Publications

2013

• A fast random walk algorithm for computing diffusion-weighted NMR signals in multi-scale porous media: A feasibility study for a Menger sponge
  
  • Grebenkov Denis S
  • Nguyen Hang Tuan
  • Li Jing-Rebecca
  Microporous and Mesoporous Materials, Elsevier, 2013. A fast random walk (FRW) algorithm is adapted to compute diffusion-weighted NMR signals in a Menger sponge which is formed by multiple channels of broadly distributed sizes and often considered as a model for soils and porous materials. The self-similar structure of a Menger sponge allows for rapid simulations that were not feasible by other numerical techniques. The role of multiple length scales on diffusion-weighted NMR signals is investigated. (10.1016/j.micromeso.2013.02.040)
  DOI : 10.1016/j.micromeso.2013.02.040
• Aléatoire
  
  • Chafaï Djalil
  • Giraud Christophe
  • Méléard Sylvie
  , 2013, pp.130. Les textes réunis dans ce volume présentent plusieurs aspects des mathématiques de l'aléatoire et mettent en évidence les nombreux domaines d'applications où elles sont utilisées. Sylvie Méléard s'appuie sur la modélisation en dynamique des populations pour introduire les processus markoviens de saut. Elle met en évidence, par l'étude des approximations en grande population du processus de naissance et mort logistique, un modèle d'équation différentielle ordinaire et un modèle d'équation différentielle stochastique, suivant les échelles de temps et de taille de population considérées. Christophe Giraud propose une introduction aux fondements mathématiques de la classification automatique, théorie qui intervient tout autant dans le filtrage de pourriels de messagerie que dans la recherche automatique de molécules actives en médecine. Son texte présente diverses techniques mathématiques utilisées lors de l'estimation de la probabilité d'erreur de classification par un algorithme. Djalil Chafaï développe quelques aspects de la théorie des matrices aléatoires, qui est un domaine des mathématiques à l'intersection de la théorie des probabilités et de l'algèbre linéaire. Cette théorie a des applications aussi bien dans les domaines appliqués que dans les parties les plus fondamentales des mathématiques.
• On the motion planning of the ball with a trailer
  
  • Boizot Nicolas
  • Gauthier Jean-Paul
  Mathematical Control and Related Fields, AIMS, 2013, 3 (3), pp.269 - 286. (10.3934/mcrf.2013.3.269)
  DOI : 10.3934/mcrf.2013.3.269
• Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions
  
  • Santacesaria Matteo
  Applicable Analysis, Taylor & Francis, 2013, 92 (8), pp.1666-1681. We study the inverse problem of determining a real-valued potential in the two-dimensional Schrödinger equation at negative energy from the Dirichlet-to-Neumann map. It is known that the problem is ill-posed and a stability estimate of logarithmic type holds. In this paper we prove three new stability estimates. The main feature of the first one is that the stability increases exponentially with respect to the smoothness of the potential, in a sense to be made precise. The others show how the first estimate depends on the energy, for low and high energies (in modulus). In particular it is found that for high energies the stability estimate changes, in some sense, from logarithmic type to Lipschitz type: in this sense the ill-posedness of the problem decreases when increasing the energy (in modulus). (10.1080/00036811.2012.698006)
  DOI : 10.1080/00036811.2012.698006
• Sensitivity analysis for relaxed optimal control problems with final-state constraints
  
  • Bonnans Joseph Frederic
  • Pfeiffer Laurent
  • Serea Oana Silvia
  Nonlinear Analysis: Theory, Methods and Applications, Elsevier, 2013, 89, pp.55-80. In this article, we compute a second-order expansion of the value function of a family of relaxed optimal control problems with final-state constraints, parameterized by a perturbation variable. The sensitivity analysis is performed for controls that we call R-strong solutions. They are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given R and having an associated trajectory in a small neighborhood for the uniform norm. In this framework, relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function. (10.1016/j.na.2013.04.013)
  DOI : 10.1016/j.na.2013.04.013
• Variation artistique sur deux ensembles de Julia/Mandelbrot dans le corps des quaternions -section tridimensionnelle
  
  • Colonna Jean-François
  , 2013. Artistic variation on two quaternionic Julia/Mandelbrot sets -tridimensional cross-section- (Variation artistique sur deux ensembles de Julia/Mandelbrot dans le corps des quaternions -section tridimensionnelle-)
• Strong order of convergence of a semidiscrete scheme for the stochastic Manakov equation
  
