Publications

2014

• La singularité voilée
  
  • Perez Jérôme
  • Alimi Jean-Michel
  Pour la science, Pour la Science, 2014, Dossier N°83, pp.p 123. La relativité générale prédit l'existence de points de densité infinie où les lois de la physique s'effondrent : les singularités. Certaines sont tapies au cœur des trous noirs et nous ne pouvons les observer. Qu'en est-il de la singularité initiale, le Big Bang ? Elle est, elle aussi, isolée, car la relativité générale masque cet étrange événement en introduisant le chaos à l'origine de l'Univers.
• Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
  
  • Bayen Térence
  • Bonnans J. Frederic
  • Silva Francisco J.
  Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (4), pp.2063--2087. In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions. (10.1090/S0002-9947-2013-05961-2)
  DOI : 10.1090/S0002-9947-2013-05961-2
• Optimal control problems on stratifiable state constraints sets.
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  , 2014. We consider an infinite horizon problem with state constraints K : inf Z 1 0 e t'(yx;u(t); u(t))dt u : [ 0 ;+1) ! A measurable yx;u(t) 2 K 8t 0 (P) : where > 0 is fixed and yx;u( ) is a trajectory of the control system ( y_ = f (y; u) a.e. t 0 y(0) = x 2 K We are mainly concerned with a characterization of the value function of (P) as the bilateral solution to a Hamilton-Jacobi-Bellman equation.
• Second-order sufficient conditions for strong solutions to optimal control problems
  
  • Bonnans Joseph Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (03), pp.704-724. In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem. (10.1051/cocv/2013080)
  DOI : 10.1051/cocv/2013080
• T-coercivity for the Maxwell problem with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2014. In this paper, we study the time-harmonic Maxwell problem with sign-changing permittivity and/or permeability, set in a domain of R^3. We prove, using the T-coercivity approach, that the well-posedness of the two canonically associated scalar problems, with Dirichlet and Neumann boundary conditions, implies the well-posedness of the Maxwell problem. This allows us to give simple and sharp criteria, obtained in the study of the scalar cases, to ensure that the Maxwell transmission problem between a classical dielectric material and a negative metamaterial is well-posed.
• Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control
  
  • Fabbri Giorgio
  • Russo Francesco
  , 2014. The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$, taking values in a Hilbert space $H$, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of "mild solution" for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.
• Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian
  
  • Graber P. Jameson
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2014, 70 (2), pp.185-224. We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games (10.1007/s00245-014-9239-3)
  DOI : 10.1007/s00245-014-9239-3
• Numerical modeling of nonlinear acoustic waves in a tube connected with an array of Helmholtz resonators
  
  • Lombard Bruno
  • Mercier Jean-François
  Journal of Computational Physics, Elsevier, 2014, 259 (15). (10.1016/j.jcp.2013.11.036)
  DOI : 10.1016/j.jcp.2013.11.036
• Mathematical modelling of multi conductor cables
  
  • Beck Geoffrey
  • Imperiale Sebastien
  • Joly Patrick
  Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 2014, pp.26. This paper proposes a formal justification of simplified 1D models for the propagation of electromagnetic waves in thin non-homogeneous lossy conductor cables. Our approach consists in deriving these models from an asymptotic analysis of 3D Maxwell’s equations. In essence, we extend and complete previous results to the multi-wires case. (10.3934/dcdss.2015.8.521)
  DOI : 10.3934/dcdss.2015.8.521
• Space-time focusing of acoustic waves on unknown scatterers
  
  • Cassier Maxence
  • Hazard Christophe
  Wave Motion, Elsevier, 2014, pp.19. Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit-receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy-Lax model. In this context, several mathematical and numerical arguments that support our idea are explored. (10.1016/j.wavemoti.2014.07.009)
  DOI : 10.1016/j.wavemoti.2014.07.009
• Numerical modeling of nonlinear acoustic waves in a tube connected with Helmholtz resonators
  
