Publications

2024

• Substructuring based FEM-BEM coupling for Helmholtz problems
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2024. This talk concerns the solution of the Helmholtz equation in a medium composed of a bounded heterogeneous domain and an unbounded homogeneous one. Such problems can be expressed using classical FEM-BEM coupling techniques. We solve these coupled formulations using iterative solvers based on substructuring Domain Decomposition Methods (DDM), and aim to develop a convergence theory, with fast and guaranteed convergence. A recent article of Xavier Claeys proposed a substructuring Optimized Schwarz Method, with a nonlocal exchange operator, for Helmholtz problems on a bounded domain with classical conditions on its boundary (Dirichlet, Neumann, Robin). The variational formulation of the problem can be written as a bilinear application associated with the volume and another with the surface, for which, under certain sufficient assumptions, convergence of the DDM strategy is guaranteed. In this presentation we show how some specific FEM-BEM coupling methods fit, or not, the previous framework, in which we consider Boundary Integral Equations (BIEs) instead of classical boundary conditions. In particular, we prove that the symmetric Costabel coupling satisfies the framework assumptions, implying that the convergence is guaranteed. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Fast and accurate boundary integral equation methods for the multi-layer transmission problem
  
  • Cortes Elsie A
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  , 2024. We consider a multi-layer transmission problem, which can be used for example to describe the light scattering in meta-materials (assemblings of various concentric penetrable materials). Our goal is to solve the multi-layer problem accurately with optimal discretization. Generally, the costs to solve this problem grow as more layers are introduced - solving this problem is thus particularly challenging for 3D models. For this reason, we use boundary integral equation (BIE) methods: they reduce the dimensionality of the problem and can provide high order accuracy. However, BIE methods suffer from the so-called close evaluation problem. We address it using modified representations. We further examine how to improve the speed of our method by optimizing the accuracy over number of discretization points ratio. In particular, we investigate whether the usual rule of thumb to mesh interfaces, based on the most constraining material, is necessary for the multi-layer transmission problem. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• High-Resolution Seismic Imaging for Dam-Rock Interface using Full-Waveform Inversion
  
  • Aziz Boukraa Mohamed
  • Audibert Lorenzo
  • Bonazzoli Marcella
  • Haddar Houssem
  • Vautrin Denis
  , 2024. We are interested in imaging the interface between the concrete structure of the hydroelectric dam and the underlying rock using non-destructive seismic waves. Our method, combining "Full Waveform Inversion" with shape optimization, produces high-resolution images. Numerical results from realistic synthetic data demonstrate the method ability to accurately recover the interface with a limited number of measurements. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Efficient methods for the solution of boundary integral equations on fractal antennas
  
  • Joly Patrick
  • Kachanovska Maryna
  • Moitier Zoïs
  , 2024. This work focuses on construction of efficient numerical methods for wave scattering by fractal antennas, see [3]. It builds on the theoretical basis proposed in the recent work [1], which establishes boundary integral (BIE) formulations for solving sound-soft Helmholtz scattering problems on fractal screens. An important feature of such formulations is the use of the Hausdorff measure on fractals instead of the standard Lebesgue’s measure. This adds an extra dimension to the two classical difficulties encountered with numerical BEM simulations, namely the evaluation of boundary integrals and the fact that the underlying matrices are dense. Our idea is to exploit the Hausdorff measure’s self-similar structure in order to deal with these difficulties. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Multistage stochastic optimization of a mono-site hydrogen infrastructure by decomposition techniques
  
  • Lefgoum Raian
  • Afsar Sezin
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  Journal of Optimization Theory and Applications, Springer Verlag, 2024, 207 (3), pp.49. The development of hydrogen infrastructures requires to reduce their costs. In this paper, we develop a multistage stochastic optimization model for the management of a hydrogen infrastructure which consists of an electrolyser, a compressor and a storage to serve a transportation demand. This infrastructure is powered by three different sources: on-site photovoltaic panels (PV), renewable energy through a power purchase agreement (PPA) and the power grid. We consider uncertainties affecting on-site photovoltaic production and hydrogen demand. Renewable energy sources are emphasized in the hydrogen production process to ensure eligibility for a subsidy, which is awarded if the proportion of nonrenewable electricity usage stays under a predetermined threshold. We solve the multistage stochastic optimization problem using a decomposition method based on Lagrange duality. The numerical results indicate that the solution to this problem, formulated as a policy, achieves a small duality gap, thus proving the effectiveness of this approach. (10.1007/s10957-025-02795-1)
  DOI : 10.1007/s10957-025-02795-1
• Contributions to Efficient Finite Element Solvers for Time-Harmonic Wave Propagation Problems
  
