Publications

2018

• Stochastic decomposition applied to large-scale hydro valleys management
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • Leclère Vincent
  • Pacaud François
  European Journal of Operational Research, Elsevier, 2018. We are interested in optimally controlling a discrete time dynamical system that can be influenced by exogenous uncertainties. This is generally called a Stochas-tic Optimal Control (SOC) problem and the Dynamic Programming (DP) principle is one of the standard way of solving it. Unfortunately, DP faces the so-called curse of dimensionality: the complexity of solving DP equations grows exponentially with the dimension of the variable that is sufficient to take optimal decisions (the so-called state variable). For a large class of SOC problems, which includes important practical applications in energy management, we propose an original way of obtaining near optimal controls. The algorithm we introduce is based on Lagrangian relaxation, of which the application to decomposition is well-known in the deterministic framework. However, its application to such closed-loop problems is not straightforward and an additional statistical approximation concerning the dual process is needed. The resulting methodology is called Dual Approximate Dynamic Programming (DADP). We briefly present DADP, give interpretations and enlighten the error induced by the approximation. The paper is mainly devoted to applying DADP to the management of large hydro valleys. The modeling of such systems is presented, as well as the practical implementation of the methodology. Numerical results are provided on several valleys, and we compare our approach with the state of the art SDDP method. (10.1016/j.ejor.2018.05.025)
  DOI : 10.1016/j.ejor.2018.05.025
• Compact MILP formulations for the p-center problem
  
  • Alès Zacharie
  • Elloumi Sourour
  , 2018.
• Compact MILP formulations for the p-center problem
  
  • Alès Zacharie
  • Elloumi Sourour
  , 2018, 10856, pp.14-25. The p-center problem consists in selecting p centers among M to cover N clients, such that the maximal distance between a client and its closest selected center is minimized. For this problem we propose two new and compact integer formulations. Our first formulation is an improvement of a previous formulation. It significantly decreases the number of constraints while preserving the optimal value of the linear relaxation. Our second formulation contains less variables and constraints but it has a weaker linear relaxation bound. We besides introduce an algorithm which enables us to compute strong bounds and significantly reduce the size of our formulations. Finally, the efficiency of the algorithm and the proposed formulations are compared in terms of quality of the linear relaxation and computation time over instances from OR-Library. (10.1007/978-3-319-96151-4_2)
  DOI : 10.1007/978-3-319-96151-4_2
• Anomalous Chained Turbulence in Actively Driven Flows on Spheres
  
  • Mickelin Oscar
  • Słomka Jonasz
  • Burns Keaton
  • Lecoanet Daniel
  • Vasil Geoffrey
  • Faria Luiz
  • Dunkel Jörn
  Physical Review Letters, American Physical Society, 2018, 120 (16). Recent experiments demonstrate the importance of substrate curvature for actively forced fluid dynamics. Yet, the covariant formulation and analysis of continuum models for non-equilibrium flows on curved surfaces still poses theoretical challenges. Here, we introduce and study a generalized covariant Navier-Stokes model for fluid flows driven by active stresses in non-planar geometries. The analytical tractability of the theory is demonstrated through exact stationary solutions for the case of a spherical bubble geometry. Direct numerical simulations reveal a curvature-induced transition from a burst phase to an anomalous turbulent phase that differs distinctly from externally forced classical 2D Kolmogorov turbulence. This new type of active turbulence is characterized by theself-assembly of finite-size vortices into linked chains of anti-ferromagnetic order, which percolatethrough the entire fluid domain, forming an active dynamic network. The coherent motion of the vortex chain network provides an efficient mechanism for upward energy transfer from smaller to larger scales, presenting an alternative to the conventional energy cascade in classical 2D turbulence. (10.1103/PhysRevLett.120.164503)
  DOI : 10.1103/PhysRevLett.120.164503
• Extraction and Clustering of Two-Dimensional Dialogue Patterns
  
