Publications

2014

• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-François
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-François
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-François
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-Francois
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-François
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Les 64 premières lignes du triangle de Pascal
  
  • Colonna Jean-Francois
  , 2014. The 64 first lines of the Pascal's Triangle (Les 64 premières lignes du triangle de Pascal)
• Long and winding central paths
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stephane
  • Joswig Michael
  , 2014.
• Liouville type results for local minimizers of the micromagnetic energy
  
  • Alouges François
  • Di Fratta Giovanni
  • Merlet Benoit
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2014, 53 (3-4), pp.525-560. We study local minimizers of the micromagnetic energy in small ferromagnetic 3d convex particles for which we justify the Stoner-Wohlfarth approximation: given a uniformly convex shape $\Omega \subset {\mathbf{R}}^3$, there exist $\delta_c$>0 and $C > 0$ such that for $0 < \delta \leq \delta_c$ any \textit{local} minimizer $\mathbf{m}$ of the micromagnetic energy in the particle $\delta \Omega$ satisfies $\|\nabla \mathbf{m} \|_{L^2} \leqslant C \delta^2$. In the case of ellipsoidal particles, we strengthen this result by proving that, for $\delta$ small enough, \tmtextit{local} minimizers are exactly spatially uniform. This last result extends W.F. Brown's fundamental theorem for fine 3d ferromagnetic particles [Brown (1968), Di Fratta et al. (2011)] which states the same result but only for \textit{global} minimizers. As a by-product of the method that we use, we establish a new Liouville type result for locally minimizing $p$-harmonic maps with values into a closed subset of a Hilbert space. Namely, we establish that in a smooth uniformly convex domain of $\mathbf{R}^d$ any local minimizer of the $p$-Dirichlet energy ($p > 1$, $p \neq d$) is constant. (10.1007/s00526-014-0757-2)
  DOI : 10.1007/s00526-014-0757-2
• Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation
  
  • Ma Jin
  • Ren Zhenjie
  • Touzi Nizar
  • Zhang Jianfeng
  , 2014. This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao \& Liu \cite{GL}, this extends the corresponding results collected in Freidlin \& Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans \& Ishii \cite{EvansIshii} to the path-dependent case, by using backward stochastic differential techniques. Similar to the Markovian case, we obtain a characterization of the action function as the unique bounded solution of a path-dependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.
• Combinatorial simplex algorithms can solve mean payoff games
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stephane
  • Joswig Michael
  , 2014.
• Global weak solutions to magnetic fluid flows with nonlinear Maxwell-Cattaneo heat transfer law
  
  • Aggoune Fatah
  • Hamdache Kamel
  • Hamroun Djamila
  , 2014. We discuss the equations describing the dynamic of the heat transfer in a magnetic fluid flow under the action of an applied magnetic field. Instead of the usual heat transfer equation we use a generalization given by the Maxwell-Cattaneo law which is a system satisfied by the temperature and the heat flux. We prove a global existence of weak solutions to the system having a finite energy.
• Combinatorial Simplex Algorithms Can Solve Mean Payoff Games
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stephane
  • Joswig Michael
  , 2014. A combinatorial simplex algorithm is an instance of the simplex method in which the pivoting depends on combinatorial data only. We show that any algorithm of this kind admits a tropical analogue which can be used to solve mean payoff games. Moreover, any combinatorial simplex algorithm with a strongly polynomial complexity (the existence of such an algorithm is open) would provide in this way a strongly polynomial algorithm solving mean payoff games (all the arithmetic operations being performed on data polynomially bounded in the size of the input, in particular). Mean payoff games are known to be in NP and co-NP; whether they can be solved in polynomial time is an open problem. Our algorithm relies on a tropical implementation of the simplex method over a real closed field of Hahn series. One of the key ingredients is a new scheme for symbolic perturbation which allows us to lift an arbitrary mean payoff game instance into a non-degenerate linear program over Hahn series.
• A semi-Lagrangian scheme for Lp-penalized minimum time problems
  
  • Falcone Maurizio
  • Kalise Dante
  • Kröner Axel
  , 2014. In this paper we consider a semi-Lagrangian scheme for minimum time problems with Lp-penalization. The minimum time function of the penalized control problem can be characterized as the solution of a Hamilton-Jacobi Bellman (HJB) equation. Furthermore, the minimum time converges with respect to the penalization parameter to the minimum time of the non-penalized problem. To solve the control problem we formulate the discrete dynamic programming principle and set up a semi-Lagrangian scheme. Various numerical examples are presented studying the effects of different choices of the penalization parameters.
• Fixed Point Sets of Payment-Free Shapley Operators and Structural Properties of Mean Payoff Games
  
  • Akian Marianne
  • Gaubert Stephane
  • Hochart Antoine
  , 2014.
• The tropical shadow-vertex algorithm solves mean payoff games in polynomial time on average
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stéphane
  , 2014, 8572, pp.12. We introduce an algorithm which solves mean payoff games in polynomial time on average, assuming the distribution of the games satisfies a flip invariance property on the set of actions associated with every state. The algorithm is a tropical analogue of the shadow-vertex simplex algorithm, which solves mean payoff games via linear feasibility problems over the tropical semiring (ℝ∪{−∞},max,+). The key ingredient in our approach is that the shadow-vertex pivoting rule can be transferred to tropical polyhedra, and that its computation reduces to optimal assignment problems through Plücker relations. (10.1007/978-3-662-43948-7_8)
  DOI : 10.1007/978-3-662-43948-7_8
• Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming.
  