  • Modave Axel
  , 2024. The numerical simulation of wave propagation phenomena is of paramount importance in many scientific and engineering disciplines. Many time-harmonic problems can be solved with finite elements in theory, but the computational cost is a strong constraint that limits the size of the problems and the accuracy of the solutions in practice. Ideally, solution techniques should provide the best accuracy at minimal computational cost for real-world problems. They should take advantage of the power of modern parallel computers, and they should be as easy as possible to use for the end user. In this HDR thesis, contributions are presented on three topics: the improvement of domain truncation techniques (i.e. high-order absorbing boundary conditions and perfectly matched layers), the acceleration of substructuring and preconditioning techniques based on domain decomposition methods (i.e. non-overlapping domain decomposition methods with interface conditions based on domain truncation techniques), and the design of a new hybridization approach for efficient discontinuous finite element solvers.
• Imaging a dam-rock interface with inversion of a full elastic-acoustic model
  
  • Boukraa Mohamed Aziz
  • Bonazzoli Marcella
  • Haddar Houssem
  • Audibert Lorenzo
  • Vautrin Denis
  , 2024. We are interested in imaging the interface between the concrete structure of hydroelectric dams and the rock foundation using non-destructive seismic waves. We present a geophysical technique for processing seismic measurements to obtain an image of the interface with metric resolution. The proposed technique is based on "Full Waveform Inversion" with a shape optimization approach. Numerical results using synthetic measurements demonstrate the method ability to accurately recover the interface with a limited number of measurement points and in the presence of noise.
• Inverse optimal control problem in the non autonomous linear-quadratic case
  
  • Jean Frédéric
  • Maslovskaya Sofya
  , 2024. Inverse optimal control problem emerges in different practical applications, where the goal is to design a cost function in order to approximate given optimal strategies of an expert. Typical application is in robotics for generation of human motions. In this paper we analyze a general class of non autonomous inverse linear quadratic problems. This class of problems is of particular interest because it arises as a linearization of a nonlinear problem around an optimal trajectory. The addressed questions are the injectivity of the inverse problem and the reconstruction. We show that the nonlinear problem admits the same characterization of the injectivity as the autonomous one. In the autonomous case we show moreover that the injectivity property is generic in the considered class. We also provide a numerical test of the reconstruction algorithm in the autonomous setting.
• Heat and momentum losses in H 2 –O2 –N 2/Ar detonations: on the existence of set-valued solutions with detailed thermochemistry
  
  • Veiga-López F.
  • Faria Luiz
  • Melguizo-Gavilanes J.
  Shock Waves, Springer Verlag, 2024, 34 (3), pp.273-283. The effect of heat and momentum losses on the steady solutions admitted by the reactive Euler equations with sink/source terms is examined for stoichiometric hydrogen–oxygen mixtures. Varying degrees of nitrogen and argon dilution are considered in order to access a wide range of effective activation energies, $$E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}$$ E a,eff / R u T 0 , when using detailed thermochemistry. The main results of the study are discussed via detonation velocity-friction coefficient ( D – $$c_{\textrm{f}}$$ c f ) curves. The influence of the mixture composition is assessed, and classical scaling for the prediction of the velocity deficits, $$D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$$ D ( c f,crit ) / D CJ , as a function of the effective activation energy, $${E}_{\textrm{a,eff}}/R_{\textrm{u}} T_{0}$$ E a,eff / R u T 0 , is revisited. Notably, a map outlining the regions where set-valued solutions exist in the $$E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}\text {--}{\alpha }$$ E a,eff / R u T 0 -- α space is provided, with $$\alpha $$ α denoting the momentum–heat loss similarity factor, a free parameter in the current study. (10.1007/s00193-024-01182-5)
  DOI : 10.1007/s00193-024-01182-5
• An algorithm for computing scattering poles based on dual characterization to interior eigenvalues
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Zilberberg Dana
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2024, 480 (2292). We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon the previous work of Cakoni et al . (2020) titled ‘A duality between scattering poles and transmission eigenvalues in scattering theory’ (Cakoni et al . 2020 Proc. A. 476 , 20200612 ( doi:10.1098/rspa.2020.0612 )), where the authors developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach. (10.1098/rspa.2024.0015)
  DOI : 10.1098/rspa.2024.0015
• Co-Contraction Embodies Uncertainty: An Optimal Feedforward Strategy for Robust Motor Control
  