  • Alès Zacharie
  • Pauchet Alexandre
  • Knippel Arnaud
  International Journal on Artificial Intelligence Tools, World Scientific Publishing, 2018, 27 (02), pp.1850001. This article proposes a two-step methodology to ease the identification of dialogue patterns in a corpus of annotated dialogues. The annotations of a given dialogue are represented within a two-dimensional array whose lines correspond to the utterances of the dialogue ordered chronologically. The first step of our methodology consists in extracting recurrent patterns. To that end, we adapt a dynamic programming algorithm used to align two-dimensional arrays by reducing its complexity and improving its trace-back procedure. During the second step, the obtained patterns are clustered using various heuristics from the literature. As evaluation process, our method is applied onto a corpus of annotated dialogues between a parent and her child in a storytelling context. The obtained partitions of dialogue patterns are evaluated by an expert in child development of language to assess how the methodology helps the expert into explaining the child behaviors. The influence of the method parameters (clustering heuristics, minimum extraction score, number of clusters and substitution score array) are studied. Dialogue patterns that manual extractions have failed to detect are highlighted by the method and the most efficient values of the parameters are therefore determined. (10.1142/s021821301850001x)
  DOI : 10.1142/s021821301850001x
• Perfect transmission invisibility for waveguides with sound hard walls
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Nazarov Sergei
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2018. We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at −∞ and probes the resulting scattered field at −∞ and +∞. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted R and T. In [9], a technique has been proposed to construct waveguides with smooth walls such that R = 0 and |T | = 1 (non reflection). However the approach fails to ensure T = 1 (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound k depending on the geometry, we cannot have T = 1 so that the observer can detect the presence of the defect if he/she is able to measure the phase at +∞. In particular, if the perturbation is smooth and small (in amplitude and in width), k is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain T = 1. In this case, the scattered field is exponentially decaying both at −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects.
• Modelling resonant arrays of the Helmholtz type in the time domain
  
  • Maurel Agnès
  • Marigo Jean-Jacques
  • Mercier Jean-François
  • Pham Kim
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2018, 474 (2210), pp.20170894. We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought. (10.1098/rspa.2017.0894)
  DOI : 10.1098/rspa.2017.0894
• Comparaison de formulations pour le problème du p-centre
  
  • Alès Zacharie
  • Elloumi Sourour
  , 2018.
• Résolution du problème de suites binaires avec faible autocorrélation à l'aide d'une reformulation quadratique convexe
  
  • Lambert Amélie
  • Elloumi Sourour
  • Lazare Arnaud
  , 2018.
• Formulations for designing robust networks. An application to wind power collection
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre-Louis
  • Ridremont Thomas
  Electronic Notes in Discrete Mathematics, Elsevier, 2018, 64, pp.365-374. (10.1016/j.endm.2018.02.011)
  DOI : 10.1016/j.endm.2018.02.011
• Minimizing the weighted sum of completion times under processing time uncertainty
  
  • Alès Zacharie
  • Nguyen Thi Sang
  • Poss Michael
  Electronic Notes in Discrete Mathematics, Elsevier, 2018, 64, pp.15 - 24. We address the robust counterpart of a classical single machine scheduling problem by considering a budgeted uncertainty and an ellipsoidal uncertainty. We prove that the problem is N P-hard for arbitrary ellipsoidal uncertainty sets. Then, a mixedinteger linear programming reformulations and a second order cone programming reformulations are provided. We assess the reformulations on randomly generated instances, comparing them with branch-and-cut algorithms. (10.1016/j.endm.2018.01.003)
  DOI : 10.1016/j.endm.2018.01.003
• Extenseurs hamiltoniens minimaux.
  
