Publications

2013

• L'optimisation du déploiement des réseaux optiques. Considérations sur l'incertitude de la demande.
  
  • Hervet Cédric
  , 2013. L'augmentation des besoins en bande passante dans les réseaux de télécommunications pousse les opérateurs à déployer de nouvelles infrastructures. Pour le réseau d'accès fixe, la fibre optique est la technologie envisagée. Du fait des enjeux financiers et de la complexité qui vont de pair avec ce déploiement, il est crucial d'optimiser son coût tout en respectant à la fois les attentes en qualité de service et les règles d'ingénierie du déploiement. Cette thèse fait suite à des travaux antérieurs, à l'issue desquels le problème avait été modélisé sous la forme d'un programme linéaire en nombres entiers. Un travail conséquent quant à l'amélioration de la résolution de ce problème avait été fourni, et de nombreuses pistes de recherches avaient été envisagées pour faire suite à ces travaux. Parmi ces pistes, il y avait le traitement de l'incertitude sur la demande qui occupe une grande partie de cette étude. En effet, les futurs clients ne s'étant pas encore déclarés, il n'est plus possible de dimensionner le réseau par rapport à des données connues et fixées. Dans ce cas, le problème devient un problème d'optimisation combinatoire dans l'incertain. Le choix a été fait de le traiter sous l'angle de l'optimisation robuste. Cette approche permet de se prémunir contre l'incertitude en garantissant la faisabilité des solutions dans tous les cas ainsi qu'une optimisation du " pire cas ". Le formalisme qui en découle rend souvent les problèmes étudiés complexes à résoudre. En effet, ils font intervenir des formulations à plusieurs niveaux où les décisions sont prises en séquence, avant ou après la réalisation du scénario incertain. Des algorithmes adaptés ont été développés pour permettre l'application de la robustesse au déploiement des réseaux de fibres optiques. Ces algorithmes, exacts ou approchés, ont permis, via leurs résultats, d'obtenir une connaissance stratégique réelle pour les déploiements à venir. A la suite de ces investigations sur le problème du déploiement optique, certains résultats ont pu être étendus et généralisés à d'autres problèmes d'optimisation robuste, comme par exemple des bornes de probabilité sur la pertinence des ensembles d'incertitudes ou une estimation probabiliste des coûts futurs dans les problèmes d'optimisation robuste en deux étapes. En marge de ces travaux sur l'incertitude qui occupent la plus grande partie de cette étude, d'autres travaux ont été réalisés sur ce problème. En effet, dans le but d'améliorer la prise en compte des coûts futurs du réseau (maintenance, gestion, etc.) qui sont, sur le long terme, les plus importants, une approche a été développée qui permet de prendre en compte les " bonnes pratiques " de déploiement directement dans l'optimisation. L'intégration de ces considérations, regroupées sous le terme d'OA&M (pour Organisation, Administration et Maintenance), a été validée par le développement de macro-modèles de coûts, à même d'estimer les gains futurs à attendre de ces nouvelles contraintes. Enfin, nos efforts ont porté sur la résolution d'une version particulière du problème, dans des graphes qui sont des arbres, avec la prise en compte des contraintes de câblage dans l'optimisation. Pour ce problème qui avait déjà été étudié, un nouvel algorithme de programmation dynamique a été proposé. Il s'appuie fortement sur les propriétés du problème et les utilise pour n'explorer qu'un nombre très limité de solutions tout en restant exact. Les performances de l'algorithme ont montré une nette amélioration du temps de calcul par rapport à des approches de type programmation linéaire en nombres entiers. L'ensemble de ces travaux a permis de découvrir d'autres pistes de recherche, notamment sur des versions alternatives du traitement de l'incertitude, ainsi que sur une prise en compte plus fine du câblage dans l'optimisation.
• Design of Optimal Experiments for Parameter Estimation of Microalgae Growth Models
  
  • Munoz Tamayo Rafael
  • Martinon Pierre
  • Bougaran Gaël
  • Mairet Francis
  • Bernard Olivier
  , 2013. Mathematical models are expected to play a pivotal role for driving microalgal production towards a profitable process of renewable energy generation. To render models of microalgae growth useful tools for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. In this paper, we provide guidelines for the design of experiments with high informative content that allows an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was solved as an optimal control problem. E-optimal experiments were obtained by using two discretization approaches namely sequential and simultaneous. The results showed that an adequate parameterization of the experimental inputs provided optimal solutions very close to those provided by the simultaneous discretization. Simulation results showed the relevance of determining optimal experimental inputs for achieving an accurate parameter estimation.
• A geometrical approach for the resolution of an optimal control problem
  