  • Gazeau Maxime
  , 2013. It is well accepted by physicists that the Manakov PMD equation is a good model to describe the evolution of nonlinear electric fields in optical fibers with randomly varying birefringence. In the regime of the diffusion approximation theory, an effective asymptotic dynamics has recently been obtained to describe this evolution. This equation is called the stochastic Manakov equation. In this article, we propose a semidiscrete version of a Crank Nicolson scheme for this limit equation and we analyze the strong error. Allowing sufficient regularity of the initial data, we prove that the numerical scheme has strong order 1/2.
• Time-reversal algorithms in viscoelastic media
  
  • Bretin Elie
  • Ammari Habib
  • Garnier Josselin
  • Wahab Abdul
  European Journal of Applied Mathematics, Cambridge University Press (CUP), 2013, 24 (04), pp.565 - 600. (10.1017/S0956792513000107)
  DOI : 10.1017/S0956792513000107
• A Shooting Algorithm for Optimal Control Problems with Singular Arcs
  
  • Aronna Maria Soledad
  • Bonnans J. Frederic
  • Martinon Pierre
  Journal of Optimization Theory and Applications, Springer Verlag, 2013, 158 (2), pp.419-459. In this article we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss-Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system) we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated to the perturbed problem. We present numerical tests that validate our method. (10.1007/s10957-012-0254-8)
  DOI : 10.1007/s10957-012-0254-8
• Universal current fluctuations in the symmetric exclusion process and other diffusive systems
  
  • Akkermans Eric
  • Bodineau Thierry
  • Derrida Bernard
  • Shpielberg Ohad
  EPL - Europhysics Letters, European Physical Society / EDP Sciences / Società Italiana di Fisica / IOP Publishing, 2013, 103 (2). We show, using the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim, that the statistics of the current of the symmetric simple exclusion process (SSEP) connected to two reservoirs are the same on an arbitrary large finite domain in dimension $d$ as in the one dimensional case. Numerical results on squares support this claim while results on cubes exhibit some discrepancy. We argue that the results of the macroscopic fluctuation theory should be recovered by increasing the size of the contacts. The generalization to other diffusive systems is straightforward.
• Lyapunov and minimum-time path planning for drones
  
  • Maillot Thibault
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Serres Ulysse
  , 2013. In this paper, we study the problem of controlling an unmanned aerial vehicle (UAV) to provide a target supervision and/or to provide convoy protection to ground vehicles. We first present a control strategy based upon a Lyapunov-LaSalle stabilization method to provide supervision of a stationary target. The UAV is expected to join a pre-designed admissible circular trajectory around the target which is itself a fixed point in the space. Our strategy is presented for both HALE (High Altitude Long Endurance) and MALE (Medium Altitude Long Endurance) types UAVs. A UAV flying at a constant altitude (HALE type) is modeled as a Dubins vehicle (i.e. a planar vehicle with constrained turning radius and constant forward velocity). For a UAV that might change its altitude (MALE type), we use the general kinematic model of a rigid body evolving in $\R^3$. Both control strategies presented are smooth and unlike what is usually proposed in the literature these strategies asymptotically track a circular trajectory of exact minimum turning radius. We also present the time-optimal control synthesis for tracking a circle by a Dubins vehicle. This optimal strategy, although much simpler than the point-to-point time-optimal strategy obtained by P. Souéres and J.-P. Laumond in the 1990s (see [45]), is very rich. Finally, we propose control strategies to provide supervision of a moving target, that are based upon the previous ones.
• A highly anisotropic nonlinear elasticity model for vesicles. II. Derivation of the thin bilayer bending theory
  
  • Merlet Benoit
  , 2013. We study the thin-shell limit of the nonlinear elasticity model for vesicles introduced in part I. We consider vesicles of width 2eps ↓ 0 with elastic energy of order eps^3. In this regime, we show that the limit model is a bending theory for generalized hypersurfaces -- namely, codimension 1 oriented varifolds without boundary. Up to a positive factor, the limit functional is the Willmore energy. In the language of Gamma -convergence, we establish a compactness result, a lower bound result and the matching upper bound in the smooth case.
• O temps tes pyramides' -un hommage à Jorge Luis Borges
  
  • Colonna Jean-François
  , 2013. O temps tes pyramides' -a tribute to Jorge Luis Borges- ('O temps tes pyramides' -un hommage à Jorge Luis Borges-)
• Singular arcs in the optimal control of a parabolic equation
  
  • Bonnans Joseph Frederic
  , 2013. We present 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.
• Contraction of Riccati Flows Applied to the Convergence Analysis of the Max-Plus Curse of Dimensionality Free Method
  
