Publications

2022

• Construction d'arbres de décision optimaux
  
  • Huré Valentine
  • Alès Zacharie
  • Lambert Amélie
  , 2022. Construction d'arbres de décision optimaux
• AutoExpe.jl : Ne coder que les méthodes de résolution
  
  • Alès Zacharie
  , 2022. AutoExpe.jl est un package julia permettant d'automatiser la réalisation d'expérimentations numériques et la génération de tableaux de résultats afin de se concentrer sur l'essentiel : l'implémentation des méthodes de résolution et la comparaison de leurs performances. Lien : https://github.com/ZacharieALES/AutoExpe
• Modélisations d'arbres de décision optimaux
  
  • Alès Zacharie
  • Huré Valentine
  • Lambert Amélie
  , 2022.
• An efficient Benders decomposition for the p-median problem
  
  • Durán Mateluna Cristian
  • Alès Zacharie
  • Elloumi Sourour
  , 2022.
• Optimisation du dimensionnement d'une flotte de véhicules électriques et de leurs bornes de recharge par des méthodes de décomposition
  
  • Girault Ludovic
  • Triboulet Thomas
  • Wan Cheng
  • Dupuis Guilhem
  • Griset Rodolphe
  , 2022. Optimisation du dimensionnement d'une flotte de véhicules électriques et de leurs bornes de recharge par des méthodes de décomposition
• Planification optimisée du déploiement d'un réseau de télécommunication multitechnologie par dispositifs aéroportés sur un théâtre d'opérations extérieures
  
  • Alès Zacharie
  • Elloumi Sourour
  • Naghmouchi M. Yassine
  • Pass-Lanneau Adèle
  • Thuillier Owein
  , 2022.
• Maxwell's equations in presence of metamaterials
  
  • Rihani Mahran
  , 2022. The main subject of this thesis is the study of time-harmonic electromagnetic waves in a heterogeneous medium composed of a dielectric and a negative material (i.e. with a negative dielectric permittivity ε and/or a negative magnetic permeability μ) which are separated by an interface with a conical tip. Because of the sign-change in ε and/or μ, the Maxwell’s equations can be ill-posed in the classical L2 −frameworks. On the other hand, we know that when the two associated scalar problems, involving respectively ε and μ, are well-posed in H1, the Maxwell’s equations are well-posed. By combining the T-coercivity approach with the Mellin analysis in weighted Sobolev spaces, we present, in the first part of this work, a detailed study of these scalar problems. We prove that for each of them, the well-posedeness in H1 is lost iff the associated contrast belong to some critical set called the critical interval. These intervals correspond to the sets of negative contrasts for which propagating singularities, also known as black hole waves, appear at the tip. Contrary to the case of a 2D corner, for a 3D tip, several black hole waves can exist. Explicit expressions of these critical intervals are obtained for the particular case of circular conical tips. For critical contrasts, using the Mandelstam radiation principle, we construct functional frameworks in which well-posedness of the scalar problems is restored. The physically relevant framework is selected by a limiting absorption principle. In the process, we present a new numerical strategy for 2D/3D scalar problems in the non-critical case. This approach, presented in the second part of this work, contrary to existing ones, does not require additional assumptions on the mesh near the interface. The third part of the thesis concerns Maxwell’s equations with one or two critical coefficients. By using new results of vector potentials in weighted Sobolev spaces, we explain how to construct new functional frameworks for the electric and magnetic problems, directly related to the ones obtained for the two associated scalar problems. If one uses the setting that respects the limiting absorption principle for the scalar problems, then the settings provided for the electric and magnetic problems are also coherent with the limiting absorption principle. Finally, the last part is devoted to the homogenization process for time-harmonic Maxwell’s equations and associated scalar problems in a 3D domain that contains a periodic distribution of inclusions made of negative material. Using the T-coercivity approach, we obtain conditions on the contrasts such that the homogenization results is possible for both the scalar and the vector problems. Interestingly, we show that the homogenized matrices associated with the limit problems are either positive definite or negative definite.
• Improvement of hierarchical matrices for 3D elastodynamic problems with a complex wavenumber
  
