Publications

2013

• The shooting approach to optimal control problems
  
  • Bonnans Joseph Frederic
  , 2013, pp.281-292. We give an overview of the shooting technique for solving deterministic optimal control problems. This approach allows to reduce locally these problems to a finite dimensional equation. We first recall the basic idea, in the case of unconstrained or control constrained problems, and show the link with second-order optimality conditions and the analysis or discretization errors. Then we focus on two cases that are now better undestood: state constrained problems, and affine control systems. We end by discussing extensions to the optimal control of a parabolic equation. (10.3182/20130703-3-FR-4038.00158)
  DOI : 10.3182/20130703-3-FR-4038.00158
• FE heterogeneous multiscale method for long-time wave propagation
  
  • Abdulle Assyr
  • Grote Marcus J.
  • Stohrer Christian
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2013, 351, pp.495-499. A new finite element heterogeneous multiscale method (FE-HMM) is proposed for the numerical solution of the wave equation over long times in a rapidly varying medium. Our FE-HMM captures long-time dispersive effects of the true solution at a cost similar to that of a standard numerical homogenization scheme which, however, only captures the short-time macroscale behavior of the wave field. (10.1016/j.crma.2013.06.002)
  DOI : 10.1016/j.crma.2013.06.002
• Energy management method for an electric vehicle
  
  • Granato Giovanni
  • Bonnans J. Frederic
  • Aouchiche K.
  • Grégory Rousseau
  • Zidani Hasnaa
  , 2013. 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.
• Seismic elastic modeling for seismic imaging
  
  • Virieux Jean
  • Brossier Romain
  • Chaillat Stéphanie
  • Duchkov A. D.
  • Etienne Vincent
  • Lombard Bruno
  • Operto Stéphane
  , 2013.
• Parallel local time-stepping for elastodynamic equations
  
  • Dudouit Yohann
  • Giraud Luc
  • Millot Florence
  • Pernet Sébastien
  , 2013.
• A discontinuous Galerkin scheme for front propagation with obstacles
  
  • Bokanowski Olivier
  • Cheng Yingda
  • Shu Chi-Wang
  Numerische Mathematik, Springer Verlag, 2013, 126 (1), pp.1-31. We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu, SIAM J. Scient. Comput., 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of (Bokanowski, Forcadel and Zidani, SIAM J. Control Optim. 2010), leading to a level set formulation driven by $\min(u_t + H(x,\nabla u), u-g(x))=0$, where $g(x)$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $H$ is a linear function of $\nabla u$, corresponding to linear convection problems in presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in (Zhang and Shu, SIAM J. Control and Optim., 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost. (10.1007/s00211-013-0555-3)
  DOI : 10.1007/s00211-013-0555-3
• Modeling the grand piano
  
  • Chaigne Antoine
  • Chabassier Juliette
  • Joly Patrick
  , 2013. A global model of a piano is presented. Its aim is to reproduce the main vibratory and acoustic phenomena involved in the generation of a piano sound from the initial blow of the hammer against the strings to the radiation from soundboard to the air. One first originality of the work is due to the string model which takes both geometrical nonlinear effects and stiffness into account. Other significant improvements are due to the combined modeling of the three main couplings between the constitutive parts of the instrument: hammer-string, string-soundboard and soundboard-air coupling.
• Numerical modeling of nonlinear acoustic waves with fractional derivatives
  
  • Lombard Bruno
  • Mercier Jean-François
  , 2013.
• A rigorous approach to the propagation of electromagnetic waves in co-axial cables
  
  • Beck Geoffrey
  • Joly Patrick
  • Imperiale Sébastien
  , 2013. We investigate the question of the electromagnetic propagation in thin electric cables from a mathemat- ical point of view via an asymptotic analysis with respect to the (small) transverse dimension of the ca- ble: as it has been done in the past in mechanics for the beam theory from 3D elasticity, we use such an approach for deriving simplified effective 1D models from 3D Maxwell’s equations.
• Modeling the piano. Numerical Aspects.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Duruflé Marc
  • Joly Patrick
  , 2013. This paper deals with the discretization of our global piano model. We have to solve a complex system of coupled equations, where each subsystem has different spatial dimensions, which poses specific difficulties. The hammer-strings part is a 1D system governed by nonlinear equations. The soundboard is a 2D system with diagonal damping. The acoustic field is a 3D problem in an unbounded domain. Energy based methods allow to build an accurate and a priori stable scheme.
• Computation of leaky modes in three-dimensional open elastic waveguides
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2013, pp.2p.. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. When the guiding structure is embedded into a solid matrix, waveguides are open and waves can be trapped or leaky. With numerical methods, one of the difficulty is that leaky modes attenuate along the axis (complex wavenumber) and exponentially grow along the transverse direction. The goal of this work is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) technique.
• Effective Transmission Conditions for Thin-Layer Transmission Problems in Elastodynamics
  
  • Bonnet Marc
  • Burel Aliénor
  • Joly Patrick
  , 2013. This research is motivated by the numerical modelling of ultrasonic non-destructive testing experiments. Some tested media feature thin layers which are difficult to handle in numerical computations due to the very small element size required for meshing them. To overcome these difficulties, one idea consists in using effective transmission conditions (ETCs) across the two interfaces bounding the layer. This work aims at establishing such ETCs by means of a formal asymptotic analysis with respect to the (small) layer thickness.
• Hamilton-Jacobi-Bellman Equations on Multi-Domains
  
