Publications

2014

• L'éponge de Menger -itération 2
  
  • Colonna Jean-François
  , 2014. The Menger sponge -iteration 2- (L'éponge de Menger -itération 2-)
• Une coupe très intéressante dans l'éponge de Menger -itération 1
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the Menger sponge -iteration 1- (Une coupe très intéressante dans l'éponge de Menger -itération 1-)
• L'éponge de Menger -itération 3
  
  • Colonna Jean-François
  , 2014. The Menger sponge -iteration 3- (L'éponge de Menger -itération 3-)
• Une coupe très intéressante dans l'éponge de Menger -itération 2
  
  • Colonna Jean-François
  , 2014. An interesting cross-section inside the Menger sponge -iteration 2- (Une coupe très intéressante dans l'éponge de Menger -itération 2-)
• Optimal transportation under controlled stochastic dynamics
  
  • Tan Xiaolu
  • Touzi Nizar
  The Annals of Probability, Institute of Mathematical Statistics, 2014, 41 (5), pp.3201-3240. We consider an extension of the Monge-Kantorovitch optimal transportation problem. The mass is transported along a continuous semimartingale, and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale. The optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions. Our first main result is an extension of the Kantorovitch duality to this context. We also suggest a finite-difference scheme combined with the gradient projection algorithm to approximate the dual value. We prove the convergence of the scheme, and we derive a rate of convergence. We finally provide an application in the context of financial mathematics, which originally motivated our extension of the Monge-Kantorovitch problem. Namely, we implement our scheme to approximate no-arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity.
• Le tapis de Sierpinski -itération 5
  
  • Colonna Jean-François
  , 2014. The Sierpinski carpet -iteration 5- (Le tapis de Sierpinski -itération 5-)
• Le tapis de Sierpinski -itération 4
  
  • Colonna Jean-François
  , 2014. The Sierpinski carpet -iteration 4- (Le tapis de Sierpinski -itération 4-)
• Le tapis de Sierpinski -itération 3
  
  • Colonna Jean-François
  , 2014. The Sierpinski carpet -iteration 3- (Le tapis de Sierpinski -itération 3-)
• Le tapis de Sierpinski -itération 1
  
  • Colonna Jean-François
  , 2014. The Sierpinski carpet -iteration 1- (Le tapis de Sierpinski -itération 1-)
• Le tapis de Sierpinski -itération 2
  
  • Colonna Jean-François
  , 2014. The Sierpinski carpet -iteration 2- (Le tapis de Sierpinski -itération 2-)
• Krein--Milman's and Choquet's theorems in the max-plus world
  
  • Akian Marianne
  • Poncet Paul
  , 2014.
• Contraction of Riccati flows applied to the convergence analysis of a max-plus curse of dimensionality free method
  
  • Qu Zheng
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (5), pp.2677-2709. Max-plus based methods have been recently explored for solution of first-order Hamilton--Jacobi--Bellman equations by several authors. Among several max-plus numerical methods, McEneaney's curse-of-dimensionality--free method applies to the equations where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. In previous works of McEneaney and Kluberg, the approximation error of the method was shown to be $O(1/(N\tau))$+$O(\sqrt{\tau})$, where $\tau$ is the time discretization step and $N$ is the number of iterations. Here we use a recently established contraction result for the indefinite Riccati flow in Thompson's part metric to show that under different technical assumptions, still covering an important class of problems, the error is only of order $O(e^{-\alpha N\tau})+O(\tau)$ for some $\alpha>0$. This also allows us to obtain improved estimates of the execution time and to tune the precision of the pruning procedure, which in practice is a critical element of the method. (10.1137/130906702)
  DOI : 10.1137/130906702
• On the Development of High Order Realizable Schemes for the Eulerian Simulation of Disperse Phase Flows: A Convex-State Preserving Discontinuous Galerkin Method
  
