Publications

2005

• Une structure fractale tridimensionnelle
  
  • Colonna Jean-François
  , 2005. A tridimensional fractal structure (Une structure fractale tridimensionnelle)
• Optimisation de structures par la méthode des lignes de niveau
  
  • Allaire Grégoire
  • de Gournay Frédéric
  • Jouve François
  , 2005. Nous proposons une méthode numérique d'optimisation de structures basée sur la combinaison des techniques classiques de dérivation par rapport au domaine et de courbe de niveau pour la propagation de fronts. La dérivée de forme est utilisée comme vitesse de transport dans une équation d'Hamilton-Jacobi pour modifier la forme. Cet algorithme de level set est un peu coûteux en temps de calcul car il capture la forme sur un maillage eulérien fixe. Il prend facilement en compte certaines variations de topologie. Il est couplé avec la méthode du gradient topologique afin de permettre la création de nouveaux trous et tenter d'éviter sa convergence vers des minima locaux.
• Mathematics and finance
  
  • Gobet Emmanuel
  • Pagès Gilles
  • Yor Marc
  , 2008, pp.63-76. Since the beginning of the 1990s, mathematics, and more particularly the theory of probability, have taken an increasing role in the banking and insurance industries. This motivated the authors to present here some interactions between Mathematics and Finance and their consequences at the level of research and training in France in these domains. (10.1007/978-3-540-75265-3_7)
  DOI : 10.1007/978-3-540-75265-3_7
• Moderate deviations for particle filtering
  
  • Douc Randal
  • Guillin A.
  • Najim J.
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2005, 15 (1B), pp.587-614. (10.1214/105051604000000657)
  DOI : 10.1214/105051604000000657
• Le paradoxe de Bertrand
  
  • Colonna Jean-François
  , 2005. The Bertrand paradox (Le paradoxe de Bertrand)
• Le paradoxe de Bertrand généralisé
  
  • Colonna Jean-François
  , 2005. The extended Bertrand paradox (Le paradoxe de Bertrand generalise)
• Le paradoxe de Bertrand
  
  • Colonna Jean-François
  , 2005. The Bertrand paradox (Le paradoxe de Bertrand)
• Conformal mappings and inverse boundary value problems
  
  • Haddar Houssem
  • Kress Rainer
  Inverse Problems, IOP Publishing, 2005, 21 (3), pp.935--953. (10.1088/0266-5611/21/3/009)
  DOI : 10.1088/0266-5611/21/3/009
• TIME-DELAY REGULARIZATION OF ANISOTROPIC DIFFUSION AND IMAGE PROCESSING
  
  • Belahmidi Abdelmounim
  • Chambolle Antonin
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2005, Vol.39, No.2., pp.pp. 231-251. We study a time-delay regularization of the anisotropic diffusion model for image denoising of Malik and Perona, which has been proposed by Nitzberg and Shiota. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images. (10.1051/m2an:2005010)
  DOI : 10.1051/m2an:2005010
• Functional central limit theorems for a large network in which customers join the shortest of several queues
  
  • Graham Carl
  Probability Theory and Related Fields, Springer Verlag, 2005, 131 (1), pp.97-120. We consider N single server infinite buffer queues with service rate \beta. Customers arrive at rate N\alpha, choose L queues uniformly, and join the shortest. We study the processes R^N for large N, where R^N_t(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R^N_0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R^N in equilibrium under the stability condition \alpha < \beta, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits for large times and for large sizes N by a compactness-uniqueness method. We deduce a posteriori the CLT for R^N_0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit. (10.1007/s00440-004-0372-9)
  DOI : 10.1007/s00440-004-0372-9
• A MUSIC algorithm for locating small inclusions buried in a half space from the scattering amplitude at a fixed frequency.
  
  • Ammari Habib
  • Iakovleva Ekaterina
  • Lesselier Dominique
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2005, 3 (3), pp.597-628. In this paper a MUSIC (standing for multiple signal classification) algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency is developed. The underlying application area which motivated this work is the imaging of antipersonnel mines from electromagnetic data, formulated as an inverse scattering problem for the Helmholtz equation. The algorithm makes use of an asymptotic expansion of the scattering amplitude. A derivation of the leading-order term in this asymptotic expansion and its application for designing a MUSIC type of algorithm are presented. The viability of this algorithm is documented by a variety of numerical results. (10.1137/040610854)
  DOI : 10.1137/040610854
• Almost periodic solutions of forced vectorial Liénard equations
  
  • Cieutat Philippe
  Journal of Differential Equations, Elsevier, 2005, 209 (2), pp.302-328. We describe the set of bounded or almost periodic solutions of the following Liénard system: $u′′+d/dt[∇F(u)]+Cu=e(t)$, where $e$ : $\mathbb{R}$ ⟶ $\mathbb{R}$^N is almost periodic, $C$ : $\mathbb{R}$ ⟶ $\mathbb{R}$^N is a symmetric and nonsingular linear operator, and $\nabla F$ denotes the gradient of the convex function $F$ on $\mathbb{R}$ ⟶ $\mathbb{R}$^N. Then, we state a result of existence and uniqueness of almost periodic solution. (10.1016/j.jde.2004.07.006)
  DOI : 10.1016/j.jde.2004.07.006
• Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control
  
  • Baudouin Lucie
  • Kavian Otared
  • Puel Jean-Pierre
  Journal of Differential Equations, Elsevier, 2005, 216 (1), pp.188-222. We study a Schrödinger equation with a coulombian potential, singular at finite distance, and an electric potential, possibly unbounded. The two potentials are real valued and may depend on space and time variables. We prove that if the electric potential is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.
• The convex back-scattering support
  
