Publications

2016

• Stochastic mean payoff games are tropical semidefinite programs
  
  • Gaubert Stéphane
  , 2016.
• Adaptive Simultaneous-FETI: scalability results, robustness assessments and application to finite displacement problems
  
  • Bovet Christophe
  • Parret-Fréaud Augustin
  • Spillane Nicole
  • Gosselet Pierre
  , 2016. Classic Domain Decomposition methods like FETI are known to slowly converge when applied to real engineering problems involving for instance, automatic partitioning, quasi-incompressible materials or large heterogeneities. To ensure robustness in those cases, the Simultaneous-FETI (S-FETI) algorithm was recently proposed [1] as an alternative to GENEO approaches [2] where an augmentation space is preprocessed by solving a generalized eigenvalue problem. The S-FETI method is essentially a multi preconditioned conjugate gradient that exploits the additive structure of the preconditioner. At each iteration, the S-FETI algorithm generates as many search directions as there are domains in the decomposition. In this contribution, the parallel scalability and the robustness of the S-FETI method are assessed. Then, observing that the search space may be oversized in some cases, an adaptive variant is introduced. At each iteration, the Adaptive Simultaneous-FETI (AS-FETI) tries to compose the smallest search space which ensures a sufficient decrease of the residual. The theoretical result driving the adaptivity process was lately proven by one of the authors [3]. Finally, slight modifications allowing SFETI to deal with large displacement finite element problems are proposed, and we present the application of these methods to real engineering tests cases.
• Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group
  
  • Barilari Davide
  • Boscain Ugo
  • Neel Robert W.
  , 2016. By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an r-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on ℝ5 depending on two real parameters α1 and α2. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic (α1=α2) and the non-isotropic cases (α1≠α2). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.
• On scaling to an integer matrix and graphs with integer weighted cycles
  
  • Maccaig Marie
  Linear Algebra and its Applications, Elsevier, 2016, 498, pp.490--520. Between 1970 and 1982 Hans Schneider and co-authors produced a number of results regarding matrix scalings. They demonstrated that a matrix has a diagonal similarity scaling to any matrix with entries in the subgroup generated by the cycle weights of the associated digraph, and that a matrix has an equivalent scaling to any matrix with entries related to the weights of cycles in an associated bipartite graph. Further, given matrices A and B, they produced a description of all diagonal X such that X−1AX=B. In 2005 Butkovič and Schneider used max-algebra to give a simple and efficient description of this set of scalings. In this paper we focus on the additive group of integers, and work in the max-plus algebra to give a full description of all scalings of a real matrix A to any integer matrix. We do this for four types of scalings; beginning with the familiar X−1AX, XAY and XAX scalings and finishing with a new scaling which we call a signed similarity scaling. This is a scaling of the form XAY where we specify for each row i , either xi=yi or xi=−yi. In all of our results we use necessary and sufficient conditions for existence which are based on integer weighted cycles in the associated digraph, or associated bipartite graph, of the matrix. (10.1016/j.laa.2016.01.018)
  DOI : 10.1016/j.laa.2016.01.018
• Identifiability and non-identifiability in acoustic tomography of moving fluid
  
  • Agaltsov Alexey
  • Novikov Roman
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2016, 24 (3), pp.333-340. We consider a model time-harmonic wave equation of acoustic tomography of moving fluid in an open bounded domain in $\mathbb R^d$, $d \geq 2$, with variable sound speed $c$, density $\rho$, fluid velocity $v$ and absorption coefficient $\alpha$. We give global uniqueness results for related inverse boundary value problem for the cases of boundary measurements given for two and for three fixed frequencies. Besides, we also give a non-uniqueness result for this inverse problem for the case of boundary measurements given for all frequencies. (10.1515/jiip-2015-0051)
  DOI : 10.1515/jiip-2015-0051
• Adaptive kernel estimation of the baseline function in the Cox model with high-dimensional covariates
  
  • Guilloux Agathe
  • Lemler Sarah
  • Taupin Marie-Luce
  Journal of Multivariate Analysis, Elsevier, 2016, 148, pp.141–159. We propose a novel kernel estimator of the baseline function in a general high-dimensional Cox model, for which we derive non-asymptotic rates of convergence. To construct our estimator, we first estimate the regression parameter in the Cox model via a LASSO procedure. We then plug this estimator into the classical kernel estimator of the baseline function, obtained by smoothing the so-called Breslow estimator of the cumulative baseline function. We propose and study an adaptive procedure for selecting the bandwidth, in the spirit of Goldenshluger and Lepski (2011). We state non-asymptotic oracle inequalities for the final estimator, which leads to a reduction in the rate of convergence when the dimension of the covariates grows. (10.1016/j.jmva.2016.03.002)
  DOI : 10.1016/j.jmva.2016.03.002
• Non-archimedean valuations of eigenvalues of matrix polynomials
  
