Publications

2023

• COMPARISON OF BOUNDARY ELEMENT BASED AND PLANE WAVE APPROXIMATION COMPUTATIONS OF TARGET ECHO STRENGTHS
  
  • Pacaut Louise
  • Mercier Jean-François
  • Serre Gilles
  • Chaillat Stéphanie
  , 2023. In naval defence applications, the knowledge of the Target echo strength (TES) of a submarine is of major interest, in order to optimize the scattered pressure that can be measured by an active sonar. In this contribution, we consider a rigid target and compute the TES using two methods: (i) the solution of the Helmholtz equation by reformulating it into a boundary integral equation with either a full space Green's function or a tailored Green's function, and (ii) the use of a plane wave approximation, well-suited for medium to high frequencies. In the first case, the use of a tailored Green's function adapted to the presence of a target reduces the cost of the numerical model. However, an integral equation still has to be solved. It is not the case with the plane wave approximation where the boundary pressure is not calculated but is considered proportional to the incoming wave. Numerical tests are performed to compare the efficiency and accuracy of each approach with respect to available numerical models developed on the submarine model "BeTSSi" -for Benchmark Target Strength Simulation -, under rigid hypothesis.
• Stability of the P1nc-(P0+P1) element
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Sauter Stefan
  , 2025, 153, pp.494-503. We solve the Stokes problem numerically. We analyse the P1nc-(P0+P1) mixed finite element method which exhibits interesting numerical features. However, only an incomplete proof of the inf-sup condition is available. We prove here this condition and the stability of the method.
• Incorporating interface permeability into the diffusion MRI signal representation while using impermeable Laplace eigenfunctions
  
  • Yang Zheyi
  • Fang Chengran
  • Li Jing-Rebecca
  Physics in Medicine and Biology, IOP Publishing, 2023, 68 (17), pp.175036. Abstract Objective . The complex-valued transverse magnetization due to diffusion-encoding magnetic field gradients acting on a permeable medium can be modeled by the Bloch–Torrey partial differential equation. The diffusion magnetic resonance imaging (MRI) signal has a representation in the basis of the Laplace eigenfunctions of the medium. However, in order to estimate the permeability coefficient from diffusion MRI data, it is desirable that the forward solution can be calculated efficiently for many values of permeability. Approach . In this paper we propose a new formulation of the permeable diffusion MRI signal representation in the basis of the Laplace eigenfunctions of the same medium where the interfaces are made impermeable. Main results. We proved the theoretical equivalence between our new formulation and the original formulation in the case that the full eigendecomposition is used. We validated our method numerically and showed promising numerical results when a partial eigendecomposition is used. Two diffusion MRI sequences were used to illustrate the numerical validity of our new method. Significance. Our approach means that the same basis (the impermeable set) can be used for all permeability values, which reduces the computational time significantly, enabling the study of the effects of the permeability coefficient on the diffusion MRI signal in the future. (10.1088/1361-6560/acf022)
  DOI : 10.1088/1361-6560/acf022
• Relaxed-inertial proximal point algorithms for problems involving strongly quasiconvex functions
  
  • Grad Sorin-Mihai
  • Lara Felipe
  • Marcavillaca Raul Tintaya
  , 2023.
• Path-dependent Hamilton-Jacobi-Bellman equation: Uniqueness of Crandall-Lions viscosity solutions
  
  • Cosso Andrea
  • Gozzi Fausto
  • Rosestolato Mauro
  • Russo Francesco
  , 2023. We formulate a path-dependent stochastic optimal control problem under general conditions, for which we prove rigorously the dynamic programming principle and that the value function is the unique Crandall- Lions viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. Compared to the literature, the proof of our core result, that is the comparison theorem, is based on the fact that the value function is bigger than any viscosity subsolution and smaller than any viscosity supersolution. It also relies on the approximation of the value function in terms of functions defined on finite-dimensional spaces as well as on regularity results for parabolic partial differential equations.
• A combination of Kohn-Vogelius and DDM methods for a geometrical inverse problem
  
