Publications

2018

• Dual Particle Output Feedback Control based on Lyapunov drifts for nonlinear systems
  
  • Flayac Emilien
  • Dahia Karim
  • Hérissé Bruno
  • Jean Frédéric
  , 2018, pp.250-255. This paper presents a dual receding horizon output feedback controller for a general non linear stochastic system with imperfect information. The novelty of this controller is that stabilization is treated, inside the optimization problem, as a negative drift constraint on the control that is taken from the theory of stability of Markov chains. The dual effect is then created by maximizing information over the stabilizing controls which makes the global algorithm easier to tune than our previous algorithm. We use a particle filter for state estimation to handle nonlinearities and multimodality. The performance of this method is demonstrated on the challenging problem of terrain aided navigation. (10.1109/CDC.2018.8619640)
  DOI : 10.1109/CDC.2018.8619640
• Inverse Optimal Control Problem: The Linear-Quadratic Case
  
  • Jean Frédéric
  • Maslovskaya Sofya
  , 2018. A common assumption in physiology about human motion is that the realized movements are done in an optimal way. The problem of recovering of the optimality principle leads to the inverse optimal control problem. Formally, in the inverse optimal control problem we should find a cost function such that under the known dynamical constraint the observed trajectories are minimizing for such cost. In this paper we analyze the inverse problem in the case of finite horizon linear-quadratic problem. In particular, we treat the injectivity question, i.e. whether the cost corresponding to the given data is unique, and we propose a cost reconstruction algorithm. In our approach we define the canonical class on which the inverse problem is either unique or admit a special structure, which can be used in cost reconstruction. (10.1109/CDC.2018.8619204)
  DOI : 10.1109/CDC.2018.8619204
• Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Vohralík Martin
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (5), pp.2037-2064. We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments. (10.1051/m2an/2018034)
  DOI : 10.1051/m2an/2018034
• Gravitational Dynamical Systems
  
  • Simon-Petit Alicia
  , 2018. Dynamical systems have a centuries-long history with roots going back to the mathematical development for astronomy. In the modern formalism, the present thesis investigates dynamical properties of gravitation at different astrophysical or cosmological scales.In potential theory, isochrony often refers to harmonic oscillations of pendulums. In 1959, the mathematician and astronomer Michel Hénon introduced an extended definition of isochrony to characterize orbital oscillations of stars around the center of the system to which they belong. In that case, the period of oscillations can depend on the energy of the star. Today, Michel Hénon’s isochrone potential is mainly used for its integrable property in numerical simulations, but is not widely known. In this thesis, we revisit his geometrical characterization of isochrony and complete the family of isochrone potentials in physics. The classification of this family under different mathematical group actions highlights a particular relation between the isochrones. The actual Keplerian nature of isochrones is pointed out and stands at the heart of the new isochronerelativity, which are presented together.The consequences of this relativity in celestial mechanics — a generalization of Kepler’sThird law, Bohlin or Levi-Civita transformation, Bertrand’s theorem — are applied to analyze the result of a gravitational collapse. By considering dynamical orbital properties, an isochrone analysis is developed to possibly characterize a quasi-stationary state of isolated self-gravitating systems, such as dynamically young stellar clusters or galaxies.At a cosmological scale, the dynamics of the universe depends on its energy content. Its evolution can be expressed as an ecological dynamical system, namely a conservative generalized Lotka-Volterra model. In this framework of a spatially homogeneous and isotropic spacetime, named Jungle Universe, the dynamical impact of a non-gravitational interaction between the energy components is analyzed. As a result, effective dynamical behaviors could account for an accelerated expansion of the universe without dark energy.
• Srishti Dhar Chatterji, my Ph.D. advisor
  
  • Russo Francesco
  Expositiones Mathematicae, Elsevier, 2018, 36 (3-4), pp.257-258. (10.1016/j.exmath.2018.09.002)
  DOI : 10.1016/j.exmath.2018.09.002
• A MILP Formulation for Adaptive Solutions in Railway Scheduling
  