  • Haddar Houssem
  • Kusiak Steven
  • Sylvester John
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2005, 66 (2), pp.591--615. (10.1137/040616231)
  DOI : 10.1137/040616231
• A common formalism for the Integral formulations of the forward EEG problem
  
  • Kybic J.
  • Clerc M.
  • Abboud T.
  • Faugeras O.
  • Keriven R.
  • Papadopoulo T.
  IEEE Transactions on Medical Imaging, Institute of Electrical and Electronics Engineers, 2005, 24 (1), pp.12-28. The forward electro-encephalography (EEG) problem involves finding a potential V from the Poisson equation ∇ · (σ∇V ) = f , in which f represents electrical sources in the brain, and σ the conductivity of the head tissues. In the piecewise constant conductivity head model, this can be accomplished by the Boundary Element Method (BEM) using a suitable integral formulation. Most previous work uses the same integral formulation, corresponding to a double-layer potential. In this article we present a conceptual framework based on a well-known theorem (Theorem 1) that characterizes harmonic functions defined on the complement of a bounded smooth surface. This theorem says that such harmonic functions are completely defined by their values and those of their normal derivatives on this surface. It allows us to cast the previous BEM approaches in a unified setting and to develop two new approaches corresponding to different ways of exploiting the same theorem. Specifically, we first present a dual approach which involves a single-layer potential. Then, we propose a symmetric formulation, which combines single and double-layer potentials, and which is new to the field of EEG, although it has been applied to other problems in electromagnetism. The three methods have been evaluated numerically using a spherical geometry with known analytical solution, and the symmetric formulation achieves a significantly higher accuracy than the alternative methods. Additionally, we present results with realistically shaped meshes. Beside providing a better understanding of the foundations of BEM methods, our approach appears to lead also to more efficient algorithms. (10.1109/tmi.2004.837363)
  DOI : 10.1109/tmi.2004.837363
• Comparison of Resampling Schemes for Particle Filtering
  
  • Douc Randal
  • Cappé Olivier
  • Moulines Eric
  , 2005, pp.64-69. This contribution is devoted to the comparison of various resampling approaches that have been proposed in the literature on particle filtering. It is first shown using simple arguments that the so-called residual and stratified methods do yield an improvement over the basic multinomial resampling approach. A simple counter-example showing that this property does not hold true for systematic resampling is given. Finally, some results on the large-sample behavior of the simple bootstrap filter algorithm are given. In particular, a central limit theorem is established for the case where resampling is performed using the residual approach.
• Dynamic asset pricing theory with uncertain time-horizon.
  
  • Blanchet-Scalliet Christophette
  • El Karoui Nicole
  • Martellini Lionel
  Journal of Economic Dynamics and Control, Elsevier, 2005, 29 (10), pp.1737-1764.
• Inférence dans les HMM hiérarchiques et factorisés: changement de représentation vers le formalisme des réseaux bayésien
  
  • Gelly Sylvain
  • Bredeche Nicolas
  • Sebag Michèle
  , 2005, pp.pp.35-44. This short paper deals with extending HMM with both hierarchical and factorial features. A new inference algorithm is proposed to deal with such a representation that is based on a representation change towards bayesian network representation formalism.
• Recovering Volatility from Option Prices by Evolutionary Optimization
  
  • Hamida Sana Ben
  • Cont Rama
  The Journal of Computational Finance, Incisive Media, 2005. We propose a probabilistic approach for estimating parameters of an option pricing model from a set of observed option prices. Our approach is based on a stochastic optimization algorithm which generates a random sample from the set of global minima of the in-sample pricing error and allows for the existence of multiple global minima. Starting from an IID population of candidate solutions drawn from a prior distribution of the set of model parameters, the population of parameters is updated through cycles of independent random moves followed by "selection" according to pricing performance. We examine conditions under which such an evolving population converges to a sample of calibrated models. The heterogeneity of the obtained sample can then be used to quantify the degree of ill-posedness of the inverse problem: it provides a natural example of a coherent measure of risk, which is compatible with observed prices of benchmark ("vanilla") options and takes into account the model uncertainty resulting from incomplete identification of the model. We describe in detail the algorithm in the case of a diffusion model, where one aims at retrieving the unknown local volatility surface from a finite set of option prices, and illustrate its performance on simulated and empirical data sets of index options. (10.2139/ssrn.546882)
  DOI : 10.2139/ssrn.546882
• Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise
  
  • de Bouard Anne
  • Debussche Arnaud
  The Annals of Probability, Institute of Mathematical Statistics, 2005, 33 (3), pp.1078-1110. We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability (10.1214/009117904000000964)
  DOI : 10.1214/009117904000000964
• Convergence of discontinuous value functions of singularly perturbed control systems with terminal or infimum cost
  
  • Watbled Frédérique
  • Serea Oana
  Differential Equations and Applications, Element, 2005, pp.233--249.
• 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.
• An application of the reciprocity gap functional to inverse scattering theory
  
  • Colton David
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2005, 21 (1), pp.383--398. (10.1088/0266-5611/21/1/023)
  DOI : 10.1088/0266-5611/21/1/023
• On the use of the reciprocity-gap functional in inverse scattering from planar cracks
  
  • Ben Abda Amel
  • Delbary Fabrice
  • Haddar Houssem
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2005, 15 (10), pp.1553-1574. (10.1142/S0218202505000819)
  DOI : 10.1142/S0218202505000819
• 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.