Publications

2015

• Solutions of the time-harmonic wave equation in periodic waveguides : asymptotic behaviour and radiation condition
  
  • Fliss Sonia
  • Joly Patrick
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2015, 219 (1), pp.10.1007/s00205-015-0897-3. In this paper, we give the expression and the asymptotic behaviour of the physical solution of a time harmonic wave equation set in a periodic waveguide. This enables us to define a radiation condition and show well-posedness of the Helmholtz equation set in a periodic waveguide. (10.1007/s00205-015-0897-3)
  DOI : 10.1007/s00205-015-0897-3
• Asymptotic analysis for the multiscale modeling of defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  , 2015. This research is a first step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. The proposed strategy aims at taking into account the modification induced by the presence of a small defect through displacement field correction using an asymptotic analysis. Such an approach would allow to assess the criticality of defects by introducing trial micro-defects with varying positions, sizes and mechanical properties.
• A Wideband Fast Multipole Method for the Helmholtz Kernel: Theoretical Developments
  
  • Chaillat Stéphanie
  • Collino Francis
  Computers & Mathematics with Applications, Elsevier, 2015, pp.to appear. This work presents a new Fast Multipole Method (FMM) based on plane wave expansions (PWFMM), combining the advantages of the low and high frequency formulations. We revisit the method of Greengard et al. [1] devoted to the low frequency regime and based on the splitting of the Green's function into a propagative and an evanescent part. More precisely, we give an explicit formula of the filtered translation function for the propagative part, we derive a new formula for the evanescent part and we provide a new interpolation algorithm. At all steps, we check the accuracy of the method by providing error estimates. These theoretical developments are used to propose a wideband FMM based entirely on plane wave expansions. The numerical efficiency and accuracy of this broadband PWFMM are illustrated with a numerical example. (10.1016/j.camwa.2015.05.019)
  DOI : 10.1016/j.camwa.2015.05.019
• Local controllability of the two-link magneto-elastic swimmer
  
  • Giraldi Laetitia
  • Pomet Jean-Baptiste
  , 2015. A recent promising technique for moving a robotic micro-swimmers is to apply an external magnetic field. This paper focuses on a simple micro-swimmer model with two magnetized segments connected by an elastic joint, which is able to move in a plane by using a magnetic field. By considering the latter as control functions, we prove that the swimmer is locally controllable around the straight position.
• Probabilistic representation of a class of non conservative nonlinear Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  , 2015. 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. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution of a regularized version of PDE.
• On some stochastic control problems with state constraints
  
  • Picarelli Athena
  , 2015. This thesis deals with Hamilton-Jacobi-Bellman (HJB) approach for some stochastic control problems in presence of state-constraints. This class of problems arises in many challenging applications, and a wide literature has already analysed such problems under some strong compatibility conditions. The main features of the present thesis is to provide new ways to face the presence of constraints without assuming any controllability condition. The first contribution of the thesis in this direction is obtained by exploiting the existing link between backward reachability and optimal control problems. It is shown that by considering a suitable auxiliary unconstrained optimal control problem, the level set approach can be extended to characterize the backward reachable sets under state-constrained. On the other hand the value function associated with a general state constrained stochastic optimal control problem is characterized by means of a state constrained backward reachable set, enabling the application of the level set method for handling the presence of the state constraints. This link between optimal control problems and reachability sets led to the theoretical and numerical analysis of HJB equations with oblique derivative boundary conditions and problems with unbounded controls. Error estimates for Markov-chain approximation represent another contribution of this manuscript. Furthermore, the properties of asymptotic controllability of a stochastic system have also been studied. A generalization of the Zubov method to state constrained stochastic systems is presented. In the last part of the thesis an ergodic optimal control problems in presence of state-constraints are considered.
• Complexity of control-affine motion planning
  
  • Jean Frédéric
  • Prandi Dario
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2015, 53 (2), pp.816-844. In this paper we study the complexity of the motion planning problem for control- affine systems. Such complexities are already defined and rather well-understood in the particular case of nonholonomic (or sub-Riemannian) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time- rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities. As a byproduct, we are able to treat the long time local controllability problem, giving quanti- tative estimates on the cost of stabilizing the system near a non-equilibrium point of the drift. (10.1137/130950793)
  DOI : 10.1137/130950793
• Perfectly matched layers in negative index metamaterials and plasmas
  
