Publications

2014

• Statistical mechanics of self-gravitating systems: Mixing as a criterion for indistinguishability
  
  • Beraldo Leandro
  • Lima Marcos
  • Sodré Laerte
  • Perez Jérôme
  Physical Review D, American Physical Society, 2014, 90 (12), pp.123004. We propose an association between the phase-space mixing level of a self-gravitating system and the indistinguishability of its constituents (stars or dark matter particles). This represents a refinement in the study of systems exhibiting incomplete violent relaxation. Within a combinatorial analysis similar to that of Lynden-Bell, we make use of this association to obtain a distribution function that deviates from the Maxwell-Boltzmann distribution, increasing its slope for high energies. Considering the smallness of the occupation numbers for large distances from the center of the system, we apply a correction to Stirling's approximation which increases the distribution slope also for low energies. The distribution function thus obtained presents some resemblance to the " S " shape of distributions associated with cuspy density profiles (as compared to the distribution function obtained from the Einasto profile), although it is not quite able to produce sharp cusps. We also argue how the association between mixing level and indistinguishability can provide a physical meaning to the assumption of particle-permutation symmetry in the N-particle distribution function, when it is used to derive the one-particle Vlasov equation, which raises doubts about the validity of this equation during violent relaxation. (10.1103/PhysRevD.90.123004)
  DOI : 10.1103/PhysRevD.90.123004
• Energy management method for an electric vehicle
  
  • Granato Giovanni
  • Bonnans J. Frederic
  • Aouchiche K.
  • Grégory Rousseau
  • Zidani Hasnaa
  , 2014. The invention relates to a method for managing energy consumption for an automobile having an electric battery and a heat engine, said method making it possible to select the use phases of said engine along a given route so as to minimize the fuel consumption of said vehicle. The main characteristic of the method according to the invention is that it includes the following steps: a step of cutting the road network, which is taken into consideration for a given route, into a plurality of segments, each segment being defined by an input node and by an output node; a step of calculating, from a speed associated with said segment, the probability of a speed transition between a speed at an input node and a speed at an output node of a segment, while taking a plurality of speeds at the input node and a plurality speeds at the output node into consideration, said step being carried out gradually over all of the segments of the route; a step of applying a stochastic optimization algorithm taking all the possible transition scenarios between each input node and each output node, and the probability associated therewith, into account, and taking a fuel consumption model between two successive nodes into account, said step being carried out over all of the segments of the route; and a step of selecting use phases of the heat engine along the route.
• Qualitative and Asymptotic Theory of Detonations
  
  • Faria Luiz
  , 2014. Shock waves in reactive media possess very rich dynamics: from formation of cells in multiple dimensions to oscillating shock fronts in one-dimension. Because of the extreme complexity of the equations of combustion theory, most of the current understanding of unstable detonation waves relies on extensive numerical simulations of the reactive compressible Euler/Navier-Stokes equations. Attempts at a simplified theory have been made in the past, most of which are very successful in describing steady detonation waves. In this work we focus on obtaining simplified theories capable of capturing not only the steady, but also the unsteady behavior of detonation waves. The first part of this thesis is focused on qualitative theories of detonation, where ad hoc models are proposed and analyzed. We show that equations as simple as a forced Burgers equation can capture most of the complex phenomena observed in detonations. In the second part of this thesis we focus on rational theories, and derive a weakly nonlinear model of multi-dimensional detonations. We also show, by analysis and numerical simulations, that the asymptotic equations provide good quantitative predictions.
• Singular perturbation of optimal control problems on multi-domains
  
  • Forcadel Nicolas
  • Rao Zhiping
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (5), pp.2917–2943. The goal of this paper is to study a singular perturbation problem in the framework of optimal control on multi-domains. We consider an optimal control problem in which the controlled system contains a fast and a slow variables. This problem is reformulated as an Hamilton-Jacobi-Bellman (HJB) equation. The main difficulty comes from the fact that the fast variable lives in a multi-domain. The geometric singularity of the multi-domains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variables, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approache and by means of the tools of the control theory. (10.1137/130916709)
  DOI : 10.1137/130916709
• Finite Element Heterogeneous Multiscale Method for the Classical Helmholtz Equation
  
  • Ciarlet Patrick
  • Stohrer Christian
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2014, 352 (9), pp.755–760. We show that the standard Finite Element Heterogeneous Multiscale Method (FE-HMM) can be used to approximate the effective behavior of solutions to the classical Helmholtz equation in highly oscillatory media. Using a novel combination of well-known results about FE-HMM and the notion of T-coercivity, we derive an a priori error bound. Numerical experiments corroborate the analytical findings. (10.1016/j.crma.2014.07.006)
  DOI : 10.1016/j.crma.2014.07.006
• Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides
  
