Publications

2016

• Functional representation of deformable surfaces for geometry processing
  
  • Corman Etienne
  , 2016. Creating and understanding deformations of surfaces is a recurring theme in geometry processing. As smooth surfaces can be represented in many ways from point clouds to triangle meshes, one of the challenges is being able to compare or deform consistently discrete shapes independently of their representation. A possible answer is choosing a flexible representation of deformable surfaces that can easily be transported from one structure to another.Toward this goal, the functional map framework proposes to represent maps between surfaces and, to further extents, deformation of surfaces as operators acting on functions. This approach has been recently introduced in geometry processing but has been extensively used in other fields such as differential geometry, operator theory and dynamical systems, to name just a few. The major advantage of such point of view is to deflect challenging problems, such as shape matching and deformation transfer, toward functional analysis whose discretization has been well studied in various cases. This thesis investigates further analysis and novel applications in this framework. Two aspects of the functional representation framework are discussed.First, given two surfaces, we analyze the underlying deformation. One way to do so is by finding correspondences that minimize the global distortion. To complete the analysis we identify the least and most reliable parts of the mapping by a learning procedure. Once spotted, the flaws in the map can be repaired in a smooth way using a consistent representation of tangent vector fields.The second development concerns the reverse problem: given a deformation represented as an operator how to deform a surface accordingly? In a first approach, we analyse a coordinate-free encoding of the intrinsic and extrinsic structure of a surface as functional operator. In this framework a deformed shape can be recovered up to rigid motion by solving a set of convex optimization problems. Second, we consider a linearized version of the previous method enabling us to understand deformation fields as acting on the underlying metric. This allows us to solve challenging problems such as deformation transfer are solved using simple linear systems of equations.
• A probabilistic max-plus numerical method for solving stochastic control problems
  
  • Fodjo Eric
  , 2016.
• Nonlinear Perron-Frobenius theory and mean-payoff zero-sum stochastic games
  
  • Hochart Antoine
  , 2016. Zero-sum stochastic games have a recursive structure encompassed in their dynamic programming operator, so-called Shapley operator. The latter is a useful tool to study the asymptotic behavior of the average payoff per time unit. Particularly, the mean payoff exists and is independent of the initial state as soon as the ergodic equation - a nonlinear eigenvalue equation involving the Shapley operator - has a solution. The solvability of the latter equation in finite dimension is a central question in nonlinear Perron-Frobenius theory, and the main focus of the present thesis. Several known classes of Shapley operators can be characterized by properties based entirely on the order structure or the metric structure of the space. We first extend this characterization to "payment-free" Shapley operators, that is, operators arising from games without stage payments. This is derived from a general minimax formula for functions homogeneous of degree one and nonexpansive with respect to a given weak Minkowski norm. Next, we address the problem of the solvability of the ergodic equation for all additive perturbations of the payment function. This problem extends the notion of ergodicity for finite Markov chains. With bounded payment function, this "ergodicity" property is characterized by the uniqueness, up to the addition by a constant, of the fixed point of a payment-free Shapley operator. We give a combinatorial solution in terms of hypergraphs to this problem, as well as other related problems of fixed-point existence, and we infer complexity results. Then, we use the theory of accretive operators to generalize the hypergraph condition to all Shapley operators, including ones for which the payment function is not bounded. Finally, we consider the problem of uniqueness, up to the addition by a constant, of the nonlinear eigenvector. We first show that uniqueness holds for a generic additive perturbation of the payments. Then, in the framework of perfect information and finite action spaces, we provide an additional geometric description of the perturbations for which uniqueness occurs. As an application, we obtain a perturbation scheme allowing one to solve degenerate instances of stochastic games by policy iteration.
• Solving Generic Nonarchimedean Semidefinite Programs using Stochastic Game Algorithms
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Skomra Mateusz
  , 2016.
• Price incentives in mobile networks: a tropical approach
  
  • Akian Marianne
  • Bouhtou Mustapha
  • Eytard Jean Bernard
  • Gaubert Stéphane
  , 2016.
• Optimal Skorokhod embedding given full marginals and Azéma -Yor peacocks *
  
