Publications

2005

• New constructions of domain decomposition methods for systems of PDEs
  
  • Dolean Victorita
  • Nataf Frédéric
  • Rapin Gerd
  Comptes Rendus de l'Académie des Sciences - Series III - Sciences de la Vie, Elsevier, 2005, Ser. I 340, pp.693-696. We propose new domain decomposition methods for systems of partial differential equations in two and three dimensions. The algorithms are derived with the help of the Smith factorization of the operator. This could also be validated by numerical experiments.
• Image decomposition into a bounded variation component and an oscillating component
  
  • Aujol Jean-François
  • Aubert Gilles
  • Blanc-Féraud Laure
  • Chambolle Antonin
  Journal of Mathematical Imaging and Vision, Springer Verlag, 2005, 22 (1), pp.pages 71-88. We construct an algorithm to split an image into a sum $u+v$ of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from a recent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternately in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration.
• A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models
  
  • Cont Rama
  • Voltchkova Ekaterina
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2005, 43 (4), pp.1596-1626. We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions. (10.1137/S0036142903436186)
  DOI : 10.1137/S0036142903436186
• Concentration for independent random variables with heavy tails
  
  • Barthe Franck
  • Cattiaux Patrick
  • Roberto Cyril
  AMRX Appl.Math.Res.Express, 2005 (2), pp.39-60. If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of $n$ independent copies, with good dependence in $n$.