  • Fliss Sonia
  • Klindworth Dirk
  • Schmidt Kersten
  BIT Numerical Mathematics, Springer Verlag, 2014, 55 (1), pp.35. The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides. (10.1007/s10543-014-0521-1)
  DOI : 10.1007/s10543-014-0521-1
• Optimal control of a semilinear parabolic equation with singular arcs
  
  • Bonnans Joseph Frederic
  Optimization Methods and Software, Taylor & Francis, 2014, 29 (2), pp.964-978. This paper develops a theory of singular arc, and the corresponding second order necessary and sufficient conditions, for the optimal control of a semilinear parabolic equation with scalar control applied on the r.h.s. We obtain in particular an extension of Kelley's condition, and the characterization of a quadratic growth property for a weak norm. (10.1080/10556788.2013.830220)
  DOI : 10.1080/10556788.2013.830220
• Commande optimale en temps minimal d'un procédé biologique d'épuration de l'eau
  
  • Bouafs Walid
  • Abdellatif Nahla
  • Jean Frédéric
  • Jérôme Harmand
  Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, African Society in Digital Science, 2014, Volume 18, 2014, pp.35-51. In this work, we consider an optimal control problem of a biological sequencing batch reactor for the treatment of pollutants. This model includes two biological reactions, one being aerobic while the other is anoxic. We are first interested in a problem of optimal control in time and then, in both time and energy. The existence of the optimal trajectories is proven and the corresponding optimal controls are derived in each case. (10.46298/arima.1978)
  DOI : 10.46298/arima.1978
• On the identification of defects in a periodic waveguide from far field data
  
  • Bourgeois Laurent
  • Fliss Sonia
  Inverse Problems, IOP Publishing, 2014, 30 (9). The aim of this paper is to apply the Linear Sampling Method and the Factorization Method to retrieve some defects in a known periodic 2D waveguide from scattering data. More precisely, some far field approximations of these two sampling methods are derived. They amount to consider the so-called propagating Floquet modes as incident waves. The efficiency of the far field formulation of the LSM is shown with the help of some numerical experiments. (10.1088/0266-5611/30/9/095004)
  DOI : 10.1088/0266-5611/30/9/095004
• Gaussian and non-Gaussian processes of zero power variation, and related stochastic calculus.
  
  • Russo Francesco
  • Viens Frederi
  , 2014. We consider a class of stochastic processes $X$ defined by $X\left( t\right) =\int_{0}^{T}G\left( t,s\right) dM\left( s\right) $ for $t\in\lbrack0,T]$, where $M$ is a square-integrable continuous martingale and $G$ is a deterministic kernel. Let $m$ be an odd integer. Under the assumption that the quadratic variation $\left[ M\right] $ of $M$ is differentiable with $\mathbf{E}\left[ \left\vert d\left[ M\right] (t)/dt\right\vert ^{m}\right] $ finite, it is shown that the $m$th power variation $$ \lim_{\varepsilon\rightarrow0}\varepsilon^{-1}\int_{0}^{T}ds\left( X\left( s+\varepsilon\right) -X\left( s\right) \right) ^{m} $$ exists and is zero when a quantity $\delta^{2}\left( r\right) $ related to the variance of an increment of $M$ over a small interval of length $r$ satisfies $\delta\left( r\right) =o\left( r^{1/(2m)}\right) $. When $M$ is the Wiener process, $X$ is Gaussian; the class then includes fractional Brownian motion and other Gaussian processes with or without stationary increments. When $X$ is Gaussian and has stationary increments, $\delta$ is $X$'s univariate canonical metric, and the condition on $\delta$ is proved to be necessary. In the non-stationary Gaussian case, when $m=3$, the symmetric (generalized Stratonovich) integral is defined, proved to exist, and its Itô formula is established for all functions of class $C^{6}$.
• Recent Advances in the Numerical Analysis of Optimal Hybrid Control Problems
  
  • Ferretti Roberto
  • Sassi Achille
  • Zidani Hasnaa
  , 2014.
• Convergence of discontinuous Galerkin schemes for front propagation with obstacles
  
  • Bokanowski Olivier
  • Cheng Yingda
  • Shu Chi-Wang
  , 2014. We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form $\min(u_t + c u_x, u - g(x))=0$, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These ``low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.
• Contributions à la simulation numérique en élastodynamique : découplage des ondes P et S, modèles asymptotiques pour la traversée de couches minces
  