  • Lombard Bruno
  • Mercier Jean-François
  Journal of Computational Physics, Elsevier, 2014, 259, pp.421-443. Acoustic wave propagation in a one-dimensional waveguide connected with Helmholtz resonators is studied numerically. Finite amplitude waves and viscous boundary layers are considered. The model consists of two coupled evolution equations: a nonlinear PDE describing nonlinear acoustic waves, and a linear ODE describing the oscillations in the Helmholtz resonators. The thermal and viscous losses in the tube and in the necks of the resonators are modeled by fractional derivatives. A diffusive representation is followed: the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations. A splitting method is then applied to the evolution equations: their propagative part is solved using a standard TVD scheme for hyperbolic equations, whereas their diffusive part is solved exactly. Various strategies are examined to compute the coefficients of the diffusive representation; finally, an optimization method is preferred to the usual quadrature rules. The numerical model is validated by comparisons with exact solutions. The properties of the full nonlinear solutions are investigated numerically. In particular, the existence of acoustic solitary waves is confirmed. (10.1016/j.jcp.2013.11.036)
  DOI : 10.1016/j.jcp.2013.11.036
• High-order asymptotic expansion for the acoustics in viscous gases close to rigid walls
  
  • Schmidt Kersten
  • Anastasia Thöns-Zueva
  • Joly Patrick
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, pp.1823. (10.1142/S0218202514500080)
  DOI : 10.1142/S0218202514500080
• Quasi-local transmission conditions for non-overlapping domain decomposition methods for the Helmholtz equation
  
  • Collino Francis
  • Joly Patrick
  • Lecouvez Matthieu
  • Stupfel Bruno
  Comptes Rendus. Physique, Académie des sciences (Paris), 2014, 15 (5), pp.403-414. In this article, we present new transmission conditions for a domain decomposition method, applied to a scattering problem. Unlike other conditions used in the literature, the conditions developed here are non-local, but can be written as an integral operator (as a Riesz potential) on the interface between two domains. This operator, of order View the MathML source12, leads to an exponential convergence of the domain decomposition algorithm. A spectral analysis of the influence of the operator on simple cases is presented, as well as some numerical results and comparisons. (10.1016/j.crhy.2014.04.005)
  DOI : 10.1016/j.crhy.2014.04.005
• Wave propagation through penetrable scatterers in a waveguide and through a penetrable gratings
  
  • Maurel Agnès
  • Mercier Jean-François
  • Félix Simon
  Journal of the Acoustical Society of America, Acoustical Society of America, 2014, 135 (1), pp.165-174. A multimodal method based on the admittance matrix is used to analyze wave propagation through scatterers of arbitrary shape. Two cases are considered: a waveguide containing scatterers, and the scattering of a plane wave at oblique incidence to an infinite periodic row of scatterers. In both cases, the problem reduces to a system of two sets of first-order differential equations for the modal components of the wavefield, similar to the system obtained in the rigorous coupled wave analysis. The system can be solved numerically using the admittance matrix, which leads to a stable numerical method, the basic properties of which are discussed (convergence, reciprocity, energy conservation). Alternatively, the admittance matrix can be used to get analytical results in the weak scattering approximation. This is done using the plane wave approximation, leading to a generalized version of the Webster equation and using a perturbative method to analyze the Wood anomalies and Fano resonances. (10.1121/1.4836075)
  DOI : 10.1121/1.4836075
• GKW representation theorem and linear BSDEs under restricted information. An application to risk-minimization.
  
  • Ceci Claudia
  • Cretarola Alessandra
  • Russo Francesco
  Stochastics and Dynamics, World Scientific Publishing, 2014, 14 (2), pp.1350019. In this paper we provide Galtchouk-Kunita-Watanabe representation results in the case where there are restrictions on the available information. This allows to prove existence and uniqueness for linear backward stochastic differential equations driven by a general càdlàg martingale under partial information. Furthermore, we discuss an application to risk-minimization where we extend the results of Föllmer and Sondermann (1986) to the partial information framework and we show how our result fits in the approach of Schweizer (1994). (10.1142/S0219493713500196)
  DOI : 10.1142/S0219493713500196
• A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation
  
