Publications

2023

• Optimizing Variational Circuits for Higher-Order Binary Optimization
  
  • Verchère Zoé
  • Elloumi Sourour
  • Simonetto Andrea
  , 2023. Variational quantum algorithms have been advocated as promising candidates to solve combinatorial optimization problems on near-term quantum computers. Their methodology involves transforming the optimization problem into a quadratic unconstrained binary optimization (QUBO) problem. While this transformation offers flexibility and a ready-to-implement circuit involving only two-qubit gates, it has been shown to be less than optimal in the number of employed qubits and circuit depth, especially for polynomial optimization. On the other hand, strategies based on higher-order binary optimization (HOBO) could save qubits, but they would introduce additional circuit layers, given the presence of higher-than-two-qubit gates. In this paper, we study HOBO problems and propose new approaches to encode their Hamiltonian into a ready-to-implement circuit involving only two-qubit gates. Our methodology relies on formulating the circuit design as a combinatorial optimization problem, in which we seek to minimize circuit depth. We also propose handy simplifications and heuristics that can solve the circuit design problem in polynomial time. We evaluate our approaches by comparing them with the state of the art, showcasing clear gains in terms of circuit depth.
• 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
  , 2024. 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
• Decomposition Methods for Monotone Two-Time-Scale Stochastic Optimization Problems
  
  • Rigaut Tristan
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Computational Management Science, Springer Verlag, 2023, 21 (1), pp.28. It is common that strategic investment decisions are made at a slow time-scale, whereas operational decisions are made at a fast time-scale. Hence, the total number of decision stages may be huge. In this paper, we consider multistage stochastic optimization problems with two time-scales, and we propose a time block decomposition scheme to address them numerically. More precisely, i) we write recursive Bellman-like equations at the slow time-scale and ii), under a suitable monotonicity assumption, we propose computable upper and lower bounds — relying respectively on primal and dual decomposition — for the corresponding slow time-scale Bellman functions. With these functions, we are able to design policies. We assess the methods tractability and validate their efficiency by solving a battery management problem where the fast time-scale operational decisions have an impact on the storage current capacity, hence on the strategic decisions to renew the battery at the slow time-scale. (10.1007/s10287-024-00510-5)
  DOI : 10.1007/s10287-024-00510-5
• 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.
• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Bitterlich Sandy
  • Grad Sorin-Mihai
  Numerical Algorithms, Springer Verlag, 2023. For minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in an Euclidean space we propose a stochastic incremental mirror descent algorithm constructed by means of the Nesterov smoothing. Further we modify the algorithm in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Next a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing is proposed in order to minimize over a nonempty closed convex set in an Euclidean space a sum of finitely many proper, convex and lower semicontinuous functions and a prox-friendly proper, convex and lower semicontinuous function. Different to the previous contributions from the literature on mirror descent methods for minimizing sums of functions, we do not require these to be (Lipschitz) continuous or differentiable. Applications in Logistics, Tomography and Machine Learning modelled as optimization problems illustrate the theoretical achievements. (10.1007/s11075-023-01574-1)
  DOI : 10.1007/s11075-023-01574-1
• Fractured meshes
  
  • Averseng Martin
  • Claeys Xavier
  • Hiptmair Ralf
  Finite Elements in Analysis and Design, Elsevier, 2023, 220, pp.103907. This work introduces “generalized meshes”, a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct “generalized” vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted to the representation of fractured domains and their boundaries. An algorithm is proposed to construct the so-called virtually inflated mesh, which correspond to a “two-sided” mesh of a fracture. Discrete -differential forms on the virtually inflated mesh are characterized as the trace space of discrete -differential forms in the surrounding volume. (10.1016/j.finel.2022.103907)
  DOI : 10.1016/j.finel.2022.103907
• 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.
• Correlation Clustering Problem under Mediation
  
  • Alès Zacharie
  • Engelbeen Céline
  • Figueiredo Rosa
  , 2023.
• 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
• 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
• 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. 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.
• 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.
• Personalized incentives as feedback design in generalized Nash equilibrium problems
  
  • Fabiani Filippo
  • Simonetto Andrea
  • Goulart Paul J.
  IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2023. We investigate both stationary and time-varying, nonmonotone generalized Nash equilibrium problems that exhibit symmetric interactions among the agents, which are known to be potential. As may happen in practical cases, however, we envision a scenario in which the formal expression of the underlying potential function is not available, and we design a semi-decentralized Nash equilibrium seeking algorithm. In the proposed two-layer scheme, a coordinator iteratively integrates the (possibly noisy and sporadic) agents' feedback to learn the pseudo-gradients of the agents, and then design personalized incentives for them. On their side, the agents receive those personalized incentives, compute a solution to an extended game, and then return feedback measurements to the coordinator. In the stationary setting, our algorithm returns a Nash equilibrium in case the coordinator is endowed with standard learning policies, while it returns a Nash equilibrium up to a constant, yet adjustable, error in the time-varying case. As a motivating application, we consider the ridehailing service provided by several companies with mobility as a service orchestration, necessary to both handle competition among firms and avoid traffic congestion, which is also adopted to run numerical experiments verifying our results. (10.1109/TAC.2023.3287218)
  DOI : 10.1109/TAC.2023.3287218
• 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
• Beyond the Fermat optimality rules
  
  • Grad Sorin-Mihai
  • Abbasi Malek
  • Théra Michel
  , 2023. This work proposes a general framework for analyzing the behavior at its extrema of an extended real-valued function assumed neither convex nor differentiable and for which the classical Fermat rules of optimality do not apply. The tools used for building this frame are the notions of sup-subdifferential, recently introduced by two of the authors together with A. Kruger, and partial sup-subdifferentials. The sup-subdifferential is always a \textit{nonempty} enlargement of the Moreau-Rockafellar subdifferential from convex optimization. It satisfies most of the fundamental properties of the Moreau-Rockafellar subdifferential and enjoys certain calculus rules. The partial sup-subdifferentials are obtained by breaking down the sup-subdifferential into one-dimensional components through basis elements and play the same role as the partial derivatives in the Fermat optimality rules.
• 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.