  • Akian Marianne
  • Gaubert Stéphane
  • Bapat Ravindra
  Linear Algebra and its Applications, Elsevier, 2016, 498, pp.592–627. We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities become equalities under genericity conditions, and that the leading coefficients of the eigenvalues are determined as the eigenvalues of auxiliary matrix polynomials. (10.1016/j.laa.2016.02.036)
  DOI : 10.1016/j.laa.2016.02.036
• Clustering and flow around a sphere moving into a grain cloud
  
  • Seguin A.
  • Lefebvre-Lepot Aline
  • Faure Sylvain
  • Gondret P.
  European Physical Journal E: Soft matter and biological physics, EDP Sciences: EPJ / Springer Nature, 2016, 39 (6). A bidimensional simulation of a sphere moving at constant velocity into a cloud of smaller spherical grains far from any boundaries and without gravity is presented with a non-smooth contact dynamics method. A dense granular “cluster” zone builds progressively around the moving sphere until a stationary regime appears with a constant upstream cluster size. The key point is that the upstream cluster size increases with the initial solid fraction phi0 but the cluster packing fraction takes an about constant value independent of phi0. Although the upstream cluster size around the moving sphere diverges when phi0 approaches a critical value, the drag force exerted by the grains on the sphere does not. The detailed analysis of the local strain rate and local stress fields made in the non-parallel granular flow inside the cluster allows us to extract the local invariants of the two tensors: dilation rate, shear rate, pressure and shear stress. Despite different spatial variations of these invariants, the local friction coefficient μ appears to depend only on the local inertial number I as well as the local solid fraction, which means that a local rheology does exist in the present non-parallel flow. The key point is that the spatial variations of I inside the cluster do not depend on the sphere velocity and explore only a small range around the value one. (10.1140/epje/i2016-16063-0)
  DOI : 10.1140/epje/i2016-16063-0
• Nonarchimedean semidefinite programming and stochastic games
  
  • Skomra Mateusz
  , 2016.
• Majorization inequalities for valuations of eigenvalues using tropical algebra
  
  • Akian Marianne
  , 2016.
• Long and winding central paths
  
  • Allamigeon Xavier
  , 2016.
• Modélisation de performance des caches basée sur l'analyse de données
  
  • Olmos Marchant Luis Felipe
  , 2016. L’Internet d’aujourd’hui a une charge de trafic de plus en plus forte à cause de la prolifération des sites de vidéo, notamment YouTube. Les serveurs Cache jouent un rôle clé pour faire face à cette demande qui croît vertigineusement. Ces serveurs sont déployés à proximité de l’utilisateur, et ils gardent dynamiquement les contenus les plus populaires via des algorithmes en ligne connus comme « politiques de cache ». Avec cette infrastructure les fournisseurs de contenu peuvent satisfaire la demande de façon efficace, en réduisant l’utilisation des ressources de réseau. Les serveurs Cache sont les briques basiques des Content Delivery Networks (CDNs), que selon Cisco fourniraient plus de 70% du trafic de vidéo en 2019.Donc, d’un point de vue opérationnel, il est très important de pouvoir estimer l’efficacité d’un serveur Cache selon la politique employée et la capacité. De manière plus spécifique, dans cette thèse nous traitons la question suivante : Combien, au minimum, doit-on investir sur un serveur cache pour avoir un niveau de performance donné?Produit d’une modélisation qui ne tient pas compte de la façon dont le catalogue de contenus évolue dans le temps, l’état de l’art de la recherche fournissait des réponses inexactes à la dernière question.Dans nos travaux, nous proposons des nouveaux modèles stochastiques, basés sur les processus ponctuels, qui permettent d’incorporer la dynamique du catalogue dans l’analyse de performance. Dans ce cadre, nous avons développé une analyse asymptotique rigoureuse pour l’estimation de la performance d’un serveur Cache pour la politique « Least Recently Used » (LRU). Nous avons validé les estimations théoriques avec longues traces de trafic Internet en proposant une méthode de maximum de vraisemblance pour l’estimation des paramètres du modèle.
• Désempilement non-paramétrique de la densité d'un processus shot-noise
  