  • Bagur Laura
  • Chaillat Stéphanie
  • Ciarlet Patrick
  Advances in Computational Mathematics, Springer Verlag, 2022, 48 (9). It is well known in the literature that standard hierarchical matrix (H-matrix) based methods, although very efficient for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous paper, we have shown that the method should nevertheless be used in the mechanical engineering community due to its still important data-compression rate and its straightforward implementation compared to H 2-matrix, or directional, approaches. Since in practice, not all materials are purely elastic it is important to be able to consider visco-elastic cases. In this context, we study the effect of the introduction of a complex wavenumber on the accuracy and efficiency of H-matrix based fast methods for solving dense linear systems arising from the discretization of the elastodynamic (and Helmholtz) Green's tensors. Interestingly, such configurations are also encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. Relying on the theory proposed in [12] for H 2-matrices for Helmholtz problems, we study the influence of the introduction of damping on the data compression rate of standard H-matrices. We propose an improvement of H-matrix based fast methods for this kind of configuration. This work is complementary to the recent work [12]. Here, in addition to addressing another physical problem, we consider standard H-matrices, derive a simple criterion to introduce additional compression and we perform extensive numerical experiments. (10.1007/s10444-021-09921-3)
  DOI : 10.1007/s10444-021-09921-3
• Stochastic Analysis of non-Markovian irregular phenomena
  
  • Teixeira Nicácio de Messias Alan
  , 2022. This thesis focuses on some particular stochastic analysis aspects of non-Markovian irregular phenomena. It formulates existence and uniqueness for some martingale problems involving two types of irregulat drifts perturbed by path-dependant functionals: the first one is related to the case which is the derivative of continuous function and it models irregular path-dependent media; the second one concerns the case when the drift is of Bessel type in low dimension. Finally the thesis also focuses on rough paths techniques and its relation with the stochastic calculus via regularization.
• Stochastic incremental mirror descent algorithms with Nesterov smoothing
  
  • Grad Sorin-Mihai
  • Bitterlich Sandy
  , 2022. We propose a stochastic incremental mirror descent method constructed by means of the Nesterov smoothing for minimizing a sum of finitely many proper, convex and lower semicontinuous functions over a nonempty closed convex set in a Euclidean space. The algorithm can be adapted in order to minimize (in the same setting) a sum of finitely many proper, convex and lower semicontinuous functions composed with linear operators. Another modification of the scheme leads to a stochastic incremental mirror descent Bregman-proximal scheme with Nesterov smoothing for minimizing the sum of finitely many proper, convex and lower semicontinuous functions with a prox-friendly proper, convex and lower semicontinuous function in the same framework. 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.
• Lp-asymptotic stability of 1D damped wave equations with localized and linear damping
  
  • Kafnemer Meryem
  • Benmiloud Mebkhout
  • Jean Frédéric
  • Chitour Yacine
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022. In this paper, we study the $L^p$-asymptotic stability of the one-dimensional linear damped wave equation with Dirichlet boundary conditions in $[0,1]$, with $p\in (1,\infty)$. The damping term is assumed to be linear and localized to an arbitrary open sub-interval of $[0,1]$. We prove that the semi-group $(S_p(t))_{t\geq 0}$ associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether $p\geq 2$ or $1<p<2$. (10.1051/cocv/2021107)
  DOI : 10.1051/cocv/2021107
• Efficient evaluation of three-dimensional Helmholtz Green's functions tailored to arbitrary rigid geometries for flow noise simulations
  
  • Chaillat Stéphanie
  • Cotté Benjamin
  • Mercier Jean-François
  • Serre Gilles
  • Trafny Nicolas
  Journal of Computational Physics, Elsevier, 2022, 452. The Lighthill's wave equation provides an accurate characterization of the hydrodynamic noise due to the interaction between a turbulent flow and an obstacle, that is needed to get in many industrial applications. In the present study, to solve the Lighthill's equation expressed as a boundary integral equation, we develop an efficient numerical method to determine the three-dimensional Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying a Neumann boundary condition. This so-called tailored Green's function is useful to reduce the computational costs to solve the Lighthill's equation. The first step consists in deriving an integral equation to express the tailored Green's function thanks to the free space Green's function. Then a Boundary Element Method (BEM) is used to compute tailored Green's functions. Furthermore, an efficient method is performed to compute the second derivatives needed for accurate flow noise determinations. The proposed approach is first tested on simple geometries for which analytical solutions can be determined (sphere, cylinder, half plane). In order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. A discussion on the numerical efficiency and accuracy of these approaches in an industrial context is finally proposed. (10.1016/j.jcp.2021.110915)
  DOI : 10.1016/j.jcp.2021.110915
• Quadratic reformulations for the optimization of pseudo-boolean functions
  