  • Rao Zhiping
  • Zidani Hasnaa
  , 2013, 164, pp.93--116. A system of Hamilton Jacobi (HJ) equations on a partition of $\R^d$ is considered, and a uniqueness and existence result of viscosity solution is analyzed. While the notion of viscosity notion is by now well known, the question of uniqueness of solution, when the Hamiltonian is discontinuous, remains an important issue. A uniqueness result has been derived for a class of problems, where the behavior of the solution, in the region of discontinuity of the Hamiltonian, is assumed to be irrelevant and can be ignored (see reference [10]) . Here, we provide a new uniqueness result for a more general class of Hamilton-Jacobi equations. (10.1007/978-3-0348-0631-2_6)
  DOI : 10.1007/978-3-0348-0631-2_6
• Second Order PDEs with Dirichlet White Noise Boundary Condition
  
  • Brzezniak Zdzislaw
  • Goldys Ben
  • Peszat Szymon
  • Russo Francesco
  , 2013. In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises including Wiener and fractional Wiener space time white noise, homogeneous noise and Lévy noise is considered.
• Computation of Dispersion Curves in Elastic Waveguides of Arbitrary Cross-section embedded in Infinite Solid Media
  
  • Nguyen Khac-Long
  • Treyssede Fabien
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  , 2013, pp.8p.. Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, guiding structures are often buried in a large domain, considered as unbounded. Waveguides are then open and waves can be trapped or leaky. Analytical tools have been developed to model open solid waveguides but these tools are limited for simple geometries (plates, cylinders). With numerical methods, a difficulty is due to the unbounded geometry. Another issue is due to the presence of leaky modes, which grow exponentially along the transverse directions. The goal of this work is to implement a numerical approach to calculate modes in three dimensional elastic open waveguides, which combines the semi-analytical finite element method and the perfectly matched layers (PML) technique. Both Cartesian and cylindrical PML are implemented.
• Introduction to Identification Methods
  
  • Bonnet Marc
  , 2013 (1), pp.223-246. (10.1002/9781118578469.ch8)
  DOI : 10.1002/9781118578469.ch8
• A Finite Difference Method for Two-Phase Parabolic Obstacle-like Problem
  
  • Arakelyan Avetik
  , 2013. In this paper for two-phase parabolic obstacle-like problem, [\Delta u -u_t=\lambda^+\cdot\chi_{{u>0}}-\lambda^-\cdot\chi_{{u<0}},\quad (t,x)\in (0,T)\times\Omega,] where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain, we will introduce a certain variational form, which allows us to define a notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of discretized scheme to a unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.
• Probabilistic representation for solutions of a porous media type equation with Neumann boundary condition: the case of the half-line.
  
  • Ciotir Ioana
  • Russo Francesco
  , 2013. The purpose of this paper consists in proposing a generalized solution for a porous media type equation on a half-line with Neumann boundary condition and prove a probabilistic representation of this solution in terms of an associated microscopic diffusion. The main idea is to construct a stochastic differential equation with reflection which has a solution in law and whose marginal law densities provide the unique solution of the porous media type equation.
• Model for Shock Wave Chaos
  
  • Kasimov Aslan
  • Faria Luiz
  • Rosales Rodolfo
  Physical Review Letters, American Physical Society, 2013, 110 (10). (10.1103/PhysRevLett.110.104104)
  DOI : 10.1103/PhysRevLett.110.104104
• Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation.
  
  • Belaribi Nadia
  • Cuvelier François
  • Russo Francesco
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2013, 1 (1), pp.3-62. The object of this paper is a multi-dimensional generalized porous media equation (PDE) with not smooth and possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R}^d)$. This work continues the study related to the one-dimensional case by the same authors. One expects that a solution of the mentioned PDE can be represented through the solution (in law) of a non-linear stochastic differential equation (NLSDE). A classical tool for doing this is a uniqueness argument for some Fokker-Planck type equations with measurable coefficients. When $\beta$ is possibly discontinuous, this is often possible in dimension $d = 1$. If $d > 1$, this problem is more complex than for $d = 1$. However, it is possible to exhibit natural candidates for the probabilistic representation and to use them for approximating the solution of (PDE) through a stochastic particle algorithm. We compare it with some numerical deterministic techniques that we have implemented adapting the method of a paper of Cavalli et al. whose convergence was established when $\beta$ is Lipschitz. Special emphasis is also devoted to the case when the initial condition is radially symmetric. On the other hand assuming that $\beta$ is continuous (even though not smooth), one provides existence results for a mollified version of the (NLSDE) and a related partial integro-differential equation, even if the initial condition is a general probability measure. (10.1007/s40072-013-0001-7)
  DOI : 10.1007/s40072-013-0001-7
• Topological derivative for qualitative inverse scattering
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2013.
• Radiation condition for a non-smooth interface between a dielectric and a metamaterial
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Claeys Xavier
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2013. We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in $H^1$. This is due to the degeneration of the two dual singularities which then behave like $r^{\pm i\eta}=e^{\pm i\eta\ln\,r}$ with $\eta\in\mathbb{R}^{\ast}$. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to $H^1$ one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.
• Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds
  
  • Ghezzi Roberta
  • Jean Frédéric
  , 2013. This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p,ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.
• A Hamilton-Jacobi approach to junction problems and application to traffic flows
  
  • Imbert Cyril
  • Monneau Régis
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp.pp 129-166. This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ''junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems. (10.1051/cocv/2012002)
  DOI : 10.1051/cocv/2012002
• Conjugate-cut loci and injectivity domains on two-spheres of revolution
  
  • Bonnard Bernard
  • Caillau Jean-Baptiste
  • Janin Gabriel
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (2), pp.533-554. In a recent article \cite{BCST2009}, we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is $g=\d\vp^{2}+m(\vp)\d\th^{2}$ to the period mapping of the $\vp$-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics related to applications to optimal control in space mechanics, quantum control and optimal transport. (10.1051/cocv/2012020)
  DOI : 10.1051/cocv/2012020