Publications

2016

• Modélisation et étude mathématique de réseaux de câbles électriques
  
  • Beck Geoffrey
  , 2016. Cette thèse porte sur la modélisation d'un réseau de câbles coaxiaux et multi-conducteurs. Ce dernier peut être mathématiquement traduit par les équations aux dérivées partielles de Maxwell qui régissent la propagation des ondes électromagnétiques en son sein ou par un modèle type circuit électrique d'inconnues - les potentiels et courants électriques- qui vérifient sur les branches du circuit l'équation des télégraphistes et sur les noeuds les lois de Kirchhoff.Si la première méthode est assez générale pour comprendre toutes sortes de défauts, elle néanmoins trop couteuse pour les applications que nous avons en tête, à savoir le contrôle non destructif. La seconde quant à elle est obtenue par une modélisation non issue de la théorie de Maxwell et est valide que si les câbles sont parfaits (cylindriques, sans pertes...). Nous avons établi diverses modèles 1D venant généraliser l'équation des télégraphistes et les lois de Kirchhoff pour y incorporer diverses défauts (géométrie, pertes, effet de peau, caractéristique des matériaux variables) tant sur les câbles que dans les jonctions. Ceux-ci sont obtenus via des analyses asymptotiques (classiques, multi-échelles, raccordées) des équations 3D de Maxwell en considérant certains paramètres (dimensions transverses des câbles par rapport à leurs longueurs, conductivité du milieu diélectrique par rapport à celle du métal des âmes, petite taille de la zone de jonction par rapport à l'ensemble du réseau) extrêmement petits.Une des difficultés mathématiques tient en ce que les domaines que nous prendrons en compte (sections des câbles, jonctions) ne sont aucunement simplement connexes, nous obligeant ainsi à remanier quelques outils standard tel les décompositions en potentiels.
• Chance constrained optimization of a three-stage launcher
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  , 2016.
• Legendre Transform and Applications to Finite and Infinite Optimization
  
  • Hermosilla Cristopher
  Set-Valued and Variational Analysis, Springer, 2016. We investigate convex constrained nonlinear optimization problems and optimal control with convex state constraints in the light of the so-called Legendre transform. We use this change of coordinate to propose a gradient-like algorithm for mathematical programs, which can be seen as a search method along geodesics. We also use the Legendre transform to study the value function of a state constrained Mayer problem and we show that it can be characterized as the unique viscosity solution of the Hamilton-Jacobi-Bellman equation. (10.1007/s11228-016-0368-5)
  DOI : 10.1007/s11228-016-0368-5
• Multi-objective optimization of automotive electrical/energy storage system
  
  • Fontaine Gauthier
  • Hammami Omar
  , 2016, pp.339-343. Hybrid Electric Vehicles (HEVs) and Electric Vehicles (EVs) are strongly emerging as potential solutions to respect the more and more restrictive laws concerning CO2 emissions. In this context, the optimization of the costly Hybrid Energy Storage System (HESS) of HEVs is a key factor in reducing both CO2 emissions and costs of HEVs. In this paper, we propose an approach to optimize the configuration of an HESS architecture using evolutionary multi-objective optimization algorithms. Multiple constraints relaxation was necessary in order to match the objectives of the study. This results in a need for a systems engineering approach to define the scope and consequences of constraints relaxation. (10.1109/ICIT.2016.7474775)
  DOI : 10.1109/ICIT.2016.7474775
• Trapped modes in thin and infinite ladder like domains: existence and asymptotic analysis
  
  • Delourme Bérangère
  • Fliss Sonia
  • Joly Patrick
  • Vasilevskaya Elizaveta
  , 2016. We are interested in a 2D propagation medium which is a localized perturbation of a reference homogeneous periodic medium. This reference medium is a "thick graph", namely a thin structure (the thinness being characterized by a small parameter $\varepsilon$ > 0) whose limit (when $\varepsilon$ tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. The question we investigate is whether such a geometrical perturbation is able to produce localized eigenmodes. We have investigated this question when the propagation model is the scalar Helmholtz equation with Neumann boundary conditions . This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We use a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough and construct an asymptotic expansion of the eigenvalues with respect to the small parameter. Following this approach, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung. Using the matched asymptotic expansion method, we obtain an asymptotic expansion at any order of the eigenvalues, which can be used for instance to compute a numerical approximations of these eigenvalues and associated eigenvectors.
• Set-valued solutions for non-ideal detonation
  