  • Billionnet Alain
  • Elloumi Sourour
  • Lambert Amélie
  Journal of Combinatorial Optimization, Springer Verlag, 2014, 28 (2), pp.376-399. Let (MQP) be a general mixed-integer quadratic program that consists of minimizing a quadratic function f(x) = x^TQx +c^Tx subject to linear constraints. Our approach to solve (MQP) is first to consider an equivalent general mixed-integer quadratic problem. This equivalent problem has additional variables y_{ij}, additional quadratic constraints y_{ij}=x_ix_j, a convex objective function, and a set of valid inequalities. Contrarily to the reformulation proposed in MIQCR, the equivalent problem cannot be directly solved by a standard solver. Here, we propose a new Branch and Bound process based on the relaxation of the non-convex constraints y_{ij}=x_ix_j to solve $(MQP)$. Computational experiences are carried out on pure- and mixed-integer quadratic instances. The results show that the solution time of most of the considered instances with up to 60 variables is improved by our Branch and Bound algorithm in comparison with MIQCR and with the general mixed-integer nonlinear solver BARON. (10.1007/s10878-012-9560-1)
  DOI : 10.1007/s10878-012-9560-1
• Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional
  
  • Warner James E.
  • Diaz Manuel I.
  • Aquino Wilkins
  • Bonnet Marc
  Computational Mechanics, Springer Verlag, 2014, 54, pp.645-659. This work focuses on the identification of heterogeneous linear elastic moduli in the context of frequency-domain, coupled acoustic-structure interaction (ASI), using either solid displacement or fluid pressure measurement data. The approach postulates the inverse problem as an optimization problem where the solution is obtained by minimizing a modified error in constitutive equation (MECE) functional. The latter measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, while incorporating the measurement data as additional quadratic error terms. We demonstrate two strategies for selecting the MECE weighting coefficient to produce regularized solutions to the ill-posed identification problem: 1) the discrepancy principle of Morozov, and 2) an error-balance approach that selects the weight parameter as the minimizer of another functional involving the ECE and the data misfit. Numerical results demonstrate that the proposed methodology can successfully recover elastic parameters in 2D and 3D ASI systems from response measurements taken in either the solid or fluid subdomains. Furthermore, both regularization strategies are shown to produce accurate reconstructions when the measurement data is polluted with noise. The discrepancy principle is shown to produce nearly optimal solutions, while the error-balance approach, although not optimal, remains effective and does not need a priori information on the noise level. (10.1007/s00466-014-1018-0)
  DOI : 10.1007/s00466-014-1018-0
• Propagation in waveguides with varying cross-section and curvature: A new light on the role of supplementary modes in multimodal methods
  
  • Maurel Agnes
  • Mercier Jean-François
  • Félix Simon
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2014, 470 (2166), pp.20130743. We present an efficient multi-modal method to describe the acoustic propagation in waveguides with varying curvature and cross section. A key feature is the use of a flexible geometrical transformation to a virtual space in which the waveguide is straight and has unitary cross section. In this new space, the pressure field has to satisfy a modified wave equation and associated modified boundary conditions. These boundary conditions are in general not satisfied by the Neumann modes, used for the series representation of the field. Following previous work, an improved modal method (MM) is presented, by means of the use of two supplementary modes. Resulting increased convergences are exemplified by comparison with the classical MM. Next, the following question is addressed: when the boundary conditions are verified by the Neumann modes, does the use of supplementary modes improve or degrade the convergence of the computed solution? Surprisingly, although the supplementary modes degrade the behaviour of the solution at the walls, they improve the convergence of the wavefield and of the scattering coefficients. This sheds a new light on the role of the supplementary modes and opens the way for their use in a wide range of scattering problems. (10.1098/rspa.2014.0008)
  DOI : 10.1098/rspa.2014.0008
• Generalized covariation for Banach space valued processes, Itô formula and applications
  
  • Di Girolami Cristina
  • Russo Francesco
  Osaka Journal of Mathematics, Osaka University, 2014, 51 (3). This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$), we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.
• Second-order necessary conditions in Pontryagin form for optimal control problems
  