  • Chaabane Slim
  • Haddar Houssem
  • Jerbi Rahma
  Inverse Problems, IOP Publishing, 2023, 39 (9), pp.095001. We consider the inverse geometrical problem of identifying the discontinuity curve of an electrical conductivity from boundary measurements. This standard inverse problem is used as a model to introduce and study a combined inversion algorithm coupling a gradient descent on the Kohn-Vogelius cost functional with a domain decomposition method that includes the unknown curve in the domain partitioning. We prove the local convergence of the method in a simplified case and numerically show its efficiency for some two dimensional experiments. (10.1088/1361-6420/ace64a)
  DOI : 10.1088/1361-6420/ace64a
• Stochastic mirror descent algorithms with Nesterov smoothing
  
  • Grad Sorin-Mihai
  • Bitterlich Sandy
  , 2023.
• Efficient linear reformulations for binary polynomial optimization problems
  
  • Elloumi Sourour
  • Verchère Zoé
  Computers and Operations Research, Elsevier, 2023, 155, pp.106240. (10.1016/j.cor.2023.106240)
  DOI : 10.1016/j.cor.2023.106240
• A survey on the discrete-time differentiators in closed-loop control systems: experiments on an electro-pneumatic system
  
  • Rasool Mojallizadeh Mohammad
  • Brogliato Bernard
  • Polyakov Andrey
  • Subischa Sevarajan
  • Michel Loïc
  • Plestan Franck
  • Ghanes Malek
  • Barbot Jean-Pierre
  • Aoustin Yannick
  Control Engineering Practice, Elsevier, 2023, 136 (105546), pp.1-23. This paper is dedicated to the experimental analysis of discrete-time differentiators implemented in closed-loop control systems. To this end, two laboratory setups, namely an electro-pneumatic system and a rotary inverted pendulum have been used to implement 25 different differentiators. Since the selected laboratory setups behave differently in the case of dynamic response and noise characteristics, it is expected that the results remain valid for a wide range of control applications. The validity of several theoretical results, which have been already reported in the literature using analytical analysis and numerical simulations, has been investigated, and several comments are provided to allow one to select an appropriate differentiation scheme in practical closed-loop control systems. (10.1016/j.conengprac.2023.105546)
  DOI : 10.1016/j.conengprac.2023.105546
• An efficient Benders decomposition for the p-median problem
  
  • Durán Mateluna Cristian
  • Alès Zacharie
  • Elloumi Sourour
  European Journal of Operational Research, Elsevier, 2023. The p-median problem is a classic discrete location problem with several applications. It aims to open p sites while minimizing the sum of the distances of each client to its nearest open site. We study a Benders decomposition of the most efficient formulation in the literature. We prove that the Benders cuts can be separated by a polynomial time algorithm. The Benders decomposition also leads to a new compact formulation for the p-median problem. We implement a branch-and-Benders-cut approach that outperforms state-of-the-art methods on benchmark instances by an order of magnitude. (10.1016/j.ejor.2022.11.033)
  DOI : 10.1016/j.ejor.2022.11.033
• Eddy-current asymptotics of the Maxwell PMCHWT formulation for multiple bodies and conductivity levels
  
  • Bonnet Marc
  • Demaldent Edouard
  Computers & Mathematics with Applications, Elsevier, 2023, 141, pp.80-101. In eddy current (EC) testing applications, ECs σE (E : electric field, σ: conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation - which neglects the displacement current - was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O(1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants. (10.1016/j.camwa.2023.03.026)
  DOI : 10.1016/j.camwa.2023.03.026
• Convergence analysis of time-domain PMLs for 2D electromagnetic wave propagation in dispersive waveguides
  
  • Bécache Eliane
  • Kachanovska Maryna
  • Wess Markus
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (4), pp.2451-2491. This work is dedicated to the analysis of generalized perfectly matched layers (PMLs) for 2D electromagnetic wave propagation in dispersive waveguides. Under quite general assumptions on frequency-dependent dielectric permittivity and magnetic permeability we prove convergence estimates in homogeneous waveguides and show that the PML error decreases exponentially with respect to the absorption parameter and the length of the absorbing layer. The optimality of this error estimate is studied both numerically and analytically. Finally, we demonstrate that in the case when the waveguide contains a heterogeneity supported away from the absorbing layer, instabilities may occur, even in the case of the non-dispersive media. Our findings are illustrated by numerical experiments. (10.1051/m2an/2023060)
  DOI : 10.1051/m2an/2023060
• A second order asymptotic model for diffusion MRI in permeable media
  
