Publications

2008

• High-order angles in almost-Riemannian geometry
  
  • Boscain Ugo
  • Sigalotti Mario
  Séminaire de Théorie Spectrale et Géométrie, Grenoble : Université de Grenoble 1, Institut Fourier, 1983-, 2008, 24, pp.41-54.
• Minimum stress optimal design with the level set method
  
  • Allaire Grégoire
  • Jouve François
  Engineering Analysis with Boundary Elements, Elsevier, 2008, 32, pp.909-918. This paper is devoted to minimum stress design instructural optimization. We propose a simple andefficient numerical algorithm for shape and topologyoptimization based on the level set method coupledwith the topological derivative. We compute ashape derivative, as well as a topological derivative,for a stress-based objective function. Using anadjoint equation we implement a gradient algorithmfor the minimization of the objective function.Several numerical examples in 2-d and 3-d are discussed.
• Fluid-Structure Interaction and multi-body contact. Application to the aortic valves
  
  • Astorino Matteo
  • Gerbeau Jean-Frédéric
  • Pantz Olivier
  • Traore Karim-Frédéric
  , 2008, pp.23. We present a partitioned procedure for fluid-structure interaction problems in which contacts among different deformable bodies can occur. A typical situation is the movement of a thin valve (e.g. the aortic valve) immersed in an incompressible viscous fluid (e.g. the blood). In the proposed strategy the fluid and structure solvers are considered as independent ``black-boxes'' that exchange forces and displacements; the structure solvers are moreover not supposed to manage contact by themselves. The hypothesis of non-penetration among solid objects defines a non-convex optimization problem. To solve the latter, we use an internal approximation algorithm that is able to directly handle the cases of thin structures and self-contacts. A numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme.
• Analyse théorique et numérique du modèle de Webster Lokshin
  
  • Haddar Houssem
  • Matignon Denis
  , 2008. Les ondes acoustiques qui se propagent dans un pavillon dont la paroi est le siège de pertes visco-thermiques et dont les deux extremités sont sujettes à des conditions de rayonnement obéissent à un modèle de Webster-Lokshin, lequel fait intervenir des dérivées fractionnaires en temps dans le milieu et des conditions aux limites dynamiques. Ce système peut s'interpréter comme le couplage de trois sous-systèmes : une équation des ondes, une réalisation diffusive de l'opérateur pseudo-différentiel en temps, et une réalisation dissipative de l'impédance par le lemme de Kalman-Yakubovich-Popov. En utilisant le théorème de Hille-Yosida, l'existence et l'unicité des solutions fortes de ce système sont établies. De plus, des schémas numériques sont proposés et leur stabilité est analysée en utilisant des techniques d'énergie ; de nombreuses simulations numériques viennent illustrer le comportement du modèle pour diverses valeurs des paramètres.
• Particle filter-based policy gradient for pomdps
  
  • Coquelin Pierre-Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2008. Our setting is a Partially Observable Markov Decision Process with continuous state, observation and action spaces. Decisions are based on a Particle Filter for estimating the belief state given past observations. We consider a policy gradient approach for parameterized policy optimization. For that purpose, we investigate sensitivity analysis of the performance measure with respect to the parameters of the policy, focusing on Finite Difference (FD) techniques. We show that the naive FD is subject to variance explosion because of the non-smoothness of the resampling procedure. We propose a more sophisticated FD method which overcomes this problem and establish its consistency.
• Generalized Fast Marching Method: Applications to Image Segmentation
  
  • Forcadel Nicolas
  • Le Guyader Carole
  • Gout Christian
  Numerical Algorithms, Springer Verlag, 2008, 48 (1-3), pp.189-211. In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan-Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266-277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm. (10.1007/s11075-008-9183-x)
  DOI : 10.1007/s11075-008-9183-x
• Implicit time discretization of the mean curvature flow with a discontinuous forcing term.
  
  • Chambolle Antonin
  • Novaga Matteo
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2008, 10 (3), pp.283--300. We consider an implicit time discretization for the motion of a hypersurface driven by its anisotropic mean curvature. We prove some convergence results of the scheme under very general assumptions on the forcing term, which include in particular the case of a typical path of the Brownian motion. We compare this limit with other available solutions, whenever they are defined. As a by-product of the analysis, we also provide a simple proof of the coincidence of the limit flow with the regular evolutions, defined for small times, in the case of a regular forcing term.
• Numerical Linear Algebra
  
  • Allaire Grégoire
  • Kaber Sidi-Mahmoud
  , 2008, 55.
• On the existence of transmission eigenvalues in an inhomogeneous medium
  
  • Cakoni Fioralba
  • Haddar Houssem
  , 2008, pp.24. We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both scalar Helmholtz equation and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that this contrast is large enough.
• Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai-Minh
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (10), pp.1787-1827. (10.1142/S0218202508003194)
  DOI : 10.1142/S0218202508003194
• Shape and topology optimization of the robust compliance via the level set method
  