  • Lucas Rémi
  • Alès Zacharie
  • Elloumi Sourour
  , 2018.
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Geuzaine Christophe
  , 2018. Solving high-frequency time-harmonic scattering problems using finite element techniques is challenging, as such problems lead to very large, complex and indefinite linear systems. Optimized Schwarz domain decomposition methods (DDMs) are currently a very promising approach, where subproblems of smaller sizes are solved in parallel using direct solvers, and are combined in an iterative procedure. It is well-known that the convergence rate of these methods strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local transmission conditions based on high-order absorbing boundary conditions (HABCs) have proved well suited [Boubendir et al, 2012; Gander et al, 2002]. They represent a good compromise between basic impedance conditions (which lead to suboptimal convergence) and the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain (which is expensive to compute). However, a direct application of this approach for domain decomposition configurations with cross-points, where more than two subdomains meet, does not provide satisfactory results. We present an improved DDM that efficiently addresses configurations with cross points. Noting that these points actually are corners for the subdomains, our strategy consists in incorporating a corner treatment developed for HABCs into the DDM procedure. After a presentation of the key aspects of the methods, the effectiveness of our approach is discussed with two-dimensional finite element results.
• A Sufficient Condition for the Absence of Two-Dimensional Instabilities of an Elastic Plate in a Duct with Compressible Flow
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (6), pp.3119-3144. We study the time-harmonic resonance of a finite-length elastic plate in a fluid in 5 uniform flow confined in a duct. Although the resonance frequencies are usually real, the combined 6 effects of plate elasticity and of a flow can create complex frequencies, different from the usual so-7 called scattering frequencies, corresponding to instabilities. We study theoretically the existence of 8 instabilities versus several problem parameters, notably the flow velocity and the ratio of densities 9 and of sound speeds between the plate and the fluid. A 3D-volume in the parameters space is defined, 10 in which no instability can develop. In particular it corresponds to a low enough velocity or a light 11 enough plate. The theoretical estimates are validated numerically. 12 (10.1137/18M1165761)
  DOI : 10.1137/18M1165761
• Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods
  
  • Chaillat Stéphanie
  • Groth Samuel P
  • Loseille Adrien
  Journal of Computational Physics, Elsevier, 2018, 372, pp.473 - 499. This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilise the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified. In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries. (10.1016/j.jcp.2018.06.048)
  DOI : 10.1016/j.jcp.2018.06.048
• Inverse Optimal Control : theoretical study
  
  • Maslovskaya Sofya
  , 2018. This PhD thesis is part of a larger project, whose aim is to address the mathematical foundations of the inverse problem in optimal control in order to reach a general methodology usable in neurophysiology. The two key questions are : (a) the uniqueness of a cost for a given optimal synthesis (injectivity) ; (b) the reconstruction of the cost from the synthesis. For general classes of costs, the problem seems very difficult even with a trivial dynamics. Therefore, the injectivity question was treated for special classes of problems, namely, the problems with quadratic cost and a dynamics, which is either non-holonomic (sub-Riemannian geometry) or control-affine. Based on the obtained results, we propose a reconstruction algorithm for the linear-quadratic problem.
• A Hamilton-Jacobi-Bellman approach for the numerical computation of probabilistic state constrained reachable sets
  
  • Assellaou Mohamed
  • Picarelli Athena
  , 2018. Aim of this work is to characterise and compute the set of initial conditions for a system of controlled diffusion processes which allow to reach a terminal target satisfying pointwise state constraints with a given probability of success. Defining a suitable auxiliary optimal control problem, the characterization of this set is related to the solution of a particular Hamilton-Jacobi-Bellman equation. A semi-Lagrangian numerical scheme is defined and its convergence to the unique viscosity solution of the equation is proved. The validity of the proposed approach is then tested on some numerical examples.
• Decoupled mild solutions of deterministic evolution problemswith singular or path-dependent coefficients, represented by backward SDEs
  