  • Berret Bastien
  • Verdel Dorian
  • Burdet Etienne
  • Jean Frédéric
  PLoS Computational Biology, PLOS, 2024, 20 (11), pp.e1012598. Despite our environment is often uncertain, we generally manage to generate stable motor behaviors. While reactive control plays a major role in this achievement, proactive control is critical to cope with the substantial noise and delays that affect neuromusculoskeletal systems. In particular, muscle co-contraction is exploited to robustify feedforward motor commands against internal sensorimotor noise as was revealed by stochastic optimal open-loop control modeling. Here, we extend this framework to neuromusculoskeletal systems subjected to random disturbances originating from the environment. The analytical derivation and numerical simulations predict a singular relationship between the degree of uncertainty in the task at hand and the optimal level of anticipatory co-contraction. This prediction is confirmed through a single-joint pointing task experiment where an external torque is applied to the wrist near the end of the reaching movement with varying probabilities across blocks of trials. We conclude that uncertainty calls for impedance control via proactive muscle co-contraction to stabilize behaviors when reactive control is insufficient for task success. (10.1101/2024.06.17.599269)
  DOI : 10.1101/2024.06.17.599269
• New computable algorithms for smooth multiobjective optimization problems
  
  • Grad Sorin-Mihai
  • Illés Tibor
  • Rigó Petra Renáta
  , 2024. We propose new practical algorithms for solving smooth multiobjective optimization problems based on determining joint decreasing directions via suitable linear programming problems. The presented iterative method is specialized for unconstrained, sign constrained and linearly constrained multiobjective optimization problems. In all cases we show that the objective function values sequence is decreasing with respect to the considered nonnegative orthant while the iterates are feasible. Furthermore, we prove that every accumulation point of the sequence generated by the algorithm, if any, is a substationary point to the considered multiobjective optimization problem, and, under convexity assumptions, it is actually a weakly Pareto efficient (also known as weakly Pareto-optimal) point. Different to similar algorithms from the literature, the ones proposed in this work involve joint decreasing directions that are easily computable in polynomial time by solving linear programming problems. The computational performance of our algorithms has been illustrated on convex unconstrained and convex linearly constrained multiobjective optimization problems.
• Generalized impedance boundary conditions with vanishing or sign-changing impedance
  
  • Bourgeois Laurent
  • Chesnel Lucas
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (3), pp.4223-4251. We consider a Laplace type problem with a generalized impedance boundary condition of the form ∂_νu = −∂_x(g∂_xu) on a flat part Γ of the boundary. Here ν is the outward unit normal vector to ∂Ω, g is the impedance function and x is the coordinate along Γ. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases g=1 or g=−1 have been investigated. In this work, we address situations where Γ contains the origin and g(x)=1_{x>0}(x)x^\alpha or g(x)=−sign(x)|x|^\alpha with \alpha≥ 0. In other words, we study cases where g vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of \alpha. For \alpha∈ [0,1), we show that the associated operators are Fredholm of index zero while it is not the case when \alpha=1. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems. (10.1137/23M1604217)
  DOI : 10.1137/23M1604217
• Time Consistency for Multistage Stochastic Optimization Problems under Constraints in Expectation
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2024, 34 (2), pp.1909-1936. We consider sequences-indexed by time (discrete stages)-of families of multistage stochastic optimization problems. At each time, the optimization problems in a family are parameterized by some quantities (initial states, constraint levels.. .). In this framework, we introduce an adapted notion of time consistent optimal solutions, that is, solutions that remain optimal after truncation of the past and that are optimal for any values of the parameters. We link this time consistency notion with the concept of state variable in Markov Decision Processes for a class of multistage stochastic optimization problems incorporating state constraints at the final time, either formulated in expectation or in probability. For such problems, when the primitive noise random process is stagewise independent and takes a finite number of values, we show that time consistent solutions can be obtained by considering a finite dimensional state variable. We illustrate our results on a simple dam management problem. (10.1137/22M151830X)
  DOI : 10.1137/22M151830X
• Optimal computation of integrals in the Half-Space Matching method for modal simulation of SHM/NDE in 3D elastic plate
  