  • Kchaou Marwa
  • Li Jing-Rebecca
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023, 57 (4), pp.1953-1980. Starting from a reference partial differential equation model of the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses, one can use periodic homogenization theory to establish macroscopic models. A previous work introduced an asymptotic model that accounted for permeable interfaces in the imaging medium. In this paper we formulate a higher order asymptotic model to treat higher values of permeability. We explicitly solved this new asymptotic model to obtain a system of ordinary differential equations that can model the diffusion MRI signal and we present numerical results showing the improved accuracy of the new model in the regime of higher permeability. (10.1051/m2an/2023043)
  DOI : 10.1051/m2an/2023043
• Construction de conditions transparentes pour les guides d’ondes électromagnétiques.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2023. Nous nous intéressons à la résolution numérique de problèmes de diffraction dans des guides d’ondes électromagnétiques fermés au moyen de méthodes d’Éléments Finis. Pour ce faire, il est nécessaire de tronquer le domaine et de créer une condition transparente adaptée sur la frontière artificielle pour éviter les réflexions parasites. Nous montrons ici comment étendre à ce cas plusieurs techniques développées en acoustique ou en élasticité. Pour écrire la condition transparente, on utilise une décomposition modale. Numériquement, il est nécessaire de tronquer la série correspondante. Pour justifier la convergence de notre approche, on montre que le problème en domaine tronqué est bien posé, et que l’erreur avec la solution exacte décroît exponentiellement avec le rang de troncature. Nous montrerons des résultats numériques obtenus à l’aide de la librairie XliFE++, qui illustrent la résolution des équations de Maxwell 3D en utilisant les éléments finis de Nédélec.
• Polynomial Approximation for Binary Nonlinear Programming
  
  • Mencarelli Luca
  • Elloumi Sourour
  , 2023.
• Propagation des ondes dans les guides partiellement enfouis : résolution du problème direct et imagerie par méthode de type échantillonnage
  
  • Fritsch Jean-François
  , 2023. Ce travail de thèse porte sur le contrôle non destructif de structures élancées partiellement enfouies ou immergées, par exemple un câble d'acier partiellement enfoui dans du béton ou une plaque d'acier partiellement immergée dans du sodium liquide. Ces structures peuvent être vues comme la jonction d'un guide fermé et d'un guide ouvert. Pour effectuer des calculs, nous avons tronqué transversalement la partie ouverte de la structure avec des PML finies. Un guide partiellement enfoui peut alors être traité comme la jonction de deux guides fermés, dont la propagation des ondes dans l'un des guides est régie par une équation impliquant des coefficients complexes liés à la présence des PML. Ce constat nous a amené à commencer par traiter dans un premier temps le cas plus simple de la jonction de deux guides acoustiques fermés. Pour ce cas simple, nous avons proposé une démarche de résolution du problème inverse adaptée aux jonctions de guides d'ondes fermés. Elle repose d'une part sur l'introduction des champs de référence, qui sont les réponses de la structure totale sans défaut à un mode provenant d'un des deux demi-guides, et d'autre part sur l'utilisation de la relation de réciprocité de la fonction de Green de la structure sans défaut. Suivant cette démarche, nous avons obtenu une formulation modale efficace de la LSM qui nous a permis d'identifier des défauts. Dans ce cas simple, nous avons tiré parti de la complétude des modes pour analyser les problèmes direct et inverse. Dans un second temps, nous avons traité le cas d'un guide acoustique partiellement enfoui. La perte de complétude des modes dans le demi-guide tronqué transversalement avec des PML nous a amenée à étudier le problème direct à l'aide de la théorie de Kondratiev. Les outils introduits pour la jonction de deux guides fermés ont été ensuite adaptés à la résolution du problème inverse. Dans un troisième temps, nous avons abordé le cas plus réaliste, mais plus complexe, d'un guide élastique partiellement immergé dans un fluide. Pour ce cas difficile, nous avons développé des outils de simulation adaptés et étendus les outils introduits précédemment pour résoudre le problème inverse.
• Optimization Filters for Stochastic Time-Varying Convex Optimization
  