  • Bouhafs Walid
  • Abdellatif Nahla
  • Jean Frédéric
  • Harmand Jérôme
  , 2013.
• Hamilton-Jacobi-Bellman approach for optimal control problems with discontinuous coefficients
  
  • Rao Zhiping
  , 2013. This thesis deals with the Dynamical Programming and Hamilton-Jacobi-Bellman approach for a general class of deterministic optimal control problems with discontinuous coefficients. The tools essentially used in this work are based on the control theory, the viscosity theory for Partial Differential Equations, the nonsmooth analysis and the dynamical systems. The first part of the thesis is concerned with the state constrained problem of discontinuous trajectories driven by impulsive dynamical systems. A characterization result of the value function of this problem has been obtained. Another contribution of this part consists of the extension of the HJB approach for the problems with time-measurable dynamical systems and in presence of time-dependent state constraints. The second part is devoted to the problem on stratified domain, which consists of a union of subdomains separated by several interfaces. One of the motivations of this work comes from the hybrid control problems. Here new transmission conditions on the interfaces have been obtained to ensure the uniqueness and the characterization of the value function. The third part investigates the homogenization of Hamilton-Jacobi equations in the framework of state-discontinuous Hamiltonians. This work considers the singular perturbation of optimal control problem on a periodic stratified structure. The limit problem has been analyzed and the associated Hamilton-Jacobi equation has been established. This equation describes the limit behavior of the value function of the perturbed problem when the scale of periodicity tends to 0.
• Geometric modeling of the movement based on an inverse optimal control approach
  
  • Jean Frédéric
  • Mason Paolo
  • Chittaro Francesca
  , 2013, pp.1816-1821. The present paper analyses a class of optimal control problems on geometric paths of the euclidean space, that is, curves parametrized by arc length. In the first part we deal with existence and robustness issues for such problems and we define the associated inverse optimal control problem. In the second part we discuss the inverse optimal control problem in the special case of planar trajectories and under additional assumptions. More precisely we define a criterion to restrict the study to a convenient class of costs based on the analysis of experimentally recorded trajectories. This method applies in particular to the case of human locomotion trajectories. (10.1109/cdc.2013.6760146)
  DOI : 10.1109/cdc.2013.6760146
• A marvelous contribution from Michel Hénon to globular cluster’s study : the isochrone cluster
  
  • Perez Jérôme
  , 2016. A survey of Michel Henon contributions to the study of globular cluster systems
• On the inverse optimal control problems of the human locomotion: stability and robustness of the minimizers
  
  • Chittaro Francesca
  • Jean Frédéric
  • Mason Paolo
  Journal of Mathematical Sciences, Springer Verlag (Germany), 2013, 195 (3), pp.269-287. In recent papers models of the human locomotion by means of an optimal control problem have been proposed. In this paradigm, the trajectories are assumed to be solutions of an optimal control problem whose cost has to be determined. The purpose of the present paper is to analyze the class of optimal control problems defined in this way. We prove strong convergence result for their solutions on the one hand for perturbations of the initial and final points (stability), and on the other hand for perturbations of the cost (robustness). (10.1007/s10958-013-1579-z)
  DOI : 10.1007/s10958-013-1579-z
• Comparison of Numerical Methods in the Contrast Imaging Problem in NMR
  
  • Bonnard Bernard
  • Claeys Mathieu
  • Cots Olivier
  • Martinon Pierre
  , 2013. In this article, the contrast imaging problem in nuclear magnetic resonance is modeled as a Mayer problem in optimal control. A first synthesis of locally optimal solutions is given in the single-input case using geometric methods based on Pontryagin's maximum principle. We then compare these results using direct methods and a moment-based approach, and make a first step towards global optimality. Finally, some preliminary results are given in the bi-input case.
• Time domain simulation of a piano. Part 1 : model description.
  
  • Chabassier Juliette
  • Chaigne Antoine
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2013, 48 (05), pp.1241-1278. The purpose of this study is the time domain modeling of a piano. We aim at explaining the vibratory and acoustical behavior of the piano, by taking into account the main elements that contribute to sound production. The soundboard is modeled as a bidimensional thick, orthotropic, heterogeneous, frequency dependent damped plate, using Reissner Mindlin equations. The vibroacoustics equations allow the soundboard to radiate into the surrounding air, in which we wish to compute the complete acoustical field around the perfectly rigid rim. The soundboard is also coupled to the strings at the bridge, where they form a slight angle from the horizontal plane. Each string is modeled by a one dimensional damped system of equations, taking into account not only the transversal waves excited by the hammer, but also the stiffness thanks to shear waves, as well as the longitudinal waves arising from geometric nonlinearities. The hammer is given an initial velocity that projects it towards a choir of strings, before being repelled. The interacting force is a nonlinear function of the hammer compression. The final piano model is a coupled system of partial differential equations, each of them exhibiting specific difficulties (nonlinear nature of the string system of equations, frequency dependent damping of the soundboard, great number of unknowns required for the acoustic propagation), in addition to couplings' inherent difficulties. (10.1051/m2an/2013136)
  DOI : 10.1051/m2an/2013136
• Contrôle optimal d'équations différentielles avec - ou sans - mémoire
  