  • Crama Yves
  • Elloumi Sourour
  • Lambert Amélie
  • Rodriguez-Heck Elisabeth
  , 2022. We investigate various solution approaches for the uncon- strained minimization of a pseudo-boolean function. More precisely, we assume that the original function is expressed as a real-valued polynomial in 0-1 variables, of degree three or more, and we consider a generic family of two-step ap- proaches for its minimization. First, a quadratic reformu- lation step aims at transforming the minimization problem into an equivalent constrained or unconstrained quadratic 0-1 minimization problem (where “equivalent” means here that a minimizer of the original function can be easily de- duced from a minimizer of the reformulation). Second, an optimization step handles the obtained equivalent quadratic problem. We provide a unified presentation of several quadratic re- formulation schemes proposed in the literature, e.g., (An- thony et al. 2017; Buchheim and Rinaldi 2007; Rodr ́ıguez- Heck 2018; Rosenberg 1975), and we review several meth- ods that can be applied in the optimization step, including a standard linearization procedure (Fortet 1959) and more elaborate convex quadratic reformulations, as in (Billionnet and Elloumi 2007; Billionnet, Elloumi, and Lambert 2012, 2016; Elloumi, Lambert, and Lazare 2021). We discuss the impact of the reformulation scheme on the efficiency of the optimization step and we illustrate our discussion with some computational results on different classes of instances.
• Structure of optimal control for planetary landing with control and state constraints
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28. This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc. (10.1051/cocv/2022065)
  DOI : 10.1051/cocv/2022065
• On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
  
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2022, 152, pp.223-257. In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived. (10.1007/s00211-022-01315-x)
  DOI : 10.1007/s00211-022-01315-x
• Kernel representation of Kalman observer and associated H-matrix based discretization
  
  • Aussal Matthieu
  • Moireau Philippe
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28, pp.78. In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization. (10.1051/cocv/2022071)
  DOI : 10.1051/cocv/2022071
• An efficient numerical method for time domain electromagnetic wave propagation in co-axial cables
  
  • Beni Hamad Akram
  • Beck Geoffrey
  • Imperiale Sébastien
  • Joly Patrick
  Computational Methods in Applied Mathematics, De Gruyter, 2022, 22 (4). In this work we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon an hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objective is to validate numerically generalized Telegrapher's models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article. (10.1515/cmam-2021-0195)
  DOI : 10.1515/cmam-2021-0195
• The Morozov's principle applied to data assimilation problems
  
  • Bourgeois Laurent
  • Dardé Jérémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022. This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation. (10.1051/m2an/2022061)
  DOI : 10.1051/m2an/2022061
• Distributed Multistage Optimization of Large-Scale Microgrids under Stochasticity
  
  • Pacaud François
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  IEEE Transactions on Power Systems, Institute of Electrical and Electronics Engineers, 2022, 37 (1). Microgrids are recognized as a relevant tool to absorb decentralized renewable energies in the energy mix. However, the sequential handling of multiple stochastic productions and demands, and of storage, make their management a delicate issue. We add another layer of complexity by considering microgrids where different buildings stand at the nodes of a network and are connected by the arcs; some buildings host local production and storage capabilities, and can exchange with others their energy surplus. We formulate the problem as a multistage stochastic optimization problem, corresponding to the minimization of the expected temporal sum of operational costs, while satisfying the energy demand at each node, for all time. The resulting mathematical problem has a large-scale nature, exhibiting both spatial and temporal couplings. However, the problem displays a network structure that makes it amenable to a mix of spatial decomposition-coordination with temporal decomposition methods. We conduct numerical simulations on microgrids of different sizes and topologies, with up to 48~nodes and 64~state variables. Decomposition methods are faster and provide more efficient policies than a state-of-the-art Stochastic Dual Dynamic Programming algorithm. Moreover, they scale almost linearly with the state dimension, making them a promising tool to address more complex microgrid optimal management problems. (10.1109/TPWRS.2021.3087775)
  DOI : 10.1109/TPWRS.2021.3087775
• Computing weakly singular and near-singular integrals over curved boundary elements
  