  • Källblad Sigrid
  • Tan Xiaolu
  • Touzi Nizar
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2016. We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0, 1]. The problem is related to the study of extremal martingales associated with a peacock (" process increasing in convex order " , by Hirsch, Profeta, Roynette and Yor [16]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labord ere, Ob lój , Spoida and Touzi [13]. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.
• Second-order shape derivatives along normal trajectories, governed by Hamilton-Jacobi equations
  
  • Allaire Grégoire
  • Cancès Eric
  • Vie Jean-Léopold
  Structural and Multidisciplinary Optimization, Springer Verlag, 2016. In this paper we introduce a new variant of shape differentiation which is adapted to the deformation of shapes along their normal direction. This is typically the case in the level-set method for shape optimization where the shape evolves with a normal velocity. As all other variants of the orginal Hadamard method of shape differentiation, our approach yields the same first order derivative. However, the Hessian or second-order derivative is different and somehow simpler since only normal movements are allowed. The applications of this new Hessian formula are twofold. First, it leads to a novel extension method for the normal velocity, used in the Hamilton-Jacobi equation of front propagation. Second, as could be expected, it is at the basis of a Newton optimization algorithm which is conceptually simpler since no tangential displacements have to be considered. Numerical examples are given to illustrate the potentiality of these two applications. The key technical tool for our approach is the method of bicharacteristics for solving Hamilton-Jacobi equations. Our new idea is to differentiate the shape along these bicharacteristics (a system of two ordinary differential equations). (10.1007/s00158-016-1514-2)
  DOI : 10.1007/s00158-016-1514-2
• Maximum likelihood estimation for Wishart processes
  
  • Alfonsi Aurélien
  • Kebaier Ahmed
  • Rey Clément
  Stochastic Processes and their Applications, Elsevier, 2016, 126 (11), pp.3243-3282. (10.1016/j.spa.2016.04.026)
  DOI : 10.1016/j.spa.2016.04.026
• Continuous-time Martingale Optimal Transport and Optimal Skorokhod Embedding
  
  • Guo Gaoyue
  , 2016. This PhD dissertation presents three research topics, the first two being independent and the last one relating the first two issues in a concrete case.In the first part we focus on the martingale optimal transport problem on the Skorokhod space, which aims at studying systematically the tightness of martingale transport plans. Using the S-topology introduced by Jakubowski, we obtain the desired tightness which yields the upper semicontinuity of the primal problem with respect to the marginal distributions, and further the first duality. Then, we provide also two dual formulations that are related to the robust superhedging in financial mathematics, and we establish the corresponding dualities by adapting the dynamic programming principle and the discretization argument initiated by Dolinsky and Soner.The second part of this dissertation addresses the optimal Skorokhod embedding problem under finitely-many marginal constraints. We formulate first this optimization problem by means of probability measures on an enlarged space as well as its dual problems. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results. We also relate these results to the martingale optimal transport on the space of continuous functions, where the corresponding dualities are derived for a special class of reward functions. Next, We provide an alternative proof of the monotonicity principle established in Beiglbock, Cox and Huesmann, which characterizes the optimizers by their geometric support. Finally, we show a stability result that is twofold: the stability of the optimization problem with respect to target marginals and the relation with another optimal embedding problem.The last part concerns the application of stochastic control to the martingale optimal transport with a payoff depending on the local time, and the Skorokhod embedding problem. For the one-marginal case, we recover the optimizers for both primal and dual problems through Vallois' solutions, and show further the optimality of Vallois' solutions, which relates the martingale optimal transport and the optimal Skorokhod embedding. As for the two-marginal case, we obtain a generalization of Vallois' solution. Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well ordered.
• Stationary solutions of discrete and continuous Petri nets with priorities
  
  • Allamigeon Xavier
  • Boeuf Vianney
  • Gaubert Stéphane
  , 2016. We study a continuous dynamics for a class of Petri nets which allows the routing at non-free choice places to be determined by priorities rules. We show that this dynamics can be written in terms of policies which identify the bottleneck places. We characterize the stationary solutions, and show that they coincide with the stationary solutions of the discrete dynamics of this class of Petri nets. We provide numerical experiments on a case study of an emergency call center, indicating that pathologies of discrete models (oscillations around a limit different from the stationary limit) vanish by passing to continuous Petri nets.
• On the smoothness of the value function for affine optimal control problems
  