  • Costa Marie-Christine
  • de Werra D
  • Picouleau Christophe
  , 2018.
• Stochastic optimal control of a domestic microgrid equipped with solar panel and battery
  
  • Pacaud François
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2018. —Microgrids are integrated systems that gather and operate energy production units to satisfy consumers demands. This paper details different mathematical methods to design the Energy Management System (EMS) of domestic microgrids. We consider different stocks coupled together — a battery, a domestic hot water tank — and decentralized energy production with solar panel. The main challenge of the EMS is to ensure, at least cost, that supply matches demand for all time, while considering the inherent uncertainties of such systems. We benchmark two optimization algorithms to manage the EMS, and compare them with a heuristic. The Model Predictive Control (MPC) is a well known algorithm which models the future uncertainties with a deterministic forecast. By contrast, Stochastic Dual Dynamic Programming (SDDP) models the future uncertainties as probability distributions to compute optimal policies. We present a fair comparison of these two algorithms to control microgrid. A comprehensive numerical study shows that i) optimization algorithms achieve significant gains compared to the heuristic, ii) SDDP outperforms MPC by a few percents, with a reasonable computational overhead.
• Méthodes d'échantillonnage appliquées à l'imagerie de défauts dans un guide d'ondes élastiques
  
  • Recoquillay Arnaud
  , 2018. De nombreuses structures utilisées industriellement peuvent être considérées comme des guides d'ondes, comme les plaques, les tuyaux ou encore le rails. La maintenance de ces structures nécessite de pouvoir détecter efficacement des défauts internes par le Contrôle Non Destructif. Nous nous intéressons dans ce manuscrit à l'application d'une méthode d'échantillonnage, la Linear Sampling Method, au CND des guides d'ondes élastiques, qui en particulier impose des sollicitations et des mesures à la surface du guide en régime temporel. La stratégie choisie repose sur une formulation modale et multi-fréquentielle de la LSM, spécifique aux guides d'ondes, qui permet une régularisation efficace et de nature physique du problème inverse, qui est par nature mal posé. Cette stratégie permet par ailleurs une optimisation du nombre et de la position des émetteurs et des récepteurs. Nous nous limitons dans un premier temps au cas scalaire du guide d'ondes acoustiques, pour ensuite s'attaquer au cas vectoriel, et par conséquent plus complexe, du guide d'ondes élastiques.L'efficacité de la méthode inverse est dans un premier temps démontrée sur des données artificielles (obtenues numériquement), puis sur des données réelles obtenues à l'aide d'expériences réalisées sur des plaques métalliques. Ces expériences confirment la faisabilité du CND par méthode d'échantillonnage dans un cadre industriel. Dans le cas où une seule sollicitation est réalisée, l'utilisation de la LSM est exclu. Nous utilisons une approche tout à fait différente et dite "extérieure", couplant une formulation mixte de quasi-réversibilité et une méthode de lignes de niveau, pour reconstruire le défaut.
• Robust capacitated trees and networks with uniform demands *
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre-Louis
  • Ridremont Thomas
  , 2018. We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in case of terminals with uniform demands. Formally, we are given a graph, capacity and cost functions on the edges, a root, a subset of nodes called terminals, and a bound k on the number of edge failures. We first study the problem where k = 1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Second, we consider the problem of computing a minimum-cost survivable network, i.e., a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set based formulation, a flow based one, and a bilevel one (with an attacker and a defender). We propose algorithms to solve each formulation and compare their efficiency.
• Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost
  
  • Kröner Axel
  • Picarelli Athena
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56 (5), pp.3296-3319. An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order Hamilton-Jacobi-Bellman (HJB) equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. These properties are verified for a specific semi-Lagrangian scheme. (10.1137/17M115253X)
  DOI : 10.1137/17M115253X
• Sub-wavelength sensing of bi-periodic materials using topological sensitivity of second-order homogenized model
  
  • Bonnet Marc
  • Cornaggia Rémi
  • Guzina Bojan B
  Journal of Physics: Conference Series, IOP Science, 2018, 1131, pp.012008. We aim to detect defects or perturbations of periodic media, e.g. due to a defective manufacturing process. To this end, we consider scalar waves in such media through the lens of a second-order macroscopic description, and we compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. Then, we present a method that permits sub-wavelength sensing of periodic media, given the (anisotropic) phase velocity of plane waves illuminating the considered medium for several angles and wavenumbers. (10.1088/1742-6596/1131/1/012008)
  DOI : 10.1088/1742-6596/1131/1/012008
• Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
  