  • Burel Aliénor
  , 2014. Cette thèse porte sur la modélisation des ondes élastodynamiques dans deux situations particulières qui pénalisent les méthodes numériques utilisées pour simuler ces phénomènes. Dans la première partie, on se place dans le cas où les ondes de pression (ondes P) se propagent à une vitesse beaucoup plus grande que celle des ondes de cisaillement (ondes S). Les modèles numériques utilisés habituellement pour traiter cette configuration sont pénalisés par la plus petite vitesse qui dicte le choix du pas du schéma. Nous proposons ici un schéma qui découple numériquement, dans le volume, les ondes P et les ondes S, pour deux types de conditions de bord en utilisant la décomposition du déplacement en potentiels de Lamé, en deux dimensions. Les conditions aux limites de Dirichlet homogènes, qui sont des conditions essentielles pour la formulation classique en déplacement, deviennent des conditions naturelles, mais non standard, pour la formulation en potentiels qui se présente comme un système de deux équations d’ondes couplées par les conditions aux limites. Cette formulation préserve une énergie équivalente à l'énergie élastodynamique. Nous construisons un schéma éléments finis en espace et utilisons un thêta-schéma en temps sur les termes de bord afin de ne pas pénaliser la CFL et mener à une condition sur le pas de temps indépendante des termes de couplage au bord. Ce schéma préserve une énergie discrète. Le cas des conditions de surface libre mène à des instabilités. Nous les avons traitées comme des perturbations des conditions de Dirichlet, ce qui permet d'obtenir de bons résultats dans le domaine fréquentiel mais donne naissance à de sévères instabilités après discrétisation en temps. La seconde partie de la thèse est consacrée à la construction, l'analyse et la validation de conditions de transmission effectives (CTE) à travers une couche mince de matériau homogène et isotrope d'épaisseur constante h. Ici, la finesse de la couche affecte les schémas explicites usuels car le maillage de la couche avec des éléments suffisamment petits entraîne une diminution analogue du pas de temps critique via la condition CFL, tandis que l'on espère avec les CTE obtenir un pas de temps indépendant de l'épaisseur de la couche. Une analyse complète du cas de la bande mince rectiligne est donnée en deux et trois dimensions. Les conditions obtenues sont stables via la conservation d'une énergie et l'ordre de l'erreur d'approximation par rapport à l'épaisseur de la couche pour les conditions d'ordre 2 est de O(h^3). Des résultats numériques sont présentés pour les configurations bi et tridimensionnelles, ils valident les résultats de stabilité, d'estimation d'erreur et de conditions de stabilité de schémas en temps proposés, qui sont des modifications du schéma explicite utilisé en l'absence de couche mince. Enfin, le traitement d'une couche curviligne est effectué dans le cas bidimensionnel. Sa stabilité est à nouveau vérifiée par conservation d'énergie et des résultats numériques sont également présentés.
• Reconstruction of Independent sub-domains in a Hamilton-Jacobi Equation and its Use for Parallel Computation
  
  • Festa Adriano
  , 2014. A previous knowledge of the domains of dependence of a dynamic programming equation can be useful in its study and approximation. Information on the nature are, in general, difficult to obtain directly from the dynamics of the problem. In this paper we introduce formally the concept of Independent Sub-Domains discussing their main properties and we provide a constructive implicit representation formula. Using these results an original approach to independent domain reconstruction is presented and its usefulness in the parallel approximation of the solution is discussed.
• Optimization of running strategies based on anaerobic energy and variations of velocity
  
  • Bonnans J. Frederic
  , 2014. 1 Keller's model 2 Variable energy recreation 3 Bounding the derivative of f
• Degenerate second order mean field games systems
  
  • Tonon Daniela
  • Cardaliaguet Pierre
  • Graber Philip J.
  • Porretta Alessio
  , 2014. We consider degenerate second order mean field games systems with a local coupling. The starting point is the idea that mean field games systems can be understood as an optimality condition for optimal control of PDEs. Developing this strategy for the degenerate second order case, we discuss the existence and uniqueness of a weak solution as well as its stability (vanishing viscosity limit). Speaker: Daniela TONON
• Optimized Transmission Conditions for Domain Decomposition Methods and Helmholtz Equation. Application to Higher Order Finite Element Methods
  