  • Barilari Davide
  • Francesco Boarotto
  , 2016. We prove that the value function associated with an affine optimal control problem with quadratic cost plus a potential is smooth on an open and dense subset of the interior of its attainable set. The result is obtained by a careful analysis of points of continuity of the value function, without assuming any condition on singular minimizers.
• Theoretical and numerical aspects of wave propagation phenomena in complex domains and applications to remote sensing
  
  • Ramaciotti Morales Pedro
  , 2016. This thesis is about some boundary integral operators defined on the unit disk in a three-dimensional spaces, their relation with the exterior Laplace and Helmholtz problems, and their application to the preconditioning of the systems arising when solving these problems using the boundary element method.We begin by describing the so-called integral method for the solution of the exterior Laplace and Helmholtz problems defined on the exterior of objects with Lipschitz-regular boundaries, or on the exterior of open two-dimensional surfaces in a three-dimensional space. We describe the integral formulation for the Laplace and Helmholtz problem in these cases, their numerical implementation using the boundary element method, and we discuss its advantages and challenges: its computational complexity (both algorithmic and memory complexity) and the inherent ill-conditioning of the associated linear systems.In the second part we show an optimal preconditioning technique (independent of the chosen discretization) based on operator preconditioning. We show that this technique is easily applicable in the case of problems defined on the exterior of objects with Lipschitz-regular boundary surfaces, but that its application fails for problems defined on the exterior of open surfaces in three-dimensional spaces. We show that the boundary integral operators associated with the resolution of the Dirichlet and Neumann problems defined on the exterior of open surfaces have inverse operators, and that these operators would provide optimal preconditioners, but that they are not easily obtainable. Then we show a technique to explicitly obtain such inverse operators for the particular case when the open surface is the unit disk in a three-dimensional space. Their explicit inverse operators will be given, however, in the form of series, and will not be immediately applicable in the use of boundary element methods.In the third part we show how some modifications to these inverse operators allow us to obtain variational explicit and closed form expressions, no longer expressed as series, but also conserve nonetheless some characteristics that are relevant for their preconditioning effect. These explicit and closed forms expressions are applicable in boundary element methods. We obtain precise variational expressions for them and propose numerical schemes to compute the integrals needed for their use with boundary elements. The proposed numerical methods are tested using identities available within the developed theory and then used to build preconditioning matrices. Their performance as preconditioners for linear systems is tested for the case of the Laplace and Helmholtz problems for the unit disk. Finally, we propose an extension of this method that allows for the treatment of cases of open surfaces other that the disk. We exemplify and study this extension in its use on a square-shaped and an L-shaped surface screen in a three-dimensional space.
• On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  Journal of Computational Physics, Elsevier, 2016, 322, pp.224-247. We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\texttt{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{loc}$ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods. (10.1016/j.jcp.2016.06.037)
  DOI : 10.1016/j.jcp.2016.06.037
• Geometric properties of solutions to the total variation denoising problem
  
  • Chambolle Antonin
  • Duval Vincent
  • Peyré Gabriel
  • Poon Clarice
  Inverse Problems, IOP Publishing, 2016. This article studies the denoising performance of total variation (TV) image regularization. More precisely, we study geometrical properties of the solution to the so-called Rudin-Osher-Fatemi total variation denoising method. The first contribution of this paper is a precise mathematical definition of the “extended support” (associated to the noise-free image) of TV denoising. It is intuitively the region which is unstable and will suffer from the staircasing effect. We highlight in several practical cases, such as the indicator of convex sets, that this region can be determined explicitly. Our second and main contribution is a proof that the TV denoising method indeed restores an image which is exactly constant outside a small tube surrounding the extended support. The radius of this tube shrinks toward zero as the noise level vanishes, and are able to determine, in some cases, an upper bound on the convergence rate. For indicators of so-called “calibrable” sets (such as disks or properly eroded squares), this extended support matches the edges, so that discontinuities produced by TV denoising cluster tightly around the edges. In contrast, for indicators of more general shapes or for complicated images, this extended support can be larger. Beside these main results, our paper also proves several intermediate results about fine properties of TV regularization, in particular for indicators of calibrable and convex sets, which are of independent interest.
• Pollen dispersal slows geographical range shift and accelerates ecological niche shift under climate change
  