  • Allouko Amond
  • Bonnet-Ben Dhia Anne-Sophie
  • Lhémery Alain
  • Baronian Vahan
  Journal of Physics: Conference Series, IOP Science, 2024, 2768, pp.012004. Simulating structural health monitoring (SHM) or nondestructive evaluation (NDE) methods based on elastic guided waves (GW) is very helpful to handle their complexity (co-existence of several GW modes, frequency dependence of wavespeed) and to further design optimal methods of inspection offering high sensitivity to the sought flaws. The half-space matching (HSM) method has been established for the development of a model that hybridizes local finite element (FE) computations for GW scattering by a flaw, with a modal semi-analytical model for GW radiation and propagation in flawless plate-like structures. Highly oscillatory Integral formulae appear in the HSM method that radiate the scattered field away from the FE zone as the superimposition of modal contributions, which computation can be time-consuming. The present work is concerned with their optimal computation. Integral of this form can be efficiently computed under the far-field approximation but this classical technique fails at predicting accurately wavefields at relatively short distances (small number of wavelengths). The method developed herein relies on the complexification of the integrals to be computed and on specific deformation of integration paths in the complex plane, as detailed in the paper. It allows the evaluation of the integrals without approximation other than that of numerical quadratures, ensuring high accuracy while offering high computing performances. It indifferently applies in the far-field and in the near-field. The method of computation is validated by comparing its predictions with a reference solution of GW scattering. Its computational performances are also demonstrated, compared to those of the standard computation of the HSM integral formulae to be computed and on specific deformation of integration paths in the complex plane, as detailed in the paper. It allows the evaluation of the integrals without approximation other than that of numerical quadratures, ensuring high accuracy while offering high computing performances. It indifferently applies in the far-field and in the near-field. The method of computation is validated by comparing its predictions with a reference solution of GW scattering. Its computational performances are also demonstrated, compared to those of the standard computation of the HSM integral formulae. (10.1088/1742-6596/2768/1/012004)
  DOI : 10.1088/1742-6596/2768/1/012004
• Automated far-field sound field estimation combining robotized acoustic measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  , 2024. The identification and reconstruction of acoustic fields radiated by unknown structures isusually performed using either Sound Field Estimation (SFE) or Near-field Acoustic Holog-raphy (NAH) techniques. The latter turns out to be especially useful when data is onlyavailable close to the source, but information throughout the whole space is needed.Yet, the lack of amendable and efficient implementations of state-of-the-art solutions, aswell as the laborious and often lengthy deployment of acoustic measurements continue to besignificant obstacles to the practical application of such methods.The purpose of this work is to address both problems. First, a completely automated metrol-ogy setup is proposed, in which a robotic arm is used to gather extensive, yet accurate ge-ometric and acoustic data without any human intervention. The impact of the robot onacoustic pressure measurements has been cautiously estimated, and proved to remain negli-gible within a defined validity frequency range.The sound field prediction is then tackled using the Boundary Element Method (BEM), andimplemented using the FreeFEM++ BEM library. Numerically simulated measurements haveallowed us to assess the method accuracy, which matches theoretically expected results, androbustness against positioning inaccuracies, provided that the robot is carefully calibrated.The overall solution has been successfully tested using actual robotized measurements of anunknown loudspeaker, with a reconstruction error of less than 30 % on the previously definedvalidity frequency range
• Multidomain FEM-BEM coupling for acoustic scattering
  
  • Bonazzoli Marcella
  • Claeys Xavier
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2024, 36 (2), pp.129-167. We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a Gårding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances. (10.1216/jie.2024.36.129)
  DOI : 10.1216/jie.2024.36.129
• Clustering data for the Optimal Classication Tree Problem
  
