencies such that there are incident fields whose energy is completely transmitted. Our approaches will mainly rely on techniques of asymptotic analysis as well as spectral theory for self-adjoint and non self-adjoint operators. Most of the results will be illustrated by numerical experiments.
• Optimized Schwarz Methods in Time for Discrete Transport Control
  
  • Bui Duc-Quang
  • Delourme Bérangère
  • Halpern Laurence
  • Kwok Felix
  , 2025. We investigate optimized Schwarz domain decomposition methods in time for the control of the 1D transport equation. In the case of an internal control over the whole domain, the optimization problem can be transformed into a system of two coupled PDEs. We then apply the time-domain decomposition (without overlap) strategy on this PDE system as well as on its discretized counterpart. Under Fourier analysis, we analyse three different iterations: the fixed point iteration, the relaxed iteration and the preconditioned GMRES method. For each case, we propose parameters for the transmission conditions that lead to fast convergence of the method. We illustrate our results by numerical examples.
• Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond
  
  • Laplace Mermoud Dylan
  • Simonetto Andrea
  • Elloumi Sourour
  , 2025. We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the cx gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, QuPer, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with 256 variables, requiring 20 qubits. (10.48550/arXiv.2505.05981)
  DOI : 10.48550/arXiv.2505.05981
• Convergence analysis of GMRES applied to Helmholtz problems near resonances
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  , 2025. In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves.
• On some coupled local and nonlocal diffusion models
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  , 2025. We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0, 1)$, are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diffusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models.
• 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
  , 2025. <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>
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic electromagnetic problems
  
  • Rappaport Ari
  • Chaumont-Frelet Théophile
  • Modave Axel
  , 2025. The CHDG method is a hybridizable discontinuous Galerkin (HDG) finite element method suitable for the iterative solution of time-harmonic wave propagation problems. Hybrid unknowns corresponding to transmission variables are introduced at the element interfaces and the physical unknowns inside the elements are eliminated, resulting in a hybridized system with favorable properties for fast iterative solution. In this paper, we extend the CHDG method, initially studied for the Helmholtz equation, to the time-harmonic Maxwell equations. We prove that the local problems stemming from hybridization are well-posed and that the fixed-point iteration naturally associated to the hybridized system is contractive. We propose a 3D implementation with a discrete scheme based on nodal basis functions. The resulting solver and different iterative strategies are studied with several numerical examples using a high-performance parallel C++ code.
• Open Review of "Normal form analysis of nonlinear oscillator equations with automated arbitrary order expansions
  
  • de Figueiredo Stabile André
  • Touzé Cyril
  • Vizzaccaro Alessandra
  • Römer Ulrich
  • Raze Ghislain
  • Chaillat Stéphanie
  , 2025.