  • Dupuis Xavier
  , 2013. La thèse porte sur des problèmes de contrôle optimal où la dynamique est donnée par des équations différentielles avec mémoire. Pour ces problèmes d'optimisation, des conditions d'optimalité sont établies ; celles du second ordre constituent une part importante des résultats de la thèse. Dans le cas - sans mémoire - des équations différentielles ordinaires, les conditions d'optimalité standards sont renforcées en ne faisant intervenir que les multiplicateurs de Lagrange pour lesquels le principe de Pontryaguine est satisfait. Cette restriction à un sous-ensemble des multiplicateurs représente un défi dans l'établissement des conditions nécessaires et permet aux conditions suffisantes d'assurer l'optimalité locale dans un sens plus fort. Les conditions standards sont d'autre part étendues au cas - avec mémoire - des équations intégrales. Les contraintes pures sur l'état du problème précédent ont été conservées et nécessitent une étude spécifique à la dynamique intégrale. Une autre forme de mémoire dans l'équation d'état d'un problème de contrôle optimal provient d'un travail de modélisation avec l'optimisation thérapeutique comme application médicale en vue. La dynamique de populations de cellules cancéreuses sous l'action d'un traitement est ramenée à des équations différentielles à retards ; le comportement asymptotique en temps long du modèle structuré en âge est également étudié.
• Sensitivity analysis for optimal control problems. Stochastic optimal control with a probability constraint
  
  • Pfeiffer Laurent
  , 2013. This thesis is divided into two parts. In the first part, we study constrained deterministic optimal control problems and sensitivity analysis issues, from the point of view of abstract optimization. Second-order necessary and sufficient optimality conditions, which play an important role in sensitivity analysis, are also investigated. In this thesis, we are interested in strong solutions. We use this generic term for locally optimal controls for the $L^1$-norm, roughly speaking. We use two essential tools: a relaxation technique, which consists in using simultaneously several controls, and a decomposition principle, which is a particular second-order Taylor expansion of the Lagrangian. Chapters 2 and 3 deal with second-order necessary and sufficient optimality conditions for strong solutions of problems with pure, mixed, and final-state constraints. In Chapter 4, we perform a sensitivity analysis for strong solutions of relaxed problems with final-state constraints. In Chapter 5, we perform a sensitivity analysis for a problem of nuclear energy production. In the second part of the thesis, we study stochastic optimal control problems with a probability constraint. We study an approach by dynamic programming, in which the level of probability is a supplementary state variable. In this framework, we show that the sensitivity of the value function with respect to the probability level is constant along optimal trajectories. We use this analysis to design numerical schemes for continuous-time problems. These results are presented in Chapter 6, in which we also study an application to asset-liability management.
• Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian
  
  • Graber Philip Jameson
  , 2013. We consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed. Keywords: Hamilton-Jacobi equations, optimal control, nonlinear PDE, viscosity solutions, front propagation, mean field games
• Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions
  
  • Graber Philip Jameson
  • Lasiecka Irena
  Semigroup Forum, Springer Verlag, 2013. We consider a linear system of PDEs of the form \begin{equation} \begin{array}{c} \begin{array}{rcl} u_{tt} - c\Delta u_t - \Delta u = 0 & \text{in} & \Omega \times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\Gamma (c \alpha u_t + u) = 0 & \text{on} & \Gamma_1 \times (0,T)\\ u = 0 & \text{on} & \Gamma_0 \times (0,T) \end{array}\\ (u(0),u_t(0),u|_{\Gamma_1}(0),u_t|_{\Gamma_1}(0)) \in \s{H} \end{array} \end{equation} on a bounded domain $\Omega$ with boundary $\Gamma = \Gamma_1 \cup \Gamma_0$. We show that the system generates a strongly continuous semigroup $T(t)$ which is analytic for $\alpha > 0$ and of Gevrey class for $\alpha = 0$. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form $\|(d/dt)T(t)\| \lesssim |t|^{-1}$ for $\alpha > 0$, $\|(d/dt)T(t)\| \lesssim |t|^{-2}$ for $\alpha = 0$. Moreover, when $\alpha = 0$ we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to $1/2$ power of the generator. (10.1007/s00233-013-9534-3)
  DOI : 10.1007/s00233-013-9534-3
• Enhanced transmission through gratings: Structural and geometrical effects
  