  • Ales Zacharie
  • Huré Valentine
  • Lambert Amélie
  , 2024. Solving the optimal classification tree problem enables to compute classifiers which are both interpretable and efficient. Most of the exact methods for this problem are based on on a Mixed Integer Linear Program (MILP) formulation. However, the efficiency of MILP solvers generally does not allow these formulations to be solved directly, once the dataset exceeds a critical size. To address this challenge, we propose in this paper an iterative exact algorithm than handles medium-sized datasets from the state-of-the-art. The basic idea is to start by solving a MILP formulation on a small subset of data points representative of the considered dataset. Then, the subset is iteratively extended until global optimality of the initial problem is reached. A key feature is to compute relevant initial subsets of data points. For this, we introduce the concept of data-partitions and design several algorithms to compute them. We then define two MILP formulations to compute optimal classification trees on data-partitions. We prove that combining our iterative algorithm with our first formulation enables to obtain an optimal solution of the original problem. We also propose an alternative method based on the second formulation which is significantly faster. We present extensive computational experiments to compare our algorithms with state-of-the-art approaches. We show that our methods constitute the best compromise between in-sample accuracy and interpretability.
• Statistical Linearization for Robust Motion Planning
  
  • Leparoux Clara
  • Bonalli Riccardo
  • Hérissé Bruno
  • Jean Frédéric
  Systems and Control Letters, Elsevier, 2024, 189, pp.105825. The goal of robust motion planning consists of designing open-loop controls which optimally steer a system to a specific target region while mitigating uncertainties and disturbances which affect the dynamics. Recently, stochastic optimal control has enabled particularly accurate formulations of the problem. Nevertheless, despite interesting progresses, these problem formulations still require expensive numerical computations. In this paper, we start bridging this gap by leveraging statistical linearization. Specifically, through statistical linearization we reformulate the robust motion planning problem as a simpler deterministic optimal control problem subject to additional constraints. We rigorously justify our method by providing estimates of the approximation error, as well as some controllability results for the new constrained deterministic formulation. Finally, we apply our method to the powered descent of a space vehicle, showcasing the consistency and efficiency of our approach through numerical experiments. (10.1016/j.sysconle.2024.105825)
  DOI : 10.1016/j.sysconle.2024.105825
• Nonlinear Optimization Filters for Stochastic Time-Varying Convex Optimization
  
  • Simonetto Andrea
  • Massioni Paolo
  International Journal of Robust and Nonlinear Control, Wiley, 2024. 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 nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case. Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum-of-squares relaxations. 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. (10.1002/rnc.7380)
  DOI : 10.1002/rnc.7380
• Spectrum of the Laplacian with mixed boundary conditions in a chamfered quarter of layer
  
  • Chesnel Lucas
  • Nazarov Sergei A.
  • Taskinen Jari
  Journal of Spectral Theory, European Mathematical Society, 2024. We investigate the spectrum of a Laplace operator with mixed boundary conditions in an unbounded chamfered quarter of layer. This problem arises in the study of the spectrum of the Dirichlet Laplacian in thick polyhedral domains having some symmetries such as the so-called Fichera layer. The geometry we consider depends on two parameters gathered in some vector κ = (κ_1,κ_2) which characterizes the domain at the edges. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. By exchanging the axes and/or modifying their orientations if necessary, it is sufficient to restrict the analysis to the cases κ_1\ge0 and κ_2∈ [−κ_1,κ_1]. We identify the essential spectrum and establish different results concerning the discrete spectrum with respect to κ. In particular, we show that for a given κ_1 > 0, there is some h(κ_1) > 0 such that discrete spectrum exists for κ_2 ∈ (−κ_1,0) ∪ (h(κ_1),κ_1) whereas it is empty for κ_2 ∈ [0; h(κ_1)]. The proofs rely on classical arguments of spectral theory such as the max-min principle. The main originality lies rather in the delicate use of the features of the geometry.
• Simulation and analysis of sign-changing Maxwell’s equations in cold plasma
  