  • Semenko R.
  • Faria Luiz
  • Kasimov A.
  • Ermolaev B.
  Shock Waves, Springer Verlag, 2016, 26 (2), pp.141-160. The existence and structure of steady gaseous detonation propagating in a packed bed of solid inert particles are analyzed in the one-dimensional approximation by taking into consideration frictional and heat losses between the gas and the particles. A new formulation of the governing equations is introduced that eliminates the well-known difficulties with numerical integration across the sonic singularity in the reactive Euler equations. The new algorithm allows us to determine that the detonation solutions as the loss factors are varied have a set-valued nature at low detonation velocities when the sonic constraint disappears from the solutions. These set-valued solutions correspond to a continuous spectrum of the eigenvalue problem that determines the velocity of the detonation. (10.1007/s00193-015-0610-3)
  DOI : 10.1007/s00193-015-0610-3
• Une vie dédié aux systèmes dynamiques
  
  • Perez Jérôme
  • Alimi Jean-Michel
  • Mohayaee Roya
  , 2016.
• Quantification du Profit Long Terme d'un Agrégateur sous Comportement Incertain des Agents *
  
  • Coló Philippe
  • Le Cadre Hélène
  , 2016. Les fournisseurs d'électricité conventionnels se trouvent aujourd'hui concurrencés par de nouveaux acteurs, les agrégateurs. Dans ce contexte concurrentiel, de nouveaux modèles économiques pour la fourniture d'électricité émergent. Citons, à titre d'exemple en France, Enercoop, Grid Pocket, Energy Pool, etc. Notre objectif, dans cette présentation, est de calculer une stratégie de tarification dynamique pour un agrégateur, intégrant un modèle de prédiction probabiliste de la demande des consommateurs puis, de simuler cette stratégie sur une étude de cas.
• Dirichlet-to-Neumann operator for diffraction problems in stratified anisotropic acoustic waveguides
  
  • Tonnoir Antoine
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2016. The purpose of this note is to construct a Dirichlet-to-Neumann operator for the diffraction problem in stratified anisotropic acoustic waveguides. The key idea consists in using an adapted change of coordinates that enables to recover the completeness and the orthogonality of the modes on " deformed " cross-sections of the waveguide. To cite this article: (10.1016/j.crma.2015.12.018)
  DOI : 10.1016/j.crma.2015.12.018
• Effective boundary conditions for thin periodic coatings
  
  • Chamaillard Mathieu
  , 2016. We have dealt with the case of the scalar Helmholtz equation. We will try to handle the case of Maxwell's equation. We also will focus on the case of meta-materials. In a first case the permittivity is negative in the thin layer and in the second case is the permeability (1/delta) ^ 2.
• Why Don't We Move Slower? The Value of Time in the Neural Control of Action
  
  • Berret Bastien
  • Jean Frédéric
  Journal of Neuroscience, Society for Neuroscience, 2016, 36 (4), pp.1056-1070. To want something now rather than later is a common attitude that reflects the brain's tendency to value the passage of time. Because the time taken to accomplish an action inevitably delays task achievement and reward acquisition, this idea was ported to neural movement control within the “cost of time” theory. This theory provides a normative framework to account for the underpinnings of movement time formation within the brain and the origin of a self-selected pace in human and animal motion. Then, how does the brain exactly value time in the control of action? To tackle this issue, we used an inverse optimal control approach and developed a general methodology allowing to squarely sample infinitesimal values of the time cost from experimental motion data. The cost of time underlying saccades was found to have a concave growth, thereby confirming previous results on hyperbolic reward discounting, yet without making any prior assumption about this hypothetical nature. For self-paced reaching, however, movement time was primarily valued according to a striking sigmoidal shape; its rate of change consistently presented a steep rise before a maximum was reached and a slower decay was observed. Theoretical properties of uniqueness and robustness of the inferred time cost were established for the class of problems under investigation, thus reinforcing the significance of the present findings. These results may offer a unique opportunity to uncover how the brain values the passage of time in healthy and pathological motor control and shed new light on the processes underlying action invigoration. (10.1523/JNEUROSCI.1921-15.2016)
  DOI : 10.1523/JNEUROSCI.1921-15.2016
• On the approximation of electromagnetic fields by edge finite elements. Part 1: Sharp interpolation results for low-regularity fields
  