  • Ilhe Paul
  • Roueff François
  • Moulines Eric
  • Souloumiac Antoine
  , 2016. Nous proposons une méthode d'estimation non-paramétrique rapide pour estimer la distribution des marques d'un processus de shot-noise en présence d'empilement à partir d'un nombre potentiellement important d'observations mais échantillonnées à basse fréquence. À partir d'une équation fonctionnelle liant la densité des marques à la fonction caractéristique des observations et sa dérivée, nous proposons un estimateur de cette densité en utilisant la base des B-splines. Nous discutons de l'implémentation pratique de l'algorithme et illustrons les performances de l'estimateur sur des données simulées.
• The Sparse Cardinal Sine Decomposition applied to Stokes integral equations
  
  • Alouges François
  • Aussal Matthieu
  • Lefebvre-Lepot Aline
  • Pigeonneau Franck
  • Sellier Antoine
  , 2016. Numerical simulations of two-phase flows driven by viscosity (e.g. for bubble motions in glass melting process) rely on the ability to efficiently compute the solutions to discretized Stokes equations. When using boundary element methods to track fluid interfaces, one usually faces the problem of solving linear systems with a dense matrix with a size proportional to the system number of degrees of freedom. Acceleration techniques, based on the compression of the underlying matrix and efficient matrix vector products are known (Fast Multipole Method, H-matrices, etc.) but are usually rather cumbersome to develop. More recently, a new method was proposed, called the " Sparse Cardinal Sine Decomposition " , in the context of acoustic problems to tackle this kind of problem in some generality (in particular with respect to the Green kernel of the problem). The proposed contribution aims at showing the potential applicability of the method in the context of viscous flows governed by Stokes equations.
• On the influence of the numerical strategy on the predictive character of Euler-Euler models for two-phase flow simulations in solid rocket motor instabilities
  
  • Dupif Valentin
  • Massot Marc
  • Dupays Joel
  • Laurent Frédérique
  • Le Touze Clément
  , 2016. Solid Rocket Motors involve strongly coupled two-phase flow. The presence of a polydisperse spray of particles resulting from the combustion of aluminized propellant has been shown to have a strong impact on stability and can eventually yield thrust oscillations. The ability to conduct predictive simulations of such a harsh environment is highly desirable. Euler-Euler models relying on moment methods for the disperse phase constitute interesting approaches due to their efficiency at coupling both phases and their ability for high performance computing. A multi-fluid model coupled to a new numerical strategy for the disperse phase is introduced in order to cope with the natural high stiffness of the resulting systems of conservation laws. The predictive character of the method is strongly related to the possibility of using accurate methods while preserving stability and robustness in the presence of intrinsic singularities occurring in the disperse phase equations. The purpose of this contribution is to stress the impact of several numerics and modeling on the solution. Relevant test cases for solid propulsion involving hydrodynamic instabilities and acoustic coupling are presented. A strategy is proposed in order to produce reliable predictions.
• Fully Eulerian simulation of 3D turbulent particle laden flow based on the Anisotropic Gaussian Closure
  
  • Sabat Macole
  • Vié Aymeric
  • Larat Adam
  • Massot Marc
  , 2016.
• Total Variation Denoising and Support Localization of the Gradient
  
  • Chambolle A.
  • Duval V.
  • Peyré G.
  • Poon C.
  , 2016, 756, pp.012007. (10.1088/1742-6596/756/1/012007)
  DOI : 10.1088/1742-6596/756/1/012007
• Ergodicity conditions in non-linear Perron-Frobenius theory and application to nonnegative tensors
  
  • Gaubert Stéphane
  , 2016.
• Measure contraction properties of Carnot groups
  
  • Rizzi Luca
  Calculus of Variations and Partial Differential Equations, Springer Verlag, 2016. We prove that any corank 1 Carnot group of dimension k + 1 equipped with a left-invariant measure satisfies the MCP(K, N) if and only if K ≤ 0 and N ≥ k + 3. This generalizes the well known result by Juillet for the Heisenberg group H k+1 to a larger class of structures, which admit non-trivial abnormal minimizing curves. The number k + 3 coincides with the geodesic dimension of the Carnot group, which we define here for a general metric space. We discuss some of its properties, and its relation with the curvature exponent (the least N such that the MCP(0, N) is satisfied). We prove that, on a metric measure space, the curvature exponent is always larger than the geodesic dimension which, in turn, is larger than the Hausdorff one. When applied to Carnot groups, our results improve a previous lower bound due to Rifford. As a byproduct, we prove that a Carnot group is ideal if and only if it is fat. (10.1007/s00526-016-1002-y)
  DOI : 10.1007/s00526-016-1002-y
• A new macroscopic model for the diffusion MRI accounting for time-dependent diffusivity
  