  • Barrasso Adrien
  , 2018. This thesis introduces a new notion of solution for deterministic non-linear evolution equations, called decoupled mild solution.We revisit the links between Markovian Brownian Backward stochastic differential equations (BSDEs) and parabolic semilinear PDEs showing that under very mild assumptions, the BSDEs produce a unique decoupled mild solution of some PDE.We extend this result to many other deterministic equations such asPseudo-PDEs, Integro-PDEs, PDEs with distributional drift or path-dependent(I)PDEs. The solutions of those equations are represented throughBSDEs which may either be without driving martingale, or drivenby cadlag martingales. In particular this thesis solves the so calledidentification problem, which consists, in the case of classical Markovian Brownian BSDEs, to give an analytical meaning to the second component Z ofthe solution (Y,Z) of the BSDE. In the literature, Y generally determinesa so called viscosity solution and the identification problem is only solved when this viscosity solution has a minimal regularity.Our method allows to treat this problem even in the case of general (even non-Markovian) BSDEs with jumps.
• The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral
  
  • Harris Daniel
  • Brun P.-T.
  • Damiano Adam
  • Faria Luiz
  • Bush John
  Chaos: An Interdisciplinary Journal of Nonlinear Science, American Institute of Physics, 2018, 28 (9), pp.096105. Millimetric droplets may walk across the surface of a vibrating fluid bath, propelled forward by their own guiding or “pilot” wave field. We here consider the interaction of such walking droplets with a submerged circular pillar. While simple scattering events are the norm, as the waves become more pronounced, the drop departs the pillar along a path corresponding to a logarithmic spiral. The system behavior is explored both experimentally and theoretically, using a reduced numerical model in which the pillar is simply treated as a region of decreased wave speed. A trajectory equation valid in the limit of weak droplet acceleration is used to infer an effective force due to the presence of the pillar, which is found to be a lift force proportional to the product of the drop’s walking speed and its instantaneous angular speed around the post. This system presents a macroscopic example of pilot-wave-mediated forces giving rise to apparent action at a distance. (10.1063/1.5031022)
  DOI : 10.1063/1.5031022
• On the spectral theory and limiting amplitude principle for Maxwells equations at the interface of a metamaterial
  
  • Cassier Maxence
  • Hazard Christophe
  • Joly Patrick
  , 2018.
• Une approche nouvelle de la modélisation mathématique et numérique en aéroacoustique par les équations de Goldstein : Applications en aéronautique
  
  • Bensalah Antoine
  , 2018. La problématique du bruit fait par les réacteurs d'avions est un des enjeux majeurs de l’industrie aéronautique.C’est dans ce contexte que l’équipe du centre de recherche d'Airbus travaille au développement du code de calcul Actipole de propagation acoustique en présence d'un écoulement porteur.L'approche consiste en un couplage FEM-BEM entre la zone de propagation loin de l'avion où l'écoulement est supposé uniforme (BEM) et la zone plus proche où l'écoulement est supposé potentiel (FEM).Les équations de l'aéroacoustique en régime harmonique se réduisent alors à la simple équation scalaire d'Helmholtz convectée.Nous étudions une reformulation des équations d'Euler linéarisées, les équations de Goldstein, prenant en compte l'interaction entre l'acoustique et l'hydrodynamique, lorsque l'écoulement n'est plus potentiel, par l'ajout d'une inconnue hydrodynamique localisée aux zones fortement rotationnelles.Les équations de Goldstein peuvent être vues comme une perturbation de l'équation d'Helmholtz convectée, couplée à une équation de transport harmonique.Nos approches théorique et numérique restent dans le cadre de cette vision perturbative en étudiant dans une premier temps la résolution de l'équation de transport.Nous montrons ainsi que sous l'hypothèse d'un écoulement domaine-remplissant, l'équation de transport harmonique peut être inversée et sous contrainte d'un faible rotationnel, le caractère Fredholm de l'équation d'Helmholtz convectée se généralise aux équations de Goldstein.Le cas général est un problème ouvert et difficile, nous montrons que l'équation de transport n'est pas toujours inversible et possède des fréquences de résonance auxquelles les og solutionsfg{} tendent à être singulières le long de lignes de recirculation de l'écoulement.Nous montrons qu'il en est de même des équations couplées qui possèdent en plus des fréquences de résonance du transport d'autres résonances, dites critiques.Nous terminons cette thèse par une étude locale des singularités, par la méthode de Frobenius, des solutions modales obtenues par absorption limite, aux fréquences de résonance du transport et critiques, au voisinage de lignes résonantes, montrant que de telles solutions sortent alors du cadre variationnelle classiques.
• Unconstrained 0-1 polynomial optimization through convex quadratic reformulation
  
