Publications

2011

• A posteriori error estimates for the effective Hamiltonian of dislocation dynamics
  
  • Cacace Simone
  • Chambolle Antonin
  • Monneau Régis
  Numerische Mathematik, Springer Verlag, 2011, pp.55 p.. We study an implicit and discontinuous scheme for a non-local Hamilton-Jacobi equation modelling dislocation dynamics. For the evolution problem, we prove an a posteriori estimate of Crandall-Lions type for the error between continuous and discrete solutions. We deduce an a posteriori error estimate for the effective Hamiltonian associated to a stationary cell problem. In dimension one and under suitable assumptions, we also give improved a posteriori estimates. Numerical simulations are provided. (10.1007/s00211-011-0430-z)
  DOI : 10.1007/s00211-011-0430-z
• Micro-Macro Modelling of an Array of Spheres Interacting Through Lubrication Forces
  
  • Lefebvre-Lepot Aline
  • Maury Bertrand
  • Lefebvre Aline
  Advances in Mathematical Sciences and Applications, Gakkōtosho Co. Ltd, 2011, 21 (2), pp.535–557. We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model, which is of the newtonian type, relies on an elongational viscosity, which is proportional to the reciprocal of the local fluid fraction. We then establish the convergence in a weak sense of solutions to the discrete problem towards the solution to the partial differential equation which we identified as the macroscopic constitutive equation.
• Stable reconstruction of generalized impedance boundary conditions
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2011, 27 (9), pp.095002. (10.1088/0266-5611/27/9/095002)
  DOI : 10.1088/0266-5611/27/9/095002
• Large Time-Step Numerical Scheme for the Seven-Equation Model of Compressible Two-Phase Flows
  
  • Chalons Christophe
  • Coquel Frédéric
  • Kokh Samuel
  • Spillane Nicole
  , 2011, pp.pp. 225-233. We consider the seven-equation model for compressible two-phase flows and propose a large time-step numerical scheme based on a time implicit-explicit Lagrange-Projection strategy introduced by Coquel et al. for Euler equations. The main objective is to get a Courant-Friedrichs-Lewy (CFL) condition driven by (slow) contact waves instead of (fast) acoustic waves. (10.1007/978-3-642-20671-9_24)
  DOI : 10.1007/978-3-642-20671-9_24
• Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W±, II
  
  • Aschbacher Walter H.
  • Barbaroux Jean-Marie
  • Faupin Jérémy
  • Guillot Jean-Claude
  Annales Henri Poincaré, Springer Verlag, 2011, 12 (8), pp.1539-1570. We do the spectral analysis of the Hamiltonian for the weak leptonic decay of the gauge bosons W+/-. Using Mourre theory, it is shown that the spectrum between the unique ground state and the first threshold is purely absolutely continuous. Neither sharp neutrino high energy cutoff nor infrared regularization are assumed. (10.1007/s00023-011-0114-3)
  DOI : 10.1007/s00023-011-0114-3
• Swing Options Valuation:a BSDE with Constrained Jumps Approach
  
  • Bernhart Marie
  • Pham Huyên
  • Tankov Peter
  • Warin Xavier
  , 2011. We introduce a new probabilistic method for solving a class of impulse control problems based on their representations as Backward Stochastic Differential Equations (BSDEs for short) with constrained jumps. As an example, our method is used for pricing Swing options. We deal with the jump constraint by a penalization procedure and apply a discrete-time backward scheme to the resulting penalized BSDE with jumps. We study the convergence of this numerical method, with respect to the main approximation parameters: the jump intensity $\lambda$, the penalization parameter $p > 0$ and the time step. In particular, we obtain a convergence rate of the error due to penalization of order $(\lambda p)^{\alpha - \frac{1}{2}}, \forall \alpha \in \left(0, \frac{1}{2}\right)$. Combining this approach with Monte Carlo techniques, we then work out the valuation problem of (normalized) Swing options in the Black and Scholes framework. We present numerical tests and compare our results with a classical iteration method.
• The Electromagnetic Interior Transmission Problem for Regions with Cavities
  
  • Cossonnière Anne
  • Haddar Houssem
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2011, 43 (4), pp.1698-1715. We consider the electromagnetic interior transmission problem in the case when the medium has cavities, i.e. regions in which the index of refraction is the same as in the host medium. We address the configuration where the electromagnetic permeability is constant while the electric permittivity is variable and may be anisotropic. In this case, using appropriate reformulation of the problem into a fourth order pde, we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. Monotonicity properties of the first eigenvalue in terms of the permittivity and the size of the cavity are established. (10.1137/100813890)
  DOI : 10.1137/100813890
• Characterization of the value function of final state constrained control problems with BV trajectories
  
  • Briani Ariela
  • Zidani Hasnaa
  Communications on Pure and Applied Mathematics, Wiley, 2011, 10 (6), pp.1567-1587. This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations. (10.3934/cpaa.2011.10.1567)
  DOI : 10.3934/cpaa.2011.10.1567
• A discontinuous Galerkin solver for front propagation
  
  • Bokanowski Olivier
  • Cheng Yingda
  • Shu Chi-Wang
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2011, 33 (2), pp.923-938. We propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation.
• Application of convex lexicographical optimization to the balance of GRTgaz gas grid
  
  • Bonnans Joseph Frederic
  • Paraisy Ruben
  • Veyrat Sébastien
  • Adam Soizic
  Journal of Global Optimization, Springer Verlag, 2011, 49 (3), pp.415--423.