Publications

2013

• Monument Valley brumeuse
  
  • Colonna Jean-François
  , 2013. Foggy Monument Valley (Monument Valley brumeuse)
• Monument Valley nuageuse
  
  • Colonna Jean-François
  , 2013. Cloudy Monument Valley (Monument Valley nuageuse)
• Nuages légers
  
  • Colonna Jean-François
  , 2013. Light clouds (Nuages légers)
• Numerical analysis of the factorization method for EIT with a piecewise constant uncertain background
  
  • Haddar Houssem
  • Migliorati Giovanni
  Inverse Problems, IOP Publishing, 2013, 29 (6), pp.065009. We extend the factorization method for electrical impedance tomography to the case of background featuring uncertainty. We first describe the algorithm for the known but inhomogeneous backgrounds and indicate expected accuracy from the inversion method through some numerical tests. Then we develop three methodologies to apply the factorization method to the more difficult case of a piecewise constant but uncertain background. The first one is based on a recovery of the background through an optimization scheme and is well adapted to relatively low-dimensional random variables describing the background. The second one is based on a weighted combination of the indicator functions provided by the factorization method for different realizations of the random variables describing the uncertain background. We show through numerical experiments that this procedure is well suited to the case where many realizations of the measurement operators are available. The third strategy is a variant of the previous one when measurements for the inclusion-free background are available. In this case, a single pair of measurements is sufficient to achieve comparable accuracy to the deterministic case. (10.1088/0266-5611/29/6/065009)
  DOI : 10.1088/0266-5611/29/6/065009
• Horofunctions and isometries of Hibert geometries
  
  • Walsh Cormac
  , 2013.
• Hybrid systems with constraints
  
  • Daafouz Jamal
  • Tarbouriech Sophie
  • Sigalotti Mario
  , 2013, pp.263. Control theory is the main subject of this title, in particular analysis and control design for hybrid dynamic systems. The notion of hybrid systems offers a strong theoretical and unified framework to cope with the modeling, analysis and control design of systems where both continuous and discrete dynamics interact. The theory of hybrid systems has been the subject of intensive research over the last decade and a large number of diverse and challenging problems have been investigated. Nevertheless, many important mathematical problems remain open. This book is dedicated mainly to hybrid systems with constraints; taking constraints into account in a dynamic system description has always been a critical issue in control. New tools are provided here for stability analysis and control design for hybrid systems with operating constraints and performance specifications. (10.1002/9781118639856)
  DOI : 10.1002/9781118639856
• Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations
  
  • Cacace Simone
  • Chambolle Antonin
  • Desimone Antonio
  • Fedeli Livio
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2013, 47 (3), pp.837-858. We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness. (10.1051/m2an/2012048)
  DOI : 10.1051/m2an/2012048
• L'île des morts -un hommage à Arnold Böcklin
  
  • Colonna Jean-François
  , 2013. Island of the Dead -a Tribute to Arnold Böcklin- (L'île des morts -un hommage à Arnold Böcklin-)
• Entrelacs
  
  • Colonna Jean-François
  , 2013. Intertwining (Entrelacs)
• An Explicit Martingale Version of Brenier's Theorem
  
  • Henry-Labordere Pierre
  • Touzi Nizar
  , 2013. By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry-LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.
• A numerical algorithm for a class of BSDE via branching process
  
  • Henry-Labordere Pierre
  • Tan Xiaolu
  • Touzi Nizar
  , 2013. We generalize the algorithm for semi-linear parabolic PDEs in Henry-Labordére \cite{Henry-Labordere_branching} to the non-Markovian case for a class of Backward SDEs (BSDEs). By simulating the branching process, the algorithm does not need any backward regression. To prove that the numerical algorithm converges to the solution of BSDEs, we use the notion of viscosity solution of path dependent PDEs introduced by Ekren, Keller, Touzi and Zhang \cite{EkrenKellerTouziZhang} and extended in Ekren, Touzi and Zhang \cite{EkrenTouziZhang1, EkrenTouziZhang2}.
• Un arbre fractal-fractal
  
  • Colonna Jean-François
  , 2013. A fractal-fractal tree (Un arbre fractal-fractal)
• Un hypercube fractal
  
  • Colonna Jean-François
  , 2013. A fractal hypercube (Un hypercube fractal)
• Entrelacs
  