  • Bécache Eliane
  • Joly Patrick
  • Kachanovska Maryna
  • Vinoles Valentin
  ESAIM: Proceedings, EDP Sciences, 2015, pp.Vol. 50, p. 113-132. This work deals with the stability of Perfectly Matched Layers (PMLs). The first part is a survey of previous results about the classical PMLs in non-dispersive media (construction and necessary condition of stability). The second part concerns some extensions of these results. We give a new necessary criterion of stability valid for a large class of dispersive models and for more general PMLs than the classical ones. This criterion is applied to two dispersive models: negative index metamaterials and uniaxial anisotropic plasmas. In both cases, classical PMLs are unstable but the criterion allows us to design new stable PMLs. Numerical simulations illustrate our purpose. (10.1051/proc/201550006)
  DOI : 10.1051/proc/201550006
• Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Nazarov Sergei
  Inverse Problems, IOP Publishing, 2015. We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce inci-dent plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coeffi-cients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a construc-tive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions.
• A Wideband Fast Multipole Method for the Helmholtz kernel: Theoretical developments
  
  • Chaillat Stéphanie
  • Collino Francis
  , 2015, pp.28. This work presents a new Fast Multipole Method (FMM) based on plane wave expansions, combining the advantages of the low and high frequency formulations. We revisit the method of Greengard et al. devoted to the low frequency regime and based on the splitting of the Green's function into a propagative and an evanescent part. More precisely, we give an explicit formula of the filtered translation function for the propagative part, we derive a new formula for the evanescent part and we provide a new interpolation algorithm. At all steps, we check the accuracy of the method by providing error estimates. These theoretical developments are used to propose a wideband FMM based entirely on plane wave expansions. The numerical efficiency and accuracy of this broadband are illustrated with a numerical example.
• Pattern selection in a biomechanical model for the growth of walled cells
  
  • Calvez Vincent
  • Giraldi Laetitia
  , 2015. In this paper, we analyse a model for the growth of three-dimensional walled cells. In this model the biomechanical expansion of the cell is coupled with the geometry of its wall. We consider that the density of building material depends on the curvature of the cell wall, thus yield-ing possible anisotropic growth. The dynamics of the axisymmetric cell wall is described by a system of nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with a Poisson equation. We develop the linear stability analysis of the spherical symmetric config-uration in expansion. We identify three critical parameters that play a role in the possible instability of the radially symmetric shape, namely the degree of nonlinearity of the coupling, the effective diffusion of the building material, and the Poisson's ratio of the cell wall. We also investigate numerically pattern selection in the nonlinear regime. All the results are also obtained for a simpler, but similar, two-dimensional model.
• Infinite horizon problems on stratifiable state-constraints sets
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2015, 258 (4), pp.1430–1460. This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the value function has not enough regularity, or can fail to be the unique constrained viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. Here, we consider the case of a set of constraints having a stratified structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing qualification hypothesis are not relevant. The discontinuous value function is then characterized by means of a system of HJB equations on each stratum that composes the state constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur. (10.1016/j.jde.2014.11.001)
  DOI : 10.1016/j.jde.2014.11.001
• Comparison of mean and osculating stability in the vicinity of the (2:1) tesseral resonant surface
  
  • Daquin Jérôme
  • Deleflie Florent
  • Perez Jérôme
  Acta Astronautica, Elsevier, 2015, 111, pp.170-177. We confront stability results over long time scales, considering alternately the averaged and the non-averaged theory to propagate the equations of motion of a celestial body orbiting the vicinity of the (2:1) tesseral resonant surface. This confrontation is performed using Fast Lyapunov Indicator stability maps. The benefit of such maps is threefold: (i) to reveal the whole phase space architecture and the consequences of the resonance overlap when several combinations of tesseral resonant parameters are accounted for, (ii) to perform a stability analysis on a whole phase space region, and (iii) to have a clear view of the possible impacts of the short-periodic effects removed during the averaging procedure. Our detailed numerical investigations conclude that the tesseral chaos is robust to the averaging procedure and the numerical methods used to propagate the equations of motion over such long time scales. (10.1016/j.actaastro.2015.02.014)
  DOI : 10.1016/j.actaastro.2015.02.014
• Optimal control problems on well-structured domains and stratified feedback controls
  