  • Collino Francis
  • Duruflé Marc
  • Joly Patrick
  • Lecouvez Matthieu
  , 2014. Domain decomposition methods for solving Helmholtz equation are considered. Such methods rely on transmission conditions set on the interfaces between subdomains. The convergence of the iterative algorithm used to solve the associated linear system depends on these transmission conditions. Optimized transmission conditions (such as proposed in [1]) usually rely on transparent boundary conditions or local operators that are an approximation of the exact transparent boundary condition. In this talk, non-local optimized transmission conditions based on Riesz potentials as detailed in [2] are studied. The non-local operators can be replaced by quasi-local operators, and the obtained rate of convergence is independent of the mesh size. These conditions are applied to higher order finite element methods.
• Finite element computation of leaky modes in straight and helical elastic waveguides
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Hazard Christophe
  • Bonnet-Ben Dhia Anne-Sophie
  , 2014, pp.4p. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, waveguides are often embedded in a large solid domain, considered as unbounded. The waves can attenuate strongly along the guide axis due to the energy leakage into the surrounding medium, which reduces the propagating distance. Searching modes with low attenuation becomes necessary. The goal of this work is to propose a numerical approach to compute modes in embedded elastic waveguides (straight and helical structures), combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) method. The application of this work is the non destructive evaluation of multi-wire strands, which constitute cables in civil structures.
• Étude de deux problèmes de propagation d'ondes transitoires : 1) Focalisation spatio-temporelle en acoustique; 2) Transmission entre un diélectrique et un métamatériau.
  
  • Cassier Maxence
  , 2014. Cette thèse comporte deux parties indépendantes. Dans la première, à l'aide d'un réseau de transducteurs, nous cherchons à émettre dans un milieu contenant des obstacles diffractants dont nous ignorons les positions une onde venant focaliser en espace et en temps au voisinage d'un de ces obstacles. La solution proposée ici est basée sur la méthode DORT (Décomposition de l'Opérateur de Retournement Temporel) qui conduit à des propriétés de focalisation spatiale en régime harmonique. Cette dernière consiste à effectuer une décomposition en valeurs singulières (SVD) de l'opérateur de diffraction qui à une distribution de signaux envoyés aux transducteurs associe la mesure de l'onde diffractée. Dans le cadre du modèle asymptotique petits obstacles de Foldy-Lax (justifié ici dans le cas bidimensionnel), nous montrons comment synchroniser les signaux fournis par la méthode DORT en introduisant une SVD particulière liée au caractère symétrique de l'opérateur de diffraction. Notre méthode est justifiée par des arguments théoriques et une étude numérique. La seconde partie est dédiée à un problème de transmission entre un diélectrique et un métamatériau. La question qui est posée ici consiste à étudier la validité du principe d'amplitude limite (PAL) dans un tel milieu. Ce principe définit le régime périodique établi comme le comportement asymptotique en temps long d'un système soumis à une excitation périodique. Nous proposons ici une réponse dans le cas d'un dioptre plan entre un diélectrique et un métamatériau particulier (modèle de Drude). Dans un tel cadre, les équations de Maxwell sont reformulées en une équation de Schrödinger dont nous réalisons l'analyse spectrale complète. Notre étude permet de voir que le PAL est vérifié sauf à une fréquence particulière, appelée fréquence plasmon, pour laquelle les rapports des valeurs prises par la permittivité et par la perméabilité de part et d'autre l'interface sont égaux à -1. Cette fréquence correspond à une résonance du système et la réponse à une telle excitation croît linéairement en temps.
• Sensitivity analysis for the outages of nuclear power plants
  
  • Barty Kengy
  • Bonnans J. Frederic
  • Pfeiffer Laurent
  Energy Systems, Springer, 2014, 5 (2), pp.371-406. Nuclear power plants must be regularly shut down in order to perform refueling and maintenance operations. The scheduling of the outages is the first problem to be solved in electricity production management. It is a hard combinatorial problem for which an exact solving is impossible. Our approach consists in modelling the problem by a two-level problem. First, we fix a feasible schedule of the dates of the outages. Then, we solve a low-level problem of optimization of elecricity production, by respecting the initial planning. In our model, the low-level problem is a deterministic convex optimal control problem. Given the set of solutions and Lagrange multipliers of the low-level problem, we can perform a sensitivity analysis with respect to dates of the outages. The approximation of the value function which is obtained could be used for the optimization of the schedule with a local search algorithm. (10.1007/s12667-013-0096-y)
  DOI : 10.1007/s12667-013-0096-y
• The Jungle Universe: coupled cosmological models in a Lotka–Volterra framework
  