  • Aguilée R.
  • Raoul Gaël
  • Rousset François
  • Ronce Ophélie
  Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 2016, 113 (39), pp.E5741 - E5748. Species may survive climate change by migrating to track favorable climates and/or adapting to different climates. Several quantitative genetics models predict that species escaping extinction will change their geographical distribution while keeping the same ecological niche. We introduce pollen dispersal in these models, which affects gene flow but not directly colonization. We show that plant populations may escape extinction because of both spatial range and ecological niche shifts. Exact analytical formulas predict that increasing pollen dispersal distance slows the expected spatial range shift and accelerates the ecological niche shift. There is an optimal distance of pollen dispersal, which maximizes the sustainable rate of climate change. These conclusions hold in simulations relaxing several strong assumptions of our analytical model. Our results imply that, for plants with long distance of pollen dispersal, models assuming niche conservatism may not accurately predict their future distribution under climate change. (10.1073/pnas.1607612113)
  DOI : 10.1073/pnas.1607612113
• Mathematical contributions for the optimization and regulation of electricity production
  
  • Heymann Benjamin
  , 2016. We present our contribution on the optimization and regulation of electricity produc- tion.The first part deals with a microgrid Energy Management System (EMS). We formulate the EMS program as a continuous time optimal control problem and then solve this problem by dynamic programming using BocopHJB, a solver developed for this application. We show that an extension of this formulation to a stochastic setting is possible. The last section of this part introduces the adaptative weights dynamic programming algorithm, an algorithm for optimization problems with different time scales. We use the algorithm to integrate the battery aging in the EMS.The second part is dedicated to network markets, and in particular wholesale electricity markets. We introduce a mechanism to deal with the market power exercised by electricity producers, and thus increase the consumer welfare. Then we study some mathematical properties of the agents’ optimization problems (producers and system operator). In the last chapter, we present some pure Nash equilibrium existence and uniqueness results for a class of Bayesian games to which some networks markets belong. In addition we introduce an algorithm to compute the equilibrium for some specific cases.We provide some additional information on BocopHJB (the numerical solver developed and used in the first part of the thesis) in the appendix.
• Application of the Sparse Cardinal Sine Decomposition to 3D Stokes flows
  
  • Alouges François
  • Aussal Matthieu
  • Lefebvre-Lepot Aline
  • Pigeonneau Franck
  • Sellier Antoine
  , 2016.
• Monte-Carlo Methods and Stochastic Processes
  
  • Gobet Emmanuel
  , 2016 (1). (10.1201/9781315368757)
  DOI : 10.1201/9781315368757
• What do we mean by identiability in mixed effects models?
  
  • Lavielle Marc
  • Aarons Leon
  Journal of Pharmacokinetics and Pharmacodynamics, Springer Verlag, 2016. We discuss the question of model identiability within the context of non-linear mixed eects models. Although there has been extensive research in the area of xed eects models, much less attention has been paid to random effects models. In this context we distinguish between theoretical identiability, in which dierent parameter values lead to non-identical probability distributions , structural identiability which concerns the algebraic properties of the structural model, and practical identiability, whereby the model may be theoretically identiable but the design of the experiment may make parameter estimation dicult and imprecise. We explore a number of pharmacokinetic models which are known to be non-identiable at an individual level but can become identiable at the population level if a number of specic assumptions on the probabilistic model hold. Essentially if the probabilistic models are different , even though the structural models are non-identiable, then they will lead to dierent likelihoods. The ndings are supported through simulations.
• Multiresolution on arbitrary domains for time dependent PDEs. Application to streamer simulations
  
  • Lee Pierre-Louis
  • Bonaventura Z.
  • Duarte Max
  • Bourdon Anne
  • Massot Marc
  , 2016.
• Stability and stabilization of linear switched systems in finite and infinite dimensions
  