  • Kalise Dante
  • Kröner Axel
  , 2014, pp.1196-1202. A numerical approach for a time-optimal feedback control problem for an advection-reaction-diffusion model is considered. Our approach is composed by three main building blocks: approximation of the abstract system dynamics, feedback computation based on dynamic programming and state observation. For the approximation of the abstract system dynamics, we consider a finite element semi-discretization in space, leading to a large-scale dynamical system, whose dimension is reduced by means of a Balanced Truncation algorithm. Next, we apply the dynamic programming principle over the reduced dynamics, and characterize the value function of the optimal control problem in terms of viscosity solutions of the resulting Hamilton-Jacobi-Bellman equation, which is solved via a semi-Lagrangian scheme. Finally, the computation of corresponding feedback controls and its insertion into the control loop is performed by implementing a Luenberger observer.
• Bundle-based pruning in the max-plus curse of dimensionality free method
  
  • Gaubert Stéphane
  • Qu Zheng
  • Sridharan Srinivas
  , 2014. Recently a new class of techniques termed the max-plus curse of dimensionality-free methods have been de- veloped to solve nonlinear optimal control problems. In these methods the discretization in state space is avoided by using a max-plus basis expansion of the value function. This requires storing only the coefficients of the basis functions used for representation. However, the number of basis functions grows exponentially with respect to the number of time steps of propagation to the time horizon of the control problem. This so called “curse of complexity” can be managed by applying a pruning procedure which selects the subset of basis functions that contribute most to the approximation of the value function. The pruning procedures described thus far in the literature rely on the solution of a sequence of high dimensional optimization problems which can become computationally expensive. In this paper we show that if the max-plus basis functions are linear and the region of interest in state space is convex, the pruning problem can be efficiently solved by the bundle method. This approach combining the bundle method and semidefinite formulations is applied to the quantum gate synthesis problem, in which the state space is the special unitary group (which is non-convex). This is based on the observation that the convexification of the unitary group leads to an exact relaxation. The results are studied and validated via examples.
• The idempotent Radon--Nikodym theorem has a converse statement
  
  • Poncet Paul
  Information Sciences, Elsevier, 2014 (271), pp.115–124. Idempotent integration is an analogue of the Lebesgue integration where $\sigma$-additive measures are replaced by $\sigma$-maxitive measures. It has proved useful in many areas of mathematics such as fuzzy set theory, optimization, idempotent analysis, large deviation theory, or extreme value theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial in all of these applications, was proved by Sugeno and Murofushi. Here we show a converse statement to this idempotent version of the Radon--Nikodym theorem, i.e. we characterize the $\sigma$-maxitive measures that have the Radon--Nikodym property. (10.1016/j.ins.2014.02.074)
  DOI : 10.1016/j.ins.2014.02.074
• Validity of some asymptotic models for eddy current inspection of highly conducting thin deposits
  
  • Haddar Houssem
  • Jiang Zixian
  , 2014. Highly conducting thin deposits may blind eddy current probes in non-destructive testing of steam generator tubes and thus should be identified. In this report, various asymptotic models are studied to model the axisymmetric thin conducting layers by effective transmission conditions depending on re-scaling parameter and asymptotic expansion order, so as to avoid the high computational cost in a full model due to those thin layers. We also select the most adapted models for practical configurations via numerical comparisons in a simplified case.
• 3D direct and inverse solvers for eddy current testing of deposits in steam generator
  
  • Haddar Houssem
  • Riahi Mohamed Kamel
  , 2014. We consider the inverse problem of estimating the shape profile of an unknown deposit from a set of eddy current impedance measurements. The measurements are acquired with an axial probe, which is modeled by a set of coils that generate a magnetic field inside the tube. For the direct problem, we validate the method that takes into account the tube support plates, highly conductive part, by a surface impedance condition. For the inverse problem, finite element and shape sensitivity analysis related to the eddy current problem are provided in order to determine the explicit formula of the gradient of a least square misfit functional. A geometrical-parametric shape inversion algorithm based on cylindrical coordinates is designed to improve the robustness and the quality of the reconstruction. Several numerical results are given in the experimental part. Numerical experiments on synthetic deposits, nearby or far away from the tube, with different shapes are considered in the axisymmetric configuration.
• Second-order BSDEs with general reflection and game options under uncertainty
  
  • Matoussi Anis
  • Piozin Lambert
  • Possamaï Dylan
  Stochastic Processes and their Applications, Elsevier, 2014, 124 (7), pp.2281-2321. The aim of this paper is twofold. First, we extend the results of Matoussi et al. (2013) concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in Crépey and Matoussi (2008), and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanić and Karatzas (1996). More precisely, we show under a technical assumption that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty. (10.1016/j.spa.2014.02.011)
  DOI : 10.1016/j.spa.2014.02.011
• Curve cuspless reconstruction via sub-Riemannian geometry
  
  • Boscain Ugo
  • Duits Remco
  • Rossi Francesco
  • Sachkov Yuri
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (3), pp.748-770. We consider the problem of minimizing for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem. (10.1051/cocv/2013082)
  DOI : 10.1051/cocv/2013082
• Monument Valley au lever du Soleil
  
  • Colonna Jean-Francois
  , 2014. Monument Valley at sunrise (Monument Valley au lever du Soleil)
• Vue artistique de Monument Valley au lever du Soleil avec la Grande Ourse
  
  • Colonna Jean-Francois
  , 2014. Artistic view of Monument Valley at sunrise with the Great Bear (Vue artistique de Monument Valley au lever du Soleil avec la Grande Ourse)
• Monument Valley au lever du Soleil
  
  • Colonna Jean-Francois
  , 2014. Monument Valley at sunrise (Monument Valley au lever du Soleil)