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
• Optimal Control of the Atmospheric Reentry of a Space Shuttle by an Homotopy Method
  
  • Hermant Audrey
  Optimal Control Applications and Methods, Wiley, 2011, 32 (6), pp.627-646. This paper deals with the optimal control problem of the atmospheric reentry of a space shuttle with a second-order state constraint on the thermal flux. We solve the problem using the shooting algorithm combined with an homotopy method which automatically determines the structure of the optimal trajectory (composed of one boundary arc and one touch point). (10.1002/oca.961)
  DOI : 10.1002/oca.961
• Hedging and Vertical Integration in Electricity Markets
  
  • Aïd René
  • Chemla Gilles
  • Porchet Arnaud
  • Touzi Nizar
  Management Science, INFORMS, 2011, 57 (8). This paper analyzes the interactions between competitive (wholesale) spot, retail, and forward markets and vertical integration in electricity markets. We develop an equilibrium model with producers, retailers, and traders to study and quantify the impact of forward markets and vertical integration on prices, risk premia, and retail market shares. We point out that forward hedging and vertical integration are two separate mechanisms for demand and spot price risk diversification that both reduce the retail price and increase retail market shares. We show that they differ in their impact on prices and firms' utility because of the asymmetry between production and retail segments. Vertical integration restores the symmetry between producers' and retailers' exposure to demand risk, whereas linear forward contracts do not. Vertical integration is superior to forward hedging when retailers are highly risk averse. We illustrate our analysis with data from the French electricity market. (10.1287/mnsc.1110.1357)
  DOI : 10.1287/mnsc.1110.1357
• Damage and fracture evolution in brittle materials by shape optimization methods
  
  • Allaire Grégoire
  • Jouve François
  • van Goethem Nicolas
  Journal of Computational Physics, Elsevier, 2011, 230 (12), pp.5010--5044. This paper is devoted to a numerical implementation of the Francfort-Marigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith-type dissipated energy. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well-separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasi-static damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack propagation , including kinking and branching. Several numerical examples in 2d and 3d are discussed.
• Lévy flights in evolutionary ecology
  
  • Jourdain Benjamin
  • Méléard Sylvie
  • Woyczynski Wojbor
  Journal of Mathematical Biology, Springer, 2011, pp.31 p.. We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of nonlocal fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such nontrivial fractional objects; their uniqueness is also proved. (10.1007/s00285-011-0478-5)
  DOI : 10.1007/s00285-011-0478-5
• 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.
• 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.
• 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.
• 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
• 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