  • Hermosilla Cristopher
  , 2015. The aim of this dissertation is to study some issues in Control Theory of ordinary differential equations. Optimal control problems with tame state-constraints and feedback controls with stratified discontinuities are of special interest. The techniques employed along the manuscript have been chiefly taken from control theory, nonsmooth analysis, variational analysis, tame geometry, convex analysis and differential inclusions theory. The first part of the thesis is devoted to provide general results and definitions required for a good understanding of the entire manuscript. In particular, a strong invariance criterion adapted to manifolds is presented. Moreover, a short insight into manifolds and stratifications is done. The notions of relatively wedged sets is introduced and in addition, some of its properties are stated. The second part is concerned with the characterization of the Value Function of an optimal control problem with state-constraints. Three cases have been taken into account. The first one treats stratifiable state-constraints, that is, sets that can be decomposed into manifolds of different dimensions. The second case is focused on linear systems with convex state-constraints, and the last one considers convex state-constraints as well, but from a penalization point of view. In the latter situation, the dynamics are nonlinear and verify an absorbing property at the boundary. The third part is about discontinuous feedbacks laws whose singularities form a stratified set on the state-space. This type of controls yields to consider stratified discontinuous ordinary differential equations, which motivates an analysis of existence of solutions and robustness with respect to external perturbation for these equations. The construction of a suboptimal continuous feedback from an optimal one is also addressed in this part. The fourth part is dedicated to investigate optimal control problems on networks. The main feature of this contribution is that no controllability assumption around the junctions is imposed. The results can also be extended to generalized notions of networks, where the junction is not a single point but a manifold.
• On the Convergence of Decomposition Methods for Multistage Stochastic Convex Programs
  
  • Girardeau Pierre
  • Leclere Vincent
  • Philpott A. B.
  Mathematics of Operations Research, INFORMS, 2015, 40 (1). We prove the almost-sure convergence of a class of sampling-based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions , and uncertainty is modelled by a scenario tree. As special cases, our results imply the almost-sure convergence of SDDP, CUPPS and DOASA when applied to problems with general convex cost functions. (10.1287/moor.2014.0664)
  DOI : 10.1287/moor.2014.0664
• Formulations par équations intégrales de surface pour la simulation numérique du contrôle non destructif par courants de Foucault
  
  • Vigneron Audrey
  , 2015. Cette thèse s'inscrit dans le contexte de la simulation numérique pour le contrôle non destructif (CND) par courants de Foucault et concerne le calcul des champs électromagnétiques induits par un capteur émetteur dans une pièce saine. Ce calcul constitue la première étape de la modélisation complète d'un procédé de contrôle dans la plateforme logicielle CIVA développée au CEA LIST. Aujourd'hui les modèles intégrés dans CIVA sont restreints à des pièces de géométrie canonique (calcul modal) ou axisymétriques. La demande de configurations plus diverses et complexes nécessite l'introduction de nouveaux outils numériques de modélisation. En pratique les capteurs peuvent être constitués d'éléments aux propriétés physiques et aux formes variées. Quant aux pièces à contrôler, elles sont conductrices et peuvent contenir des éléments diélectriques ou magnétiques. Du fait des différents matériaux présents dans une même configuration, différents régimes de modélisation (statique, quasi-statique, voire dynamique) peuvent cohabiter. Sous l'hypothèse de travail de milieux à propriétés linéaires, isotropes et homogènes par morceaux, l'approche par équations intégrales de surface (SIE) permet de ramener le problème volumique à un problème surfacique équivalent. Cependant les formulations SIE usuelles pour le problème de Maxwell souffrent en général d'un problème de robustesse numérique pour certains cas asymptotiques, en particulier à basse fréquence. L'objectif de cette étude est de déterminer une version stable pour une gamme de paramètres physique typique du CND. C'est dans ce cadre qu’un schéma itératif par blocs basé sur une décomposition liée à la physique du problème est proposé. Ce schéma est précis et bien conditionné pour le calcul des champs primaires. Une étude asymptotique du problème intégral de Maxwell est de plus effectuée. Celle-ci permet de formuler le problème intégral de l'approximation courants de Foucault comme une forme asymptotique de celui de Maxwell.
• A regularization approach to functional Itô calculus and strong-viscosity solutions to path-dependent PDEs
  
  • Cosso Andrea
  • Russo Francesco
  , 2015. First, we revisit functional Itô/path-dependent calculus started by B. Dupire, R. Cont and D.-A. Fournié, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus introduced by C. Di Girolami and the second named author are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of classical solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness. The notion of strong-viscosity solution motivates the last part of the paper which is devoted to explore this new concept of solution for general semilinear PDEs in the finite dimensional case. We prove an equivalence result between the classical viscosity solution and the new one. The definition of strong-viscosity solution for semilinear PDEs is inspired by the notion of "good" solution, and it is based again on an approximating procedure.
• Éléments de physique statistique - 2e édition
  