  • Simonetto Andrea
  • Massioni Paolo
  , 2023. We look at a stochastic time-varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation. The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a dynamical system and a measurement equation, respectively, yielding the notion of filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a linear matrix inequality condition in the constrained case. Some special cases and variations are discussed, and supporting numerical results are presented from real data sets in ride-hailing scenarios. The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant. (10.23919/ECC57647.2023.10178237)
  DOI : 10.23919/ECC57647.2023.10178237
• Homogenization of thin-structured surfaces for acoustics in the presence of a two-dimensional low Mach potential flow
  
  • Mercier Jean-François
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2023, 479 (2274). A surface homogenization method for acoustic waves over thin microstructured surfaces in the presence of a fluid in a potential flow is presented. Sound hard surfaces are considered, the flow is considered two-dimensional and slow and a low Mach approximation is introduced. We consider acoustic waves with a typical wavelength 1 / k much larger than the array spacing h and thickness e . Owing to the small parameter ε = k h , with e / h = O ( 1 ) , a matched asymptotic expansion technique is applied to the low Mach potential wave equation in the frequency domain. A boundary condition is obtained on an equivalent flat wall, which links the acoustic velocity to its normal and tangential derivatives (of the Myers type). The accuracy of the effective model is tested numerically for various periodic shapes and the accuracy of the model in O ( ε 2 ) is validated. (10.1098/rspa.2022.0697)
  DOI : 10.1098/rspa.2022.0697
• DataFlowTasks.jl
  
  • Faria Luiz
  • Févotte François
  • Sivadon Vincent
  • Plagne Laurent
  , 2023.
• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2023, 74 (3), pp.115. It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances. (10.1007/s00033-023-01989-9)
  DOI : 10.1007/s00033-023-01989-9
• Renewable-based charging in green ride-sharing
  
  • Perotti Elisabetta
  • Ospina Ana
  • Bianchin Gianluca
  • Simonetto Andrea
  • Dall’anese Emiliano
  Scientific Reports, Nature Publishing Group, 2023, 13 (1), pp.15425. Governments, regulatory bodies, and manufacturers are proposing plans to accelerate the adoption of electric vehicles (EVs), with the goal of reducing the impact of greenhouse gases and pollutants from internal combustion engines on human health and climate change. In this context, the paper considers a scenario where ride-sharing enterprises utilize a 100%-electrified fleet of vehicles, and seeks responses to the following key question: How can renewable-based EV charging be maximized without disrupting the quality of the ride-sharing services? We propose a new mechanism to promote EV charging during hours of high renewable generation, and we introduce the concept of charge request, which is issued by a power utility company. Our mechanism is inspired by a game-theoretic approach where the power utility company proposes incentives and the ride-sharing platform assigns vehicles to both ride and charge requests; the bargaining mechanism leads to prices and EV assignments that are aligned with the notion of Nash equilibria. Numerical results show that it is possible to shift the EV charging during periods of high renewable generation and adapt to intermittent generation while minimizing the impact on the quality of service. The paper also investigates how the users’ willingness to ride-share affects the charging strategy and the quality of service. (10.1038/s41598-023-42042-z)
  DOI : 10.1038/s41598-023-42042-z
• A stochastic volume approach based on tailored Green’s functions for airfoil noise prediction at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  Journal of Sound and Vibration, Elsevier, 2023, 551, pp.117603. The presence of boundary surfaces in a turbulent flow can result in the enhancement of the radiated acoustic field especially for eddies close to any geometrical singularity. At low Mach number, the best suited prediction methods consist in using an acoustic analogy solved with an integral formulation. In the present study, we focus on the Lighthill's wave equation combined with a tailored Green's function and a new semi-analytical model for the turbulence statistics in the space-frequency domain to extend acoustic analogies to geometries of arbitrary shapes. To validate the model predictions for the leading edge noise and the trailing edge noise, a NACA 0012 airfoil at zero angle of attack is considered and predictions are compared to experimental data. The volume integral approach introduced in this study allows us to study the spatial distribution of the noise sources inside the turbulence volume. In addition, the direct noise radiation associated with the turbulent boundary layer is investigated. (10.1016/j.jsv.2023.117603)
  DOI : 10.1016/j.jsv.2023.117603
• Extrapolation-based Prediction-Correction Methods for Time-varying Convex Optimization
  