  • Qu Zheng
  , 2013.
• Markov Operators on Cones and Non-Commutative Consensus
  
  • Gaubert Stéphane
  • Qu Zheng
  , 2013.
• Optimizing the anaerobic digestion of microalgae in a coupled process
  
  • Bayen Térence
  • Mairet Francis
  • Martinon Pierre
  • Sebbah Matthieu
  , 2012, pp.6. This work is devoted to maximizing the production of methane in a bioreactor coupling an anaerobic digester and a culture of micro-algae limited by light. The decision parameter is the dilution rate which is chosen as a control, and we enforce periodic constraints in order to repeat the same operation every day. The system is gathered into a three-dimensional system taking into account a day-night model of the light in the culture of micro-algae. Applying Pontryagin maximum principle, the necessary conditions on optimal trajectories indicate that the control consists of bang and/or singular arcs. We provide numerical simulations by both direct and indirect methods, which show the link between the light model and the structure of optimal solutions.
• On some open questions in bilinear quantum control
  
  • Boscain Ugo
  • Chambrion Thomas
  • Sigalotti Mario
  , 2013, pp.2080-2085. The aim of this paper is to provide a short introduction to modern issues in the control of infinite dimensional closed quantum systems, driven by the bilinear Schrödinger equation. The first part is a quick presentation of some of the numerous recent developments in the fields. This short summary is intended to demonstrate the variety of tools and approaches used by various teams in the last decade. In a second part, we present four examples of bilinear closed quantum systems. These examples were extensively studied and may be used as a convenient and efficient test bench for new conjectures. Finally, we list some open questions, both of theoretical and practical interest.
• Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Magron Victor
  • Werner Benjamin
  , 2013, pp.2244 - 2250. We consider the problem of certifying an inequality of the form $f(x)\geq 0$, $\forall x\in K$, where $f$ is a multivariate transcendental function, and $K$ is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and max-plus approximation. We assume that $f$ is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like $\cos$, $\sin$, $\exp$, etc. We bound some of these constituents by suprema or infima of quadratic forms (max-plus approximation method, initially introduced in optimal control), leading to semialgebraic optimization problems which we solve by semidefinite relaxations. The max-plus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap. Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical inequalities used in the proof of the Kepler conjecture by Thomas Hales).
• Contraction of Riccati flows applied to the convergence analysis of a max-plus curse of dimensionality free method
  
  • Qu Zheng
  , 2013. McEneaney introduced a curse-of-dimensionality free method for solving HJB equations in which the Hamiltonian is a maximum of linear/quadratic forms. The approximation error was shown to be $O(1/(N\tau))$+$O(\sqrt{\tau})$ where $\tau$ is the time discretization size and $N$ is the number of iterations. Here we use a recently established contraction result for the indefinite Riccati flow in Thompson's metric to show that under different technical assumptions, the error is only of $O(e^{-N\tau})+O(\tau)$.
• Certification of Bounds of Non-linear Functions: the Templates Method
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Magron Victor
  • Werner Benjamin
  , 2013, 7961, pp.51-65. The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq. (10.1007/978-3-642-39320-4_4)
  DOI : 10.1007/978-3-642-39320-4_4
• Tropicalizing the simplex algorithm
  
  • Allamigeon Xavier
  , 2013.
• Clavier de piano fractal
  
  • Colonna Jean-François
  , 2013. Fractal piano keyboard (Clavier de piano fractal)
• A model-free no-arbitrage price bound for variance options
  
  • Bonnans J. Frederic
  • Tan Xiaolu
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2013, 68 (1), pp.43-73. In the framework of Galichon, Henry-Labordère and Touzi, we consider the model-free no-arbitrage bound of variance option given the marginal distributions of the underlying asset. We first make some approximations which restrict the computation on a bounded domain. Then we propose a gradient projection algorithm together with a finite difference scheme to approximate the bound. The general convergence result is obtained. We also provide a numerical example on the variance swap option. (10.1007/s00245-013-9197-1)
  DOI : 10.1007/s00245-013-9197-1
• The shooting approach to optimal control problems
  
  • Bonnans Joseph Frederic
  , 2013, pp.281-292. We give an overview of the shooting technique for solving deterministic optimal control problems. This approach allows to reduce locally these problems to a finite dimensional equation. We first recall the basic idea, in the case of unconstrained or control constrained problems, and show the link with second-order optimality conditions and the analysis or discretization errors. Then we focus on two cases that are now better undestood: state constrained problems, and affine control systems. We end by discussing extensions to the optimal control of a parabolic equation. (10.3182/20130703-3-FR-4038.00158)
  DOI : 10.3182/20130703-3-FR-4038.00158