  • Bonnans J. Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (6), pp.3887-3916. In this report, we state and prove first- and second-order necessary conditions in Pontryagin form for optimal control problems with pure state and mixed control-state constraints. We say that a Lagrange multiplier of an optimal control problem is a Pontryagin multiplier if it is such that Pontryagin's minimum principle holds, and we call optimality conditions in Pontryagin form those which only involve Pontryagin multipliers. Our conditions rely on a technique of partial relaxation, and apply to Pontryagin local minima. (10.1137/130923452)
  DOI : 10.1137/130923452
• Level-set approach for Reachability Analysis of Hybrid Systems under Lag Constraints
  
  • Granato Giovanni
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (1), pp.606--628. This study aims at characterizing a reachable set of a hybrid dynamical system with a lag constraint in the switch control. The setting does not consider any controllability assumptions and uses a level-set approach. The approach consists in the introduction of on adequate hybrid optimal control problem with lag constraints on the switch control whose value function allows a characterization of the reachable set. The value function is in turn characterized by a system of quasi-variational inequalities (SQVI). We prove a comparison principle for the SQVI which shows uniqueness of its solution. A class of numerical finite differences schemes for solving the system of inequalities is proposed and the convergence of the numerical solution towards the value function is studied using the comparison principle. Some numerical examples illustrating the method are presented. Our study is motivated by an industrial application, namely, that of range extender electric vehicles. This class of electric vehicles uses an additional module -- the range extender -- as an extra source of energy in addition to its main source -- a high voltage battery. The reachability study of this system is used to establish the maximum range of a simple vehicle model. (10.1137/120874205)
  DOI : 10.1137/120874205
• BSDEs under partial information and financial applications.
  
  • Ceci Claudia
  • Cretarola Alessandra
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2014. In this paper we provide existence and uniqueness results for the solution of BSDEs driven by a general square integrable martingale under partial information. We discuss some special cases where the solution to a BSDE under restricted information can be derived by that related to a problem of a BSDE under full information. In particular, we provide a suitable version of the Föllmer-Schweizer decomposition of a square integrable random variable working under partial information and we use this achievement to investigate the local risk-minimization approach for a semimartingale financial market model. (10.1016/j.spa.2014.03.003)
  DOI : 10.1016/j.spa.2014.03.003
• Variance optimal hedging for continuous time additive processes and applications
  
  • Goutte Stéphane
  • Oudjane Nadia
  • Russo Francesco
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2014, 81 (1), pp.147--185. For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process.
This allows to provide an efficient algorithm for solving the
mean variance hedging problem.
Applications to models derived from the electricity market are performed. (10.1080/17442508.2013.774402)
  DOI : 10.1080/17442508.2013.774402
• On the absence of trapped modes in locally perturbed open waveguides
  
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2014, pp.14. This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the three-dimensional scalar wave equation is considered. (10.1093/imamat/hxu046)
  DOI : 10.1093/imamat/hxu046
• Extraordinary transmission through subwavelength dielectric gratings in the microwave range
  
  • Ahmed Akarid
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  Optics Letters, Optical Society of America - OSA Publishing, 2014, 39 (13), pp.3752-3755. We address the problem of the transmission through subwavelength dielectric gratings. Following Maurel et al. [Phys. Rev. B 88, 115416 (2013)], the problem is reduced to the transmission by an homogeneous slab, either anisotropic (for transverse magnetic waves, TM) or isotropic (for transverse electric waves, TE), and an explicit expression of the transmission coefficient is derived. The optimum angle realizing perfect impedance matching (Brewster angle) is shown to depend on the contrasts of the dielectric layers with respect to the air. Besides, we show that the Fabry–Perot resonances may be dependent on the incident angle, in addition to the dependence on the frequency. These facts depart from the case of metallic gratings usually considered; they are confirmed experimentally both for TE and TM waves in the microwave regime. (10.1364/OL.39.003752)
  DOI : 10.1364/OL.39.003752