  • Peillon Etienne
  , 2024. Nowadays, plasmas are mainly used for industrial purpose. One of the most frequently cited examples of industrial use is electric energy production via fusion nuclear reactors. Then, in order to contain plasma properly inside the reactor, a background magnetic field is imposed, and the density and temperature of the plasma must be precisely controlled. This is done by sending electromagnetic waves at specific frequencies and directions depending on the characteristics of the plasma.The first part of this PhD thesis consists in the study of the model of plasma in a strong background magnetic field, which corresponds to a hyperbolic metamaterial. The objective is to extend the existing results in 2D to the 3D-case and to derive a radiation condition. We introduce a splitting of the electric and magnetic fields resembling the usual TE and TM decomposition, then, it gives some results on the two resulting problems. The results are in a very partial state, and constitute a rough draft on the subject.The second part consists in the study of the degenerate PDE associated to the lower-hybrid resonant waves in plasma. The associated boundary-value problem is well-posed within a ``natural'' variational framework. However, this framework does not include the singular behavior presented by the physical solutions obtained via the limiting absorption principle. Notice that this singular behavior is important from the physical point of view since it induces the plasma heating mentioned before. One of the key results of this second part is the definition of a notion of weak jump through the interface inside the domain, which allows to characterize the decomposition of the limiting absorption solution into a regular and a singular parts.
• New optimization models for optimal classification trees
  
  • Alès Zacharie
  • Huré Valentine
  • Lambert Amélie
  Computers and Operations Research, Elsevier, 2024, 164, pp.106515. Interpretability is a growing concept in Machine Learning. Decision-making algorithms are more and more used in healthcare, finance or other high stakes contexts. Therefore, the need for algorithms whose decisions are understandable is of the utmost importance. Intrinsically interpretable classifiers such as decision trees are often seen as less accurate than black box models such as neural networks. For decision trees, state-of-the-art methods are recursive heuristics (e.g. CART) that may fail to find underlying characteristics in datasets. Recently, linear formulations were introduced to model the problem of the construction of the best decision tree for a given dataset. Notably, a MIO formulation, introduced by Bertsimas and al., has shown better accuracy than CART. However this model does not scale up to datasets with more than 1000 data points. Our work focuses on improvements of MIOs that speed up their resolution in order to handle larger problems. We present a quadratic formulation of the MIP devised by Bertsimas and al. as well as its linearization and another that extends a flow-formulation (from binary dataset to real-value dataset). We prove that our new formulations have stronger continuous relaxation than the MIP introduced by Bertsimas and al.. Finally, our experiments show that they have a significantly smaller resolution time than the MIP of Bertsimas and al. while maintaining or improving performances on test sets. (10.1016/j.cor.2023.106515)
  DOI : 10.1016/j.cor.2023.106515
• MAPL: Model Agnostic Peer-to-peer Learning
  
  • Mukherjee Sayak
  • Simonetto Andrea
  • Jamali-Rad Hadi
  , 2024. Effective collaboration among heterogeneous clients in a decentralized setting is a rather unexplored avenue in the literature. To structurally address this, we introduce Model Agnostic Peer-to-peer Learning (coined as MAPL) a novel approach to simultaneously learn heterogeneous personalized models as well as a collaboration graph through peer-to-peer communication among neighboring clients. MAPL is comprised of two main modules: (i) local-level Personalized Model Learning (PML), leveraging a combination of intra- and inter-client contrastive losses; (ii) network-wide decentralized Collaborative Graph Learning (CGL) dynamically refining collaboration weights in a privacy-preserving manner based on local task similarities. Our extensive experimentation demonstrates the efficacy of MAPL and its competitive (or, in most cases, superior) performance compared to its centralized model-agnostic counterparts, without relying on any central server. Our code is available and can be accessed here: https://github.com/SayakMukherjee/MAPL
• Imagerie d’interface barrage-fondation par inversion de forme d'onde complète
  
  • Boukraa Mohamed Aziz
  • Audibert Lorenzo
  • Bonazzoli Marcella
  • Haddar Houssem
  • Vautrin Denis
  , 2024, 504, pp.04002. Dans le cadre de l’étude de la stabilité des barrages, la connaissance de l’interface entre le barrage et la roche revêt une grande importance. Le recours à des techniques géophysiques peut apporter des informations complémentaires par rapport aux mesures géotechniques. Nous proposons ici une méthode de traitement des mesures sismiques, l’objectif étant d'obtenir une image de l'interface entre le béton du barrage et le rocher de la fondation avec une résolution métrique. Il s’agit d’une technique de type « Full Waveform Inversion » avec optimisation de forme. Des résultats numériques utilisant des mesures synthétiques montrent la capacité de la méthode à retrouver l'interface avec une précision satisfaisante, pour un nombre limité de points de mesure et en présence de bruit. (10.1051/e3sconf/202450404002)
  DOI : 10.1051/e3sconf/202450404002