  • Bastianello Nicola
  • Carli Ruggero
  • Simonetto Andrea
  Signal Processing, Elsevier, 2023, pp.109089. (10.1016/j.sigpro.2023.109089)
  DOI : 10.1016/j.sigpro.2023.109089
• Scaling of Free Subduction on a Sphere
  
  • Ribe Neil M.
  • Chamolly Alexander
  • Gerardi Gianluca
  • Chaillat Stéphanie
  • Li Zhong-Hai
  , 2023. Because Earth's tectonic plates are doubly curved shells, their mechanical behavior during subduction can differ significantly from that of flat plates. We use the boundary-element method (BEM) to study free (gravity-driven) subduction in axisymmetric and 3-D geometry, with a focus on determining the dimensionless parameters that control the dynamics. The axisymmetric model comprises a shell with thickness $h$ and viscosity $\eta_1$ subducting in an isoviscous planet with radius $R_0$ and viscosity $\eta_2$. The angular radius of the trench is $\theta_t$. Scaling analysis based on thin-shell theory reveals two key dimensionless parameters: a `flexural stiffness' $St = (\eta_1/\eta_2)(h/l_b)^3$ that is also relevant for flat plates, and a new `dynamical sphericity number' $\Sigma_D = (l_b/R_0)\cot\theta_t$ that is unique to spherical geometry. Here $l_b$ is the `bending length', or the sum of the lengths of the slab and of the seaward flexural bulge. The definition of $\Sigma_D$ implies that the dynamical effect of sphericity is greater for small plates than for large ones; we call this the `sphericity paradox'. By contrast, the purely geometric effect of sphericity is opposite, i.e. greater for large plates than for small ones. The dynamical and geometrical effects together imply that sphericity significantly influences subduction at all length scales. We confirm the scaling analysis using BEM numerical solutions, which show that the influence of sphericity on the slab sinking speed (up to a few tens of percent) and on the hoop stress (up to a factor of 2-3) are largest for small plates such as the Juan de Fuca, Cocos and Philippine Sea plates. We next study a 3-D model comprising a plate bounded by a ridge and a semicircular trench subducting in a three-layer earth consisting of an upper mantle, a lower mantle and an inviscid core. We examine the linear stability of the shell to longitudinal perturbations corresponding to buckling, and determine a scaling law for the most unstable wavelength that we compare with the observed shapes of northern/western Pacific trenches.
• Optimisation des grands systèmes
  
  • Carpentier Pierre
  , 2023, pp.81. Le cours "Optimisation des grands systèmes" a été donné durant de nombreuses années à l'ENSTA comme enseignement de troisième année dans le parcours dédié à l'approfondissement en optimisation et recherche opérationnelle. Ces notes constituent le support de ce cours et ont pour but de fournir aux étudiants l'essentiel de ce qu'il faut retenir du cours. Le livre "Décomposition-coordination en optimisation déterministe et stochastique", publié en 2017 chez Springer, reprend et détaille l'ensemble des notions du cours, et présente l'extension de ces notions au cas stochastique. L'objectif de ce cours est de présenter les méthodes mathématiques et algorithmes permettant d'optimiser un système dont la taille et l'hétérogénéité sont telles que les méthodes "classiques" de l'optimisation ne peuvent pas être mises en oeuvre. Dans ce cours, on se limite à la présentation des méthodes de décomposition et coordination dans le cadre de l'optimisation convexe différentiable déterministe. Le cours comprend essentiellement deux parties. Dans la première partie, on cherche à introduire les idées de la décomposition/coordination et à développer les interprétations économiques sur un modèle simple, sans se préoccuper outre mesure de généralité ou de rigueur mathématique. Dans la deuxième partie, on développe une théorie générale basée sur le principe du problème auxiliaire, permettant d'une part de lever les restrictions qui paraissaient essentielles dans la première partie, et d'autre part d'étudier dans un cadre unifié la convergence des algorithmes de coordination. Enfin, un exemple correspondant à un réseau de distribution d'eau potable de grande taille est présenté et sert à illustrer l'ensemble des méthodes présentés durant le cours.