  • Mazanti Guilherme
  , 2016. Motivated by previous work on the stabilization of persistently excited systems, this thesis addresses stability and stabilization issues for linear switched systems in finite and infinite dimensions. After a general introduction presenting the main motivations and important results from the literature, we analyze four problems.The first system we study is a linear finite-dimensional random switched system. The time spend on each subsystem i is chosen according to a probability law depending only on i, and the switches between subsystems are determined by a discrete Markov chain. We characterize the Lyapunov exponents by applying Oseledets' Multiplicative Ergodic Theorem to an associated discrete-time system, and provide an expression for the maximal Lyapunov exponent. These results are applied to a switched control system, showing that, under a controllability hypothesis, almost sure stabilization can be achieved with arbitrarily large decay rates, a situation in contrast to deterministic persistently excited systems.We next consider a system of N transport equations with intermittent internal damping, linearly coupled by their boundary conditions through a matrix M, which can be seen as a system of PDEs on a star-shaped network. We prove that, if the activity of the intermittent damping terms is determined by persistently exciting signals, then, under suitable hypotheses on M and on the rationality of the ratios between the lengths of the network edges, such system is exponentially stable, uniformly with respect to the persistently exciting signals. The proof of this result is based on an explicit representation formula for the solutions of the system, which allows one to efficiently track down the effects of the intermittent damping.The following topic we address is the asymptotic behavior of non-autonomous difference equations. We obtain an explicit representation formula for their solutions in terms of their initial conditions and some time-dependent matrix coefficients, which generalizes the one for the system of N transport equations. The asymptotic behavior of solutions is characterized in terms of the matrix coefficients. In the case of difference equations with arbitrary switching, we obtain a stability result which generalizes Hale--Silkowski criterion for autonomous systems. Using classical transformations of hyperbolic PDEs into difference equations, we apply our results to transport and wave propagation on networks.Finally, we generalize the previous representation formula to a controlled difference equation, whose controllability is then analyzed. Relative controllability is characterized in terms of an algebraic property on the matrix coefficients from the explicit formula, generalizing Kalman criterion. We also compare the relative controllability for different delays in terms of their rational dependence structure, and provide a bound on the minimal controllability time. Exact and approximate controllability for systems with commensurable delays are characterized and proved to be equivalent. We also describe exact and approximate controllability for two-dimensional systems with two delays not necessarily commensurable.
• Generic singularities of line fields on 2D manifolds
  
  • Boscain Ugo
  • Sacchelli Ludovic
  • Sigalotti Mario
  Differential Geometry and its Applications, Elsevier, 2016, Volume 49 (December 2016), pp.326–350. Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zeros of the vector fields and the genericity is considered with respect to a natural topology in the space of pairs of vector fields. Line fields at generic singularities turn out to be topologically equivalent to the Lemon, Star and Monstar singularities that one finds at umbilical points.
• On the convergence of the Sakawa-Shindo algorithm in stochastic control
  
  • Bonnans Frédéric J.
  • Gianatti Justina
  • Silva Francisco José
  Mathematical Control and Related Fields, AIMS, 2016. We analyze an algorithm for solving stochastic control problems, based on Pontryagin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution. (10.3934/mcrf.2016008)
  DOI : 10.3934/mcrf.2016008
• On the ergodic convergence rates of a first-order primal-dual algorithm.
  
  • Chambolle Antonin
  • Pock Thomas
  Mathematical Programming, Series A, Springer, 2016, 159 (1-2), pp.253–287. We revisit the proofs of convergence for a first order primal-dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms.
• Contribution to missing values & principal component methods
  
  • Josse Julie
  , 2016. This manuscript was written for the Habilitation à Diriger des Recherches and it describes my research activities. The first part of this manuscript is named "A missing values tour with principal components methods". It first focuses on performing exploratory principal components (PCA based) methods despite missing values i.e. estimating parameters scores and loadings to get biplot representations from an incomplete data set. Then, it presents the use of principal components methods as single and multiple imputation for both continuous and categorical data. The second part concerns "New practices in visualization with principal components methods." It presents regularized versions of the principal components methods in the complete case and their potential impacts on the biplot graphical outputs.The contributions are part of the more general framework of low rank matrix estimation methods. Then, it discusses notions of variability of the parameters with confidence areas for fixed effect PCA either using bootstrap and Bayesian approaches.