  • Ghilli Daria
  • Rao Zhiping
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2018. This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results in [26] in a more general framework with switching running costs and weaker controllability assumptions. The comparison principle has been established to guarantee the uniqueness and the stability results for the HJB system on such multi-domains. In the lower semi-continuous case, we characterize the value function as the unique lower semi-continuous viscosity solution of the HJB system, under a local controllability assumption. (10.1051/cocv/2018072)
  DOI : 10.1051/cocv/2018072
• Mesh requirements for the finite element approximation of problems with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2018, 138, pp.801-838. Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the T-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design. (10.1007/s00211-017-0923-5)
  DOI : 10.1007/s00211-017-0923-5
• Mathematical foundations of computational electromagnetism
  
  • Assous Franck
  • Ciarlet Patrick
  • Labrunie Simon
  , 2018. This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD). These equations and boundary conditions are discussed, including a brief review of absorbing boundary conditions. The focus then moves to well‐posedness results. The relevant function spaces are introduced, with an emphasis on boundary and topological conditions. General variational frameworks are defined for static and quasi-static problems, time-harmonic problems (including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue problems), and time-dependent problems, with or without constraints. They are then applied to prove the well-posedness of Maxwell’s equations and their simplified models, in the various settings described above. The book is completed with a discussion of dimensionally reduced models in prismatic and axisymmetric geometries, and a survey of existence and uniqueness results for the Vlasov-Poisson, Vlasov-Maxwell and MHD equations. The book addresses mainly researchers in applied mathematics who work on Maxwell’s equations. However, it can be used for master or doctorate-level courses on mathematical electromagnetism as it requires only a bachelor-level knowledge of analysis. (10.1007/978-3-319-70842-3)
  DOI : 10.1007/978-3-319-70842-3
• On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials
  
  • Bécache Eliane
  • Joly Patrick
  • Vinoles Valentin
  Mathematics of Computation, American Mathematical Society, 2018, 87, pp.2775-2810. This work deals with Perfectly Matched Layers (PMLs) in the context of dispersive media, and in particular for Negative Index Metamaterials (NIMs). We first present some properties of dispersive isotropic Maxwell equations that include NIMs. We then demonstrate numerically the inherent instabilities of the classical PMLs applied to NIMs. We propose and analyse the stability of very general PMLs for a large class of dispersive systems using a new change of variable. We give necessary criteria for the stability of such models. For dispersive isotropic Maxwell equations, this analysis is completed by giving necessary and sufficient conditions of stability. Finally, we propose new PMLs that satisfy these criteria and demonstrate numerically their efficiency. (10.1090/mcom/3307)
  DOI : 10.1090/mcom/3307
• Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  Optimal Control Applications and Methods, Wiley, 2018, 39 (5), pp.1833-1858. The goal of this paper is to show how non-parametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution , from a relatively small sample. We then show how this technique can be applied and implemented for a class of problems including the God-dard problem and the trajectory optimization of an Ariane 5-like launcher. (10.1002/oca.2445)
  DOI : 10.1002/oca.2445
• Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements
  
  • Albella Martínez Jorge
  • Imperiale Sébastien
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Scientific Computing, Springer Verlag, 2018. In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach. (10.1007/s10915-018-0768-9)
  DOI : 10.1007/s10915-018-0768-9
• Monte-Carlo Algorithms for Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  Monte Carlo Methods and Applications, De Gruyter, 2018, 24 (1), pp.55-70. The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments. (10.1515/mcma-2018-0005)
  DOI : 10.1515/mcma-2018-0005
• Error estimates for numerical approximation of Hamilton-Jacobi equations related to hybrid control systems
  
  • Ferretti Roberto
  • Sassi Achille
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2018. Hybrid control systems are dynamical systems that can be controlled by a combination of both continuous and discrete actions. In this paper we study the approximation of optimal control problems associated to this kind of systems, and in particular of the Quasi-Variational Inequality which characterizes the value function. Our main result features the error estimates between the value function of the problem and its approximation. We also focus on the hypotheses describing the mathematical model and the properties defining the class of numerical scheme for which the result holds true. (10.1007/s00245-018-9515-8)
  DOI : 10.1007/s00245-018-9515-8