  • Montanelli Hadrien
  • Aussal Matthieu
  • Haddar Houssem
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2022, 44 (6), pp.A3728-A3753. (10.1137/21M1462027)
  DOI : 10.1137/21M1462027
• A Differential game control problem with state constraints
  
  • Gammoudi Nidhal
  • Zidani Hasnaa
  Mathematical Control and Related Fields, AIMS, 2022. We study the Hamilton-Jacobi (HJ) approach for a two-person zero-sum differential game with state constraints and where controls of the two players are coupled within the dynamics, the state constraints and the cost functions. It is known for such problems that the value function may be discontinuous and its characterization by means of an HJ equation requires some controllability assumptions involving the dynamics and the set of state constraints. In this work, we characterize this value function through an auxiliary differential game free of state constraints. Furthermore, we establish a link between the optimal strategies of the constrained problem and those of the auxiliary problem and we present a general approach allowing to construct approximated optimal feedbacks to the constrained differential game for both players. Finally, an aircraft landing problem in the presence of wind disturbances is given as an illustrative numerical example. (10.3934/mcrf.2022008)
  DOI : 10.3934/mcrf.2022008
• Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations
  
  • Barrasso Adrien
  • Russo Francesco
  Journal of Stochastic Analysis, Louisiana State University, 2022, 3 (1). We discuss a class of Backward Stochastic Differential Equations (BSDEs) with no driving martingale. When the randomness of the driver depends on a general Markov process $X$, those BSDEs are denominated Markovian BSDEs and can be associated to a deterministic problem, called Pseudo-PDE which constitute the natural generalization of a parabolic semilinear PDE which naturally appears when the underlying filtration is Brownian. We consider two aspects of well-posedness for the Pseudo-PDEs: "classical" and "martingale" solutions. (10.31390/josa.3.1.03)
  DOI : 10.31390/josa.3.1.03
• Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation
  
  • Carvalho Camille
  • Ciarlet Patrick
  • Scheid Claire
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 388, pp.114207. The limiting amplitude principle states that the response of a scatterer to a harmonic light excitation is asymptotically harmonic with the same pulsation. Depending on the geometry and nature of the scatterer, there might or might not be an established theoretical proof validating this principle. In this paper, we investigate a case where the theory is missing: we consider a two-dimensional dispersive Drude structure with corners. In the non lossy case, it is well known that looking for harmonic solutions leads to an ill-posed problem for a specific range of critical pulsations, characterized by the metal’s properties and the aperture of the corners. Ill-posedness is then due to highly oscillatory resonances at the corners called black-hole waves. However, a time-domain formulation with a harmonic excitation is always mathematically valid. Based on this observation, we conjecture that the limiting amplitude principle might not hold for all pulsations. Using a time-domain setting, we propose a systematic numerical approach that allows to give numerical evidences of the latter conjecture, and find clear signature of the critical pulsa- tions. Furthermore, we connect our results to the underlying physical plasmonic resonances that occur in the lossy physical metallic case. (10.1016/j.cma.2021.114207)
  DOI : 10.1016/j.cma.2021.114207
• An extension of the proximal point algorithm beyond convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  Journal of Global Optimization, Springer Verlag, 2022, 82, pp.313–329. Abstract We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity. (10.1007/s10898-021-01081-4)
  DOI : 10.1007/s10898-021-01081-4
• CRANDALL-LIONS VISCOSITY SOLUTIONS FOR PATH-DEPENDENT PDES: THE CASE OF HEAT EQUATION
  
  • Cosso Andrea
  • Russo Francesco
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2022, 28, pp.481-503. We address our interest to the development of a theory of viscosity solutions à la Crandall-Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0, T ]; R^d). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall-Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the more delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekeland's variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fréchet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural "candidate" solution. Finally, concerning the existence part, we provide a new proof of the functional Itô formula under general assumptions, extending earlier results in the literature. (10.3150/21-BEJ1353)
  DOI : 10.3150/21-BEJ1353