  • Ciarlet Patrick
  Computers & Mathematics with Applications, Elsevier, 2016. We propose sharp results on the numerical approximation of low-regularity electromagnetic fields by edge finite elements. We consider general geometrical settings, including topologically non-trivial domains or domains with a non-connected boundary. In the model, the electric permittivity and magnetic per-meability are symmetric, tensor-valued, piecewise smooth coefficients. In all cases, the error can be bounded by h δ times a constant, where h is the mesh-size, for some exponent δ ∈]0, 1] that depends both on the geometry and on the coefficients. It relies either on classical estimates when δ > 1/2, or on a new combined interpolation operator when δ < 1/2. The optimality of the value of δ is discussed with respect to abstract shift theorems. In some simple configurations , typically for scalar-valued permittivity and permeability, the value of δ can be further characterized. This paper is the first one in a series dealing with the approximation of electromagnetic fields by edge finite elements. (10.1016/j.camwa.2015.10.020)
  DOI : 10.1016/j.camwa.2015.10.020
• A multi-step solution algorithm for Maxwell boundary integral equations applied to low-frequency electromagnetic testing of conductive objects
  
  • Vigneron Audrey
  • Demaldent Edouard
  • Bonnet Marc
  IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 2016, 52, pp.7005208. We consider the solution, using boundary elements (BE), of the surface integral equation system arising in electromagnetic testing of conducting bodies, with emphasis on situations such that $o(1) \leq \sqrt{\omega\varepsilon_{0}/\sigma} \leq O(1)$, $L \sqrt{\omega\sigma\mu_{0}} =O(1)$ which includes in particular the case of eddy current testing) and assuming $L\omega\sqrt{\varepsilon_0 \mu_{0}}\leq 2\pi$, i.e. low-frequency conditions ($L$: diameter of conducting body). Earlier approaches for dielectric objects at low frequencies are not applicable in the present context. After showing that a simple normalization of the BE system significantly improves its conditioning, we propose a multi-step solution method based on block SOR iterations, which facilitates the use of direct solvers and converges within a few iterations for the considered range of physical parameters. This novel, albeit simple, treatment allows to perform eddy current-type analyses using standard Maxwell SIE formulations, avoiding the adverse consequences of ill-conditioning for low frequencies and high conductivities. Its performance and limitations are studied on three numerical examples involfing low frequencies and high conductivities. (10.1109/TMAG.2016.2584018)
  DOI : 10.1109/TMAG.2016.2584018
• BSDEs, càdlàg martingale problems and orthogonalisation under basis risk.
  
  • Laachir Ismail
  • Russo Francesco
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2016, 7, pp.308-356. The aim of this paper is to introduce a new formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution of a semilinear PDE of parabolic type. A significant application concerns the hedging problem under basis risk of a contingent claim $g(X_T,S_T)$, where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes $(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes. (10.1137/140996239)
  DOI : 10.1137/140996239
• Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations
  
  • Cosso Andrea
  • Di Girolami Cristina
  • Russo Francesco
  , 2016 (668). The paper reminds the basic ideas of stochastic calculus via regularizations in Banach spaces and its applications to the studyof strict solutions of Kolmogorov path dependent equations associated with "windows" of diffusion processes. One makes the link between the Banach space approach and the so called functional stochastic calculus. When no strict solutions are available one describes the notion of strong-viscosity solution which alternative (in infinite dimension) to the classical notion of viscosity solution. (10.1090/conm/668/13396)
  DOI : 10.1090/conm/668/13396
• Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2016, 13, pp.1189–1233. We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. (10.30757/ALEA.v13-43)
  DOI : 10.30757/ALEA.v13-43
• Imaging an acoustic waveguide from surface data in the time domain
  
  • Baronian Vahan
  • Bourgeois Laurent
  • Recoquillay Arnaud
  Wave Motion, Elsevier, 2016, 66, pp.68 - 87. This paper deals with an inverse scattering problem in an acoustic waveguide. The data consist of time domain signals given by sources and receivers located on the boundary of the waveguide. After transforming the data to the frequency domain, the obstacle is then recovered by using a modal formulation of the Linear Sampling Method. The impact of many parameters are analyzed, such as the numbers of sources/receivers and the distance between them, the time shape of the incident wave and the number and the values of the frequencies that are used. Some numerical experiments illustrate such analysis. (10.1016/j.wavemoti.2016.05.006)
  DOI : 10.1016/j.wavemoti.2016.05.006
• Improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow
  
  • Mercier Jean-François
  • Maurel Agnes
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2016, 472 (2190). (10.1098/rspa.2016.0094)
  DOI : 10.1098/rspa.2016.0094
• Effective transmission conditions for thin-layer transmission problems in elastodynamics. The case of a planar layer model
  