  • Perez Jérôme
  • Chardin Gabriel
  • Debu Pascal
  , 2015, pp.268 pages. Cet ouvrage aborde le thème classique de la physique statistique par la méthode pédagogique du dénombrement des états d’énergie microscopiques. Après avoir passé en revue les divers résultats de la théorie des systèmes sans interactions, divers cas plus généraux sont abordés comme la transition gaz-liquide, le ferromagnétisme ou la théorie du proche équilibre. Ce cours s’insère parfaitement dans la suite logique de l’enseignement de la physique de premier cycle et met en œuvre les résultats essentiels de la théorie quantique. Il permet d’appréhender l’origine microscopique d’un grand nombre de propriétés macroscopiques essentielles d’un système qui caractérisent son état d’équilibre (température, énergie, pression, etc.). La prise en compte des interactions à l’échelle microscopique et l’étude du proche équilibre viennent compléter ce panorama pour préparer des cours plus avancés comme l’étude physique des solides ou celle des plasmas. Cet ouvrage est le fruit d’un cours donné par les auteurs à L’École Nationale Supérieure de Techniques Avancées (ENSTA ParisTech). Il contient de nombreux exercices et une synthèse des points essentiels en fin de chaque chapitre.
• Solving the hypersingular boundary integral equation for the Burton and Miller formulation
  
  • Langrenne Christophe
  • Garcia Alexandre
  • Bonnet Marc
  Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 138 (3332-3340). This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k2 + λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces. (10.1121/1.4935134)
  DOI : 10.1121/1.4935134
• Acoustic inverse scattering using topological derivative of far-field measurements-based L2 cost functionals
  
  • Bellis Cédric
  • Bonnet Marc
  • Cakoni Fioralba
  , 2015. 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.
• Classical homogenization to analyse the dispersion relations of spoof plasmons with geometrical and compositional effects
  
  • Mercier Jean-François
  • Cordero Maria-Luisa
  • Félix Simon
  • Ourir Abdelwaheb
  • Maurel Agnes
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2015, 471 (2182). We show that the classical homogenization is able to describe the dispersion relation of spoof plasmons in structured thick interfaces with periodic unit cell being at the subwavelength scale. This is because the interface in the real problem is replaced by a slab of an homogeneous birefringent medium, with an effective mass density tensor and an effective bulk modulus. Thus, explicit dispersion relation can be derived, corresponding to guided waves in the homogenized problem. Contrary to previous effective medium theories or retrieval methods, the homogenization gives effective parameters depending only on the properties of the material and on the geometry of the microstructure. Although resonances in the unit cell cannot be accounted for within this low-frequency homogenization, it is able to account for resonances occurring because of the thickness of the interface and thus, to capture the behaviour of the spoof plasmons. Beyond the case of simple grooves in a hard material, we inspect the influence of tilting the grooves and the influence of the material properties. (10.1098/rspa.2015.0472)
  DOI : 10.1098/rspa.2015.0472
• Effective birefringence to analyze sound transmission through a layer with subwavelength slits
  
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  • Ourir Abdelwaheb
  Comptes Rendus. Mécanique, Académie des sciences (Paris), 2015, 343 (12). We analyze the transmission of sound through a sound hard film or layer with periodic subwavelength slits. For wavelength comparable to or larger than the slit spacing, the transmission spectra are revisited in terms of the transmission through an equivalent birefringent layer. It is shown that the Fano-type resonances can be understood by means of the dispersion relations of guided waves within the birefringent layer in the homogenized problem, equivalent to spoof plasmons for gratings. This is done by extending the homogenization to the evanescent waves being excited in the near field of the actual perforated layer. (10.1016/j.crme.2015.07.006)
  DOI : 10.1016/j.crme.2015.07.006
• Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost
  
  • Bokanowski Olivier
  • Picarelli Athena
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2015, 71 (1), pp.125--163. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach. (10.1007/s00245-014-9255-3)
  DOI : 10.1007/s00245-014-9255-3
• Mean field games systems of first order
  
  • Cardaliaguet Pierre
  • Graber Philip Jameson
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2015, 21 (3), pp.690–722. We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem.
• Stochastic Multi-Stage Optimization
  
  • Carpentier Pierre
  • Cohen Guy
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2015, 75. (10.1007/978-3-319-18138-7)
  DOI : 10.1007/978-3-319-18138-7