  • Sabat Macole
  • Larat Adam
  • Vié Aymeric
  • Massot Marc
  The Journal of Computational Multiphase Flows, Multi-Science Publishing, 2014, 6 (3), pp.247-270. In the present work, a high order realizable scheme for the Eulerian simulation of disperse phase flows on unstructured grids is developed and tested. In the Eulerian modeling framework two approaches are studied: the monokinetic (MK) [1] and the Gaussian closures [2, 3]. The former leads to a pressureless gas dynamics system (PGD). It accurately reproduces the physics of such flows at low Stokes number, but is challenging for numerics since the resulting system is weakly hyperbolic. The latter deals with higher Stokes numbers by accounting for particle trajectory crossings (PTC) [4]. Compared to the MK closure, the resulting system of equation is hyper-bolic but has a more complex structure; realizability conditions are satisfied at the continuous level, which imply a precise framework for numerical methods. To achieve the goals of accuracy, robustness and realizability, the Discontinuous Galerkin method (DG) is a promising numerical approach [5, 6, 7, 8]. Based on the recent work of Zhang et al. [6], the DG method used is associated to a convex projection strategy, which respects the realizability constraints without affecting the accuracy. The main contribution of this work is to apply one of the latest developments in the field of numerical methods (DG) to physical models, taking into account the free transport and drag terms of the disperse phase flow, which are the building blocks for the Eulerian modeling based on moments methods. DG results are eventually compared qualitatively and quantitatively to the Lagrangian results and to the reference simulations provided by a second order structured MUSCL/HLL finite volume scheme [9, 3]. Through these comparisons, the DG method is shown to be competitive for the description of such flows. (10.1260/1757-482X.6.3.247)
  DOI : 10.1260/1757-482X.6.3.247
• Optimal control of a semilinear parabolic equation with singular arcs
  
  • Bonnans Joseph Frederic
  Optimization Methods and Software, Taylor & Francis, 2014, 29 (2), pp.964-978. This paper develops a theory of singular arc, and the corresponding second order necessary and sufficient conditions, for the optimal control of a semilinear parabolic equation with scalar control applied on the r.h.s. We obtain in particular an extension of Kelley's condition, and the characterization of a quadratic growth property for a weak norm. (10.1080/10556788.2013.830220)
  DOI : 10.1080/10556788.2013.830220
• Random maps
  
  • Abraham Céline
  • Bettinelli Jérémie
  • Collet Gwendal
  • Kortchemski Igor
  • Garivier Aurélien
  , 2015, 51, pp.133--149. This is a quick survey on some recent works done in the field of random maps. (10.1051/proc/201551008)
  DOI : 10.1051/proc/201551008
• Imaging of anomalous components in unknown background
  
  • Audibert Lorenzo
  • Girard Alexandre
  • Houssem Haddar
  , 2014. We introduce a qualitative method capable of imaging defects in an unknown complex environment using differential measurements. The main difficulty is that the back-ground medium is unknown and too complex to obtain a reliable estimate of the associated Green function. To overcome this difficulty our approach exploits two measurements of the farfield operators, one without defects and one with defects. The analysis of our method relies on the recently introduced Generalized Linear Sampling Methods (GLSM) and the link to the solutions of the interior transmission problems. We give numerical examples related to non destructive testing in concrete-like materials, illustrating the performance of our method.
• Converse Lyapunov--Krasovskii Theorems for Uncertain Time-Delay Systems
  
  • Haidar Ihab
  • Mason Paolo
  • Sigalotti Mario
  , 2014, pp.10096-10100. In this article, we give a collection of converse Lyapunov-Krasovskii theorems for uncertain time-delay systems. We show that the existence of a weakly-degenerate Lyapunov-Krasovskii functional is necessary and sufficient condition for the global exponential stability of the time-delay systems. This is carried out using the switched system transformation approach. (10.3182/20140824-6-ZA-1003.00561)
  DOI : 10.3182/20140824-6-ZA-1003.00561
• Une éponge pyramidale de Menger fractale obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS- fractal
  
  • Colonna Jean-François
  , 2014. A fractal pyramidal Menger sponge computed by means of an 'Iterated Function System' -IFS- (Une éponge pyramidale de Menger fractale obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS- fractal)
• Une éponge pyramidale de Menger fractale obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS- fractal
  
  • Colonna Jean-François
  , 2014. A fractal pyramidal Menger sponge computed by means of an 'Iterated Function System' -IFS- (Une éponge pyramidale de Menger fractale obtenue à l'aide de la méthode des 'Iterated Function Systems' -IFS- fractal)
• Un attracteur de Lorenz fractal
  
  • Colonna Jean-François
  , 2014. A fractal Lorenz attractor (Un attracteur de Lorenz fractal)
• Asymptotic eigenvalue problems
  
  • Akian Marianne
  • Gaubert Stéphane
  • Hook James
  • Marchesini Andrea
  • Tisseur Françoise
  , 2014.
• Tropical bounds for eigenvalues of matrices using Hungarian dual variables
  