  • Allaire Grégoire
  • de Gournay Frédéric
  • Jouve François
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2008, 14 (14), pp.43--70. The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it. Then, in the framework of Hadamard method, we compute the directional shape derivative of this criterion which is used in a numerical algorithm, based on the level set method, to find optimal shapes that minimize the worst-case compliance. Since this criterion is usually merely directionally differentiable, we introduce a semidefinite programming approach to select the best descent direction at each step of a gradient method. Numerical examples are given in 2-d and 3-d. Mathematics Subject Classification. 49Q10, 74P10, 74P15, 74P20. (10.1051/cocv:2007048)
  DOI : 10.1051/cocv:2007048
• Global weak solutions to the equations of motion for magnetic fluids
  
  • Amirat Youcef
  • Hamdache Kamel
  • Murat François
  Journal of Mathematical Fluid Mechanics, Springer Verlag, 2008, 10 (8), pp.326-351.
• 3D Electrostatic Hybrid Elements Model for SAW transducers modeling
  
  • Jerez-Hanckes Carlos F.
  • Laude Vincent
  • Lardat Raphael
  • Nédélec Jean-Claude
  IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, Institute of Electrical and Electronics Engineers, 2008, 55 (3), pp.686-695. In this work, the singular behavior of charges at corners and edges on the metallized areas in SAW transducers are investigated. In particular, it is demonstrated that a tensor product of the commonly used Tchebychev bases overestimates the singularities at corners, and, hence, it cannot be used in a proper boundary element method formulation. On the other hand, it is shown that a simple finite element method-like approach is impractical due to the enormous number of unknowns required to model the electrode's large length-to-width ratio. These considerations are then used for defining a hybrid element model, which combines Tchebychev and linear polynomials over differently meshed domains. Such an approach is shown to suitably account for charge singularities while greatly reducing the number of unknowns. Results are obtained for isotropic and anisotropic substrates for non-periodic configurations.
• Asymptotics for steady state voltage potentials in a bidimensional highly contrasted medium with thin layer
  
  • Poignard Clair
  Mathematical Methods in the Applied Sciences, Wiley, 2008, 31 (4), pp.443-479. We study the behavior of steady state voltage potentials in two kinds of bidimensional media composed of material of complex permittivity equal to 1 (respectively α) surrounded by a thin membrane of thickness h and of complex permittivity α (respectively 1). We provide in both cases a rigorous derivation of the asymptotic expansion of steady state voltage potentials at any order as h tends to zero, when Neumann boundary condition is imposed on the exterior boundary of the thin layer. Our complex parameter α is bounded but may be very small compared to 1, hence our results describe the asymptotics of steady state voltage potentials in all heterogeneous and highly heterogeneous media with thin layer. The asymptotic terms of the potential in the membrane are given explicitly in local coordinates in terms of the boundary data and of the curvature of the domain, while these of the inner potential are the solutions to the so-called dielectric formulation with appropriate boundary conditions. The error estimates are given explicitly in terms of h and α with appropriate Sobolev norm of the boundary data. We show that the two situations described above lead to completely different asymptotic behaviors of the potentials. (10.1002/mma.923)
  DOI : 10.1002/mma.923
• Sensitivity Analysis in Particle Filters. Application to Policy Optimization in POMDPs
  
  • Coquelin Pierre Arnaud
  • Deguest Romain
  • Munos Rémi
  , 2008. Our setting is a Partially Observable Markov Decision Process with continuous state, observation and action spaces. Decisions are based on a Particle Filter for estimating the belief state given past observations. We consider a policy gradient approach for parameterized policy optimization. For that purpose, we investigate sensitivity analysis of the performance measure with respect to the parameters of the policy, focusing on Finite Difference (FD) techniques. We show that the naive FD is subject to variance explosion because of the non-smoothness of the resampling procedure. We propose a more sophisticated FD method which overcomes this problem and establish its consistency.
• A Post-Treatment of the Homogenization Method for Shape Optimization
  
  • Pantz O.
  • Trabelsi K.
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2008, 47 (3), pp.1380-1398. We propose an alternative to the classical post-treatment of the homogenization method for shape optimization. Rather than penalize the material density once the optimal composite shape is obtained (by the homogenization method) in order to produce a workable shape close to the optimal one, we macroscopically project the microstructure of the former through an appropriate procedure that roughly consists in laying the material along the directions of lamination of the composite. We have tested our approach in the framework of compliance minimization in two-dimensional elasticity. Numerical results are provided. (10.1137/070688900)
  DOI : 10.1137/070688900
• Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems
  
  • Benner Peter
  • Li Jing-Rebecca
  • Penzl Thilo
  Numerical Linear Algebra with Applications, Wiley, 2008, 15 (9), pp.755-777. We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. (10.1002/nla.622)
  DOI : 10.1002/nla.622