  • Maurel Agnès
  • Félix Simon
  • Mercier Jean-François
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2013, 88, pp.115416. Homogenization theory is used to derive the effective properties of gratings with complex subwavelength structures. Going beyond the effect of the filling fraction, geometrical effects are analyzed using a two-step homogenization process. An explicit expression for the transmission spectrum is derived, able to predict the Fabry-Perot resonances and the Brewster angle realizing broadband extraordinary transmission. With the same filling fraction, one expects from this analytical expression that gratings with different geometries may display very different transmission properties. This sensitivity to the microstructure geometry is exemplified in the case of gratings made of hard material and made of dielectric material. The analytical results are shown to be within a few percentage points as compared to full-wave numerical simulations, paving the way for transmission properties tuned by structural and geometrical manipulations. (10.1103/PhysRevB.88.115416)
  DOI : 10.1103/PhysRevB.88.115416
• Lipschitz stability estimate in the inverse Robin problem for the Stokes system
  
  • Egloffe Anne-Claire
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2013. We are interested in the inverse problem of recovering a Robin coefficient defined on some non accessible part of the boundary from available data on another part of the boundary in the nonstationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.
• Calcul des singularités dans les méthodes d’équations intégrales variationnelles
  
  • Salles Nicolas
  , 2013. La mise en œuvre de la méthode des éléments finis de frontière nécessite l'évaluation d'intégrales comportant un intégrand singulier. Un calcul fiable et précis de ces intégrales peut dans certains cas se révéler à la fois crucial et difficile. La méthode que nous proposons consiste en une réduction récursive de la dimension du domaine d'intégration et aboutit à une représentation de l'intégrale sous la forme d'une combinaison linéaire d'intégrales mono-dimensionnelles dont l'intégrand est régulier et qui peuvent s'évaluer numériquement mais aussi explicitement. L'équation de Helmholtz 3-D sert d'équation modèle mais ces résultats peuvent être utilisés pour les équations de Laplace et de Maxwell 3-D. L'intégrand est décomposé en une partie homogène et une partie régulière ; cette dernière peut être traitée par les méthodes usuelles d'intégration numérique. Pour la discrétisation du domaine, des triangles plans sont utilisés ; par conséquent, nous évaluons des intégrales sur le produit de deux triangles. La technique que nous avons développée nécessite de distinguer entre diverses configurations géométriques ; c'est pourquoi nous traitons séparément le cas de triangles coplanaires, dans des plans sécants ou parallèles. Divers prolongements significatifs de la méthode sont présentés : son extension à l'électromagnétisme, l'évaluation de l'intégrale du noyau de Green complet pour les coefficients d'auto-influence, et le calcul de la partie finie d'intégrales hypersingulières.
• Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters
  
  • Desilles Anna
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2013, 8 (3), pp.707-726. A few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems are presented. In the considered problem the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semi-explicit form for an homogeneous lagrangian do not hold, a capture basin algorithm is proposed to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications to traffic regulation problems. (10.3934/nhm.2013.8.707)
  DOI : 10.3934/nhm.2013.8.707
• 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
• Physique MPSI-PCSI-PTSI
  
  • Perez Jérôme
  • Vincent Renvoize
  • Bellanger Eric
  • Saudrais Eddie
  • Roy Michel
  • Ducros Xavier
  , 2013. Véritable ouvrage de référence pour la préparation aux concours, ce manuel de physique est conforme au contenu et à l'esprit du nouveau programme 2013. Cette 2e édition est parfaitement adaptée au niveau des élèves et à leurs besoins : le livre, qui met l’accent sur les concepts essentiels, est progressif et comporte de nombreux rappels et des figures de qualité. Le cours s’attache dès le départ à faire ressortir les raisons d’être et le sens des notions introduites. Il est enrichi de quelques notions d’histoire de la physique.
• On countably skewed Brownian motion with accumulation point.
  
  • Ouknine Youssef
  • Russo Francesco
  • Trutnau Gerald
  , 2013. In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.
• 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
• 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.
• 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.
• Modeling and simulation of a grand piano
  
  • Chabassier Juliette
  • Joly Patrick
  • Chaigne Antoine
  Journal of the Acoustical Society of America, Acoustical Society of America, 2013, 134, pp.648. A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness and geometrical nonlinea- rity. The hammer-string interaction is governed by a nonlinear dissi- pative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers (PML). The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium and high regis- ter, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized. (10.1121/1.4809649)
  DOI : 10.1121/1.4809649
• 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