  • Perez Jérôme
  • Füzfa André
  • Carletti Timoteo
  • Mélot Laurence
  • Guedezounme Laurent
  General Relativity and Gravitation, Springer Verlag, 2014, 46, pp.1753. In this paper, we exploit the fact that the dynamics of homogeneous and isotropic Friedmann-Lemaitre universes is a special case of generalized Lotka-Volterra system where the competitive species are the barotropic fluids filling the Universe. Without coupling between those fluids, Lotka-Volterra formulation offers a pedagogical and simple way to interpret usual Friedmann-Lemaitre cosmological dynamics. A natural and physical coupling between cosmological fluids is proposed which preserve the structure of the dynamical equations. Using the standard tools of Lotka-Volterra dynamics, we obtain the general Lyapunov function of the system when one of the fluids is coupled to dark energy. This provides in a rigorous form a generic asymptotic behavior for cosmic expansion in presence of coupled species, beyond the standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we conjecture that chaos can appear for at least four interacting fluids. (10.1007/s10714-014-1753-8)
  DOI : 10.1007/s10714-014-1753-8
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2014. Originally formulated in the context of topology optimization, the concept of topological derivative has also proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. This approach remains, however, largely based on a heuristic interpretation of the topological derivative, whereas most other qualitative approaches to inverse scattering are backed by a mathematical justification. As an effort toward bridging this gap, this study focuses on a topological derivative approach applied to the L2-norm of the misfit between far-field measurements. Either an inhomogeneous medium or a finite number of point-like scatterers are considered, using either the Born approximation or a full-scattering model. Topological derivative-based imaging functionals are analyzed using a suitable factorization of the far-field operator, for each of the considered cases, in order to characterize their behavior and assess their ability to reconstruct the unknown scatterer(s). Results include the justification of the usual sign heuristic underpinning the method for (i) the Born approximation and (ii) full-scattering models limited to moderately strong scatterers. Semi-analytical and numerical examples are presented. Within the chosen framework, the topological derivative approach is finally discussed and compared to other well-known qualitative methods.
• Le piano rêvé des mathématiciens
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Duruflé Marc
  • Joly Patrick
  Interstices, INRIA, 2014.
• Modélisation numérique des guides d’onde ouverts : cas des structures élastiques courbes
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2014, pp.7p. Pour le contrôle non destructif des matériaux et structures, les ondes élastiques guidées sont intéressantes en raison de leur longue distance de propagation. Cependant pour les guides enfouis dans un milieu infini, les modes peuvent devenir à fuite. Ces modes s'atténuent selon l'axe du guide en raison du rayonnement de l'énergie, ce qui réduit leur distance de propagation. La recherche des modes les moins atténués est donc nécessaire. Le calcul numérique des modes dans les guides enfouis présente une difficulté: les amplitudes des modes à fuite croissent dans la direction transversale. Une technique consiste à combiner la méthode des éléments finis semi-analytique (SAFE) et la méthode des couches parfaitement adaptées (PML). Cette méthode a été implémentée récemment par les auteurs pour simuler des plaques multi-couches enfouies et des guides enrobés de section arbitraire. Néanmoins, ces travaux ne considèrent que des guides droits, sans courbure. Nous nous intéressons ici à la modélisation par méthode SAFE-PML des structures courbes, plus particulièrement hélicoïdales, enrobées dans un solide. La motivation de ce travail est le contrôle non destructif des torons, structures multi-brins hélicoïdales, constituant les câbles de génie civil. Dans le vide, on peut simuler les guides hélicoïdaux par méthode SAFE grâce à des coordonnées hélicoïdales. Notre travail consiste à appliquer des PML selon des directions transversales twistées pour calculer les modes à fuite. Deux cas tests sont étudiés. Le premier cas test correspond à un cylindre "twisté" dans une matrice solide et permet de valider la méthode SAFE-PML twistée. L'effet du twist sur le spectre des modes propres est présenté. Le deuxième cas consiste à simuler un brin hélicoïdal enrobé. L'effet de la courbure du guide sur l'atténuation axiale des modes est étudié. Enfin, la simulation d’un toron plongé dans un coulis de ciment sera présentée.
• A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation.
  
  • Barbu Viorel
  • Röckner Michael
  • Russo Francesco
  , 2014. The purpose of the present paper consists in proposing and discussing a double probabilistic representation for a porous media equation in the whole space perturbed by a multiplicative colored noise. For almost all random realizations $\omega$, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. The key ingredient is a uniqueness lemma for a linear SPDE of Fokker-Planck type with measurable bounded (possibly degenerated) random coefficients.