  • Lambert Amélie
  • Elloumi Sourour
  • Lazare Arnaud
  , 2018.
• Isochrony in 3D radial potentials. From Michel H\'enon ideas to isochrone relativity: classification, interpretation and applications
  
  • Simon-Petit Alicia
  • Perez Jérôme
  • Duval Guillaume T.
  Communications in Mathematical Physics, Springer Verlag, 2018, 363 (2), pp.p. 605-653. Revisiting and extending an old idea of Michel H\'enon, we geometrically and algebraically characterize the whole set of isochrone potentials. Such potentials are fundamental in potential theory. They appear in spherically symmetrical systems formed by a large amount of charges (electrical or gravitational) of the same type considered in mean-field theory. Such potentials are defined by the fact that the radial period of a test charge in such potentials, provided that it exists, depends only on its energy and not on its angular momentum. Our characterization of the isochrone set is based on the action of a real affine subgroup on isochrone potentials related to parabolas in the $\mathbb{R}^2$ plane. Furthermore, any isochrone orbits are mapped onto associated Keplerian elliptic ones by a generalization of the Bohlin transformation. This mapping allows us to understand the isochrony property of a given potential as relative to the reference frame in which its parabola is represented. We detail this isochrone relativity in the special relativity formalism. We eventually exploit the completeness of our characterization and the relativity of isochrony to propose a deeper understanding of general symmetries such as Kepler's Third Law and Bertrand's theorem.
• The Halfspace Matching Method : a new method to solve scattering problem in infinite media
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Journal of Computational and Applied Mathematics, Elsevier, 2018, 338, pp.44-68. We are interested in acoustic wave propagation in time harmonic regime in a two-dimensional medium which is a local perturbation of an infinite isotropic or anisotropic homogeneous medium. We investigate the question of finding artificial boundary conditions to reduce the numerical computations to a neighborhood of this perturbation. Our objective is to derive a method which can extend to the anisotropic elastic problem for which classical approaches fail. The idea consists in coupling several semi-analytical representations of the solution in halfspaces surrounding the defect with a Finite Element computation of the solution around the defect. As representations of the same function, they have to match in the infinite intersections of the halfspaces. It leads to a formulation which couples, via integral operators, the solution in a bounded domain including the defect and its traces on the edge of the halfspaces. A stability property is shown for this new formulation. (10.1016/j.cam.2018.01.021)
  DOI : 10.1016/j.cam.2018.01.021
• Numerical Analysis of a Non-Conforming Domain Decomposition for the Multigroup SPN Equations
  
  • Giret Léandre
  , 2018. In this thesis, we investigate the resolution of the SPN neutron transport equations in pressurized water nuclear reactor. These equations are a generalized eigenvalue problem. In our study, we first considerate the associated source problem and after we concentrate on the eigenvalue problem. A nuclear reactor core is composed of different media: the fuel, the coolant, the neutron moderator... Due to these heterogeneities of the geometry, the solution flux can have a low-regularity. We propose the numerical analysis of its approximation with finite element method for the low regular case. For the eigenvalue problem under its mixed form, we can not rely on the theories already developed. We propose here a new method for studying the convergence of the SPN neutron transport eigenvalue problem approximation with mixed finite element. When the solution has low-regularity, increasing the order of the method does not improve the approximation, the triangulation need to be refined near the singularities of the solution. Nuclear reactor cores are well-suited for Cartesian grids, but the refinement of these sort of triangulations increases rapidly their number of degrees of freedom. To avoid this drawback, we propose domain decomposition method which can handle globally non-conforming triangulations.
• Inverse acoustic scattering using high-order small-inclusion expansion of misfit function
  