  • Bonnet Marc
  • Burel Aliénor
  • Duruflé Marc
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016, 50, pp.43-75. This article is concerned with the design, analysis, numerical approximation and implementation of effective transmission conditions (ETCs) for the propagation of elastic waves through a thin planar elastic layer with small uniform thickness η which is embedded in a reference elastic medium, under transient conditions, with both materials assumed to have isotropic properties. A family of ETCs of order k (i.e. whose approximation error is of expected order O(η k+1)) is formulated by deriving and exploiting a formal asymptotic expansion in powers of η of the transmission solution inside the layer. The second-order ETCs are then retained as the main focus for the remainder of the article, and given a full justification in terms of both the stability of the resulting transient elastodynamic problem and the error analysis. The latter is performed by establishing and justifying asymptotic expansions for the solutions of both the exact transmission problem and its approximation based on the second-order ETCs. As a result, the error (in energy norm) between those two solutions is shown to be, as expected, of order O(η 3). Finally, the numerical approximation of the proposed second-order ETC within the framework of spectral element methods is studied, with special attention devoted to the selection of a robust time-stepping scheme that is mostly explicit (and conditionally stable). Among these, a scheme that is implicit only for the interfacial degrees of freedom, termed semi-implicit, is shown to be stable under the same stability condition as for the layer-less configuration. The main theoretical results of this work are illustrated and validated by 2D and 3D numerical experiments under transient elastodynamic conditions. (10.1051/m2an/2015030)
  DOI : 10.1051/m2an/2015030
• Time Domain Simulation of a Piano. Part 2 : Numerical Aspects
  
  • Chabassier Juliette
  • Duruflé Marc
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016, 50 (1), pp.93-133. This article is the second of a series of two papers devoted to the numerical simulation of the piano. It concerns the numerical aspects of the work, the implementation of a piano code and the presentation of corresponding simulations. The main difficulty is time discretisation and stability is achieved via energy methods. Numerical illustrations are provided for a realistic piano and compared to experimental recordings.
• The XJC-correspondence
  
  • Pirio Luc
  • Russo Francesco
  Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2016, 2016 (716), pp.229-250. (10.1515/crelle-2014-0052)
  DOI : 10.1515/crelle-2014-0052
• Study of a Model Equation in Detonation Theory: Multidimensional Effects
  
  • Faria Luiz
  • Kasimov A.
  • Rosales R.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (3), pp.887-909. We extend the reactive Burgers equation presented in [A. R. Kasimov, L. M. Faria,and R. R. Rosales,Phys. Rev. Lett., 110 (2013), 104104], [L. M. Faria, A. R. Kasimov, and R. R.Rosales,SIAM J. Appl. Math., 74 (2014), pp. 547–570] to include multidimensional effects. Fur-thermore, we explain how the model can be rationally justified following the ideas of the asymptotictheory developed in [L. M. Faria, A. R. Kasimov, and R. R. Rosales,J. Fluid Mech., 784 (2015),pp. 163–198]. The proposed model is a forced version of the unsteady small disturbance transonicflow equations. We show that for physically reasonable choices of forcing functions, traveling wavesolutions akin to detonation waves exist. It is demonstrated that multidimensional effects play animportant role in the stability and dynamics of the traveling waves. Numerical simulations indicatethat solutions of the model tend to form multidimensional patterns analogous to cells in gaseousdetonations. (10.1137/15M1039663)
  DOI : 10.1137/15M1039663
• FUNCTIONAL ITÔ VERSUS BANACH SPACE STOCHASTIC CALCULUS AND STRICT SOLUTIONS OF SEMILINEAR PATH-DEPENDENT EQUATIONS
  
  • Cosso Andrea
  • Russo Francesco
  Infinite Dimensional Analysis, Quantum Probability and Related Topics, World Scientific Publishing, 2016, 19 (04), pp.1650024. Functional It\^o calculus was introduced in order to expand a functional $F(t, X_{\cdot+t}, X_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X_{\cdot+t}, X_t)$ consists in considering the path $X_{\cdot+t}=\{X_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space stochastic calculus. The aim of this paper is threefold. 1) To reformulate functional It\^o calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional It\^o calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an It\^o stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation. (10.1142/S0219025716500247)
  DOI : 10.1142/S0219025716500247
• Mathematical Aspects of variational boundary integral equations for time dependent wave propagation
  
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2016.
• Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides
  
  • Baronian Vahan
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Wave Motion, Elsevier, 2016. We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite straight isotropic elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with or without an overlap between the domains. Specific transmission conditions are used, so that only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using the Fraser bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized, in particular for the extension to the anisotropic case. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.