  • Akian Marianne
  • Gaubert Stéphane
  • Marchesini Andrea
  , 2014.
• From tropical linear algebra to zero-sum games
  
  • Gaubert Stéphane
  , 2014. Recently, some relations have appeared between tropical algebra, linear programming, Perron-Frobenius theory, and zero-sum games. This talk is devoted to these relations and to their consequences. In particular, we shall make a connection between two well known unsolved questions. The first is the existence of a strongly polynomial pivoting rule in linear programming. Such a rule would allow us to solve a linear program in a number of arithmetic operations bounded only by the number of variables and the number of constraints. The second question is the existence of a polynomial algorithm to solve mean payoff games. In work with Allamigeon, Benchimol, and Joswig, we showed that a positive answer to the first question would yield a positive answer to the second, provided the pivoting rule satisfies certain conditions. This uses the equivalence between mean payoff games and tropical linear programs, established in an earlier work with Akian and Guterman. The proof of these results will give us the opportunity to see at work non-linear Perron-Frobenius theory, as well as a number of basic tropical tools: tropical eigenvalues, extensions or symmetrization of semirings, tropical analogues of Cramer theorem.
• An overview of Viscosity Solutions of Path-Dependent PDEs
  
  • Ren Zhenjie
  • Touzi Nizar
  • Zhang Jianfeng
  , 2014. This paper provides an overview of the recently developed notion of viscosity solutions of path-dependent partial di erential equations. We start by a quick review of the Crandall- Ishii notion of viscosity solutions, so as to motivate the relevance of our de nition in the path-dependent case. We focus on the wellposedness theory of such equations. In partic- ular, we provide a simple presentation of the current existence and uniqueness arguments in the semilinear case. We also review the stability property of this notion of solutions, in- cluding the adaptation of the Barles-Souganidis monotonic scheme approximation method. Our results rely crucially on the theory of optimal stopping under nonlinear expectation. In the dominated case, we provide a self-contained presentation of all required results. The fully nonlinear case is more involved and is addressed in [12].
• Impact of oxygen starvation on operation and potential gas-phase ignition of passive auto-catalytic recombiners
  
  • Meynet N.
  • Bentaïb A.
  • Giovangigli V.
  Combustion and Flame, Elsevier, 2014, 161 (8), pp.2192-2202. A large amount of hydrogen can be released into the containment of light water reactors during a severe accident. Passive Auto-catalytic Recombiners (PARs) aim to avoid flame acceleration and excessive pressure loads on the containment in case of hydrogen combustion. Their operation is based on the catalytic recombination of hydrogen into steam in the presence of oxygen. Thus, the recombiners reduce the hydrogen but also the oxygen content in the containment atmosphere. As a consequence, the oxygen/nitrogen ratio diverts more and more from the standard 21vol.% in air. This decreasing ratio may impact on the PAR efficiency. Additionally, the exothermic surface chemical mechanism leads to the overheating of the catalytic plates and activates the natural convection inside the recombiners. This heat source can also create local conditions for hydrogen combustion in the gas phase, as igniters do. Hence, the oxygen/nitrogen ratio may also determine the conditions for the gas-phase ignition inside PARs. This study deals with the numerical simulation of the impact of the oxygen starvation (i.e. low oxygen/nitrogen ratio) on the PAR efficiency and on the PAR gas-phase ignition limit. Calculations are performed with a dedicated CFD code named SPARK. We focus on the interaction of recombiners with any H2/O2/N2/H2O mixtures and thus establish a quite complete understanding of PAR operation. Calculations confirm the experimental oxygen surplus (i.e. twice more oxygen than stoichiometry) necessary to ensure an optimal PAR efficiency (XO2≈XH2) independently of the steam content. The PAR gas-phase ignition limit is then determined numerically in the classical H2/Air/H2O ternary diagram with a very good agreement with the available experimental database. It points out the importance of catalyst heat radiation, and more secondarily of species thermal diffusion (i.e. Soret effect). Finally, the PAR gas-phase ignition limit is determined for all oxygen/nitrogen ratios. The ignition domain appears to strongly contract when the oxygen content decreases, so that the steam threshold for inertization of the containment with respect to the recombiners ignition becomes very low. © 2014 The Combustion Institute. (10.1016/j.combustflame.2014.02.001)
  DOI : 10.1016/j.combustflame.2014.02.001