  • Colonna Jean-François
  , 2013. Intertwining (Entrelacs)
• Notre-Dame de la Garde fractale, Marseille
  
  • Colonna Jean-François
  , 2013. Fractal Notre-Dame de la Garde, Marseilles (Notre-Dame de la Garde fractale, Marseille)
• Notre-Dame de la Garde fractale, Marseille
  
  • Colonna Jean-François
  , 2013. Fractal Notre-Dame de la Garde, Marseilles (Notre-Dame de la Garde fractale, Marseille)
• The horofunction boundary of Teichmuller space
  
  • Walsh Cormac
  , 2013.
• An Explicit Martingale Version of Brenier's Theorem
  
  • Henry-Labordere Pierre
  • Touzi Nizar
  , 2013. By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.
• A note on solutions to controlled martingale problems and their conditioning
  
  • Claisse Julien
  • Talay Denis
  • Tan Xiaolu
  , 2013. In this note, we rigorously justify a conditioning argument which is often (explicitly or implicitly) used to prove the dynamic programming principle in the stochastic control literature. To this end, we set up controlled martingale problems in an unusual way.
• Maximum Maximum of Martingales given Marginals
  
  • Henry-Labordere Pierre
  • Obloj Jan
  • Spoida Peter
  • Touzi Nizar
  , 2013. We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We present a general duality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labordére and Touzi \cite{ght}, we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Azéma-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek \cite{hobson-klimmek} (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers \cite{brownhobsonrogers}. The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob\lój and Spoida \cite{OblSp}.
• Enhanced Controllability of Low Reynolds Number Swimmers in the Presence of a Wall
  
  • Alouges François
  • Giraldi Laetitia
  Acta Applicandae Mathematicae, Springer Verlag, 2013, 128 (1), pp.153-179. (10.1007/s10440-013-9824-5)
  DOI : 10.1007/s10440-013-9824-5
• Polynomial approximation by means of the random discrete L2 projection and application to inverse problems for PDEs with stochastic data
  
  • Migliorati Giovanni
  , 2013. The main topic of this thesis concerns the polynomial approximation of aleatory functions by means of the random discrete L2 projection, and its application to inverse problems for Partial Differential Equations (PDEs) with stochastic data. The motivations come from the parametric approximation of the solution to partial differential models. The thesis is arranged in two parts, with an introductory chapter which contains an overview of modern techniques for polynomial approximation of functions depending on random variables. In the former part, from Chapter 1 to Chapter 4, the focus is on the theoretical analysis of the random discrete L2 projection applied to solve the so-called forward problem, e.g. to approximate the moments of an aleatory function given its observations, or to compute the solution to a computational model with stochastic coefficients given initial and boundary data. The stability and optimality of the approximation error evaluated in the L2 weighted norm are addressed. In the latter part of the thesis, composed of Chapter 5 and Chapter 6, the methodology previously developed for the forward problem is applied to inverse problems for PDEs with stochastic coefficients. The factorization method is applied in the framework of Electrical Impedance Tomography, first in the case of inhomogeneous background, and then in the case of piecewise constant background, with values in each region affected by uncertainty. Finally, in Chapter 6 the variants of the Factorization Method proposed in the previous chapter are accelerated exploiting the techniques that have been presented in the first part of the thesis.
• Un arbre fractal-fractal
  
  • Colonna Jean-François
  , 2013. A fractal-fractal tree (Un arbre fractal-fractal)
• Random cascade model in the limit of infinite integral scale as the exponential of a nonstationary 1/f noise: Application to volatility fluctuations in stock markets
  
  • Muzy Jean-François
  • Baile Rachel
  • Bacry Emmanuel
  Physical Review E : Statistical, Nonlinear, and Soft Matter Physics [2001-2015], American Physical Society, 2013, 87, pp.042813. no abstract (10.1103/PhysRevE.87.042813)
  DOI : 10.1103/PhysRevE.87.042813
• Un pavage de Penrose distordu tridimensionnellement -à l'aide d'une champ de côte fractal bidimensionnel-, pseudo-périodique du Décagone d'Or
  
  • Colonna Jean-François
  , 2013. A tridimensional distorded -by means of a fractal bidimensional height field- pseudo-periodical Penrose tiling of the Golden Decagon (Un pavage de Penrose distordu tridimensionnellement -à l'aide d'une champ de côte fractal bidimensionnel-, pseudo-périodique du Décagone d'Or)