  • Bonnet Marc
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2018, 12 (4), pp.921-953. This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function J is expanded in powers of the characteristic radius a of a single small inhomo-geneity. The O(a 6) approximation J 6 of J is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of J 6 to multiple small obstacles is outlined. Simpler and more explicit expressions of J 6 are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing J 6 over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of J, is considered. (10.3934/ipi.2018039)
  DOI : 10.3934/ipi.2018039
• Well-posedness of a generalized time-harmonic transport equation for acoustics in flow
  
  • Bensalah Antoine
  • Joly Patrick
  • Mercier Jean-François
  Mathematical Methods in the Applied Sciences, Wiley, 2018, 41 (8), pp.3117 - 3137. (10.1002/mma.4805)
  DOI : 10.1002/mma.4805
• Imaging defects in an elastic waveguide using time-dependent surface data
  
  • Baronian V
  • Chapuis B
  • Recoquillay A
  • Bourgeois Laurent
  , 2018, 1131, pp.012010 (7 p.). We are interested here in applying the Linear Sampling Method to the context of Non Destructive Testing for waveguides. Specifically the Linear Sampling Method [1] in its modal form [2] is adapted to image defects in an elastic waveguide from realistic scattering data, that is data coming from sources and receivers on the surface of the waveguide in the time domain, as it has already been done in the acoustic case [16]. The obtained method is applied to artificial data and to experimental data in the special case of back-scattering. (10.1088/1742-6596/1131/1/012010)
  DOI : 10.1088/1742-6596/1131/1/012010
• Linear Sampling Method applied to Non Destructive Testing of an elastic waveguide: theory, numerics and experiments
  
  • Baronian Vahan
  • Bourgeois Laurent
  • Chapuis Bastien
  • Recoquillay Arnaud
  Inverse Problems, IOP Publishing, 2018, 34, pp.075006 (34 p.). This paper presents an application of the Linear Sampling Method to ultrasonic Non Destructive Testing of an elastic waveguide. In particular, the NDT context implies that both the solicitations and the measurements are located on the surface of the waveguide and are given in the time domain. Our strategy consists in using a modal formulation of the Linear Sampling Method at multiple frequencies, such modal formulation being justified theoretically in [1] for rigid obstacles and in [2] for cracks. Our strategy requires the inversion of some emission and reception matrices which deserve some special attention due to potential ill-conditioning. The feasibility of our method is proved with the help of artificial data as well as real data. (10.1088/1361-6420/aac21e)
  DOI : 10.1088/1361-6420/aac21e
• Reconstruction of an unknown electrical network from their reflectogram by an iterative algorithm based on local identification of peaks and inverse scattering theory
  
  • Beck Geoffrey
  , 2018, pp.1-6. We aim at recovering the topology of an unknown electrical network made out of a tree of cables with the same characteristics from only the data obtained through reflectometry. The method is based upon an iterative algorithm associating the peaks of a reflectogram with unknown scatterers which can be either junction or terminal end of the network, dispelling the ambiguities caused by the complexity of the reflectogram. To identify the peaks, we propose an new algorithm adapted to our goal. The reconstructed networks are topologically identical to the originals ones in 99 per cent of all cases. Cables lengths and terminal loads are also retrieved with high accuracy, e.g. with typical error respectively less than 5% and less than 15%. (10.1109/I2MTC.2018.8409731)
  DOI : 10.1109/I2MTC.2018.8409731
• Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Pagneux Vincent
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2018. We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of PT-symmetric problems.