  • Haddar Houssem
  • Li Jing-Rebecca
  • Schiavi Simona
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (3), pp.930–949. Diffusion Magnetic Resonance Imaging (dMRI) encodes water displacement due to diffusion and is a powerful tool to obtain information on the tissue micro-structure. An important quantity measured in dMRI in each voxel is the Apparent Diffusion Coefficient ($ADC$) and it is well-established from imaging experiments that, in the brain, {\it in-vivo}, the $ADC$ is dependent on the measured diffusion time. To aid in the understanding and interpretation of the $ADC$, using homogenization techniques, we derived a new asymptotic model for the dMRI signal from the Bloch-Torrey equation governing the water proton magnetization under the influence of diffusion-encoding magnetic gradient pulses. Our new model was obtained using a particular choice of scaling for the time, the biological cell membrane permeability, the diffusion-encoding magnetic field gradient strength, and a periodicity length of the cellular geometry. The $ADC$ of the resulting model is dependent on the diffusion time. We numerically validated this model for a wide range of diffusion times for two dimensional geometrical configurations. (10.1137/15M1019398)
  DOI : 10.1137/15M1019398
• Existence and qualitative properties of travelling waves for an epidemiological model with mutations
  
  • Griette Quentin
  • Raoul Gaël
  Journal of Differential Equations, Elsevier, 2016, 260 (10), pp.7115-7151. In this article, we are interested in a non-monotone system of logistic reaction-diffusion equations. This system of equations models an epidemics where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of minimal speed travelling waves, that are usually non monotonic. We then provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed. (10.1016/j.jde.2016.01.022)
  DOI : 10.1016/j.jde.2016.01.022
• Mechanism Design and Auctions for Electricity Network
  
  • Heymann Benjamin
  • Jofré Alejandro
  , 2016. We present some key aspects of wholesale electricity markets modeling and more specifically focus our attention on auctions and mechanism design. Some of the results arising from those models are the computation of an optimal allocation for the Independent System Operator, the study of the equilibria (existence and unicity in particular) and the design of mechanisms to increase the social surplus. From a more general perspective, this field of research provides clues to discuss how wholesale electricity market should be regulated. We start with a general introduction and then present some results the authors obtained recently. We also briefly expose some undergoing related work. As an illustrative example, a section is devoted to the computation of the Independent System Operator response function for a symmetric binodal setting with piece-wise linear production cost functions.
• Adapting the Kärger model to account for finite diffusion-encoding pulses in diffusion MRI
  
  • Haddar Houssem
  • Li Jing-Rebecca
  • Schiavi Simona
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2016. Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. If the imaging voxel can be divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be described by the Kärger model, which is a well-known model in Nuclear Magnetic Resonance. However, the Kärger model is limited to the case when the duration of the diffusion-encoding gradient pulses is short compared to the time delay between the start of the pulses. Under this assumption, the time at which to evaluate the Kärger model to obtain the dMRI signal is unambiguously the delay between the pulses. Recently, a new model of the dMRI signal, the Finite-Pulse Kärger (FPK) model, was derived for arbitrary diffusion gradient profiles. Relying on the FPK model, we show that when the duration of the gradient pulses is not short, the time at which to evaluate the Kärger model should be the time delay between the start of the pulses, shortened by one third of the pulse duration. With this choice, we show the sixth order convergence of the Kärger model to the FPK model in the non-dimensionalized pulse duration. (10.1093/imamat/hxw032)
  DOI : 10.1093/imamat/hxw032
• Limite hydrodynamique pour un dynamique sur réseau de particules actives
  
  • Erignoux Clément
  , 2016. L'étude des dynamiques collectives, observables chez de nombreuses espèces animales, a motivé dans les dernières décennies un champ de recherche actif et transdisciplinaire. De tels comportements sont souvent modélisés par de la matière active, c'est-à-dire par des modèles dans lesquels chaque individu est caractérisé par une vitesse propre qui tend à s'aligner avec celle de ses voisins.Dans un modèle fondateur proposé par Vicsek et al., ainsi que dans de nombreux modèles de matière active liés à ce dernier, une transition de phase entre un comportement chaotique à forte température, et un comportement global et cohérent à faible température, a été observée. De nombreuses preuves numériques de telles transitions de phase ont été obtenues dans le cadre des dynamiques collectives. D'un point de vue mathématique, toutefois, ces systèmes actifs sont encore mal compris. Plusieurs résultats ont été obtenus récemment sous une approximation de champ moyen, mais il n'y a encore à ce jour que peu d'études mathématiques de modèles actifs faisant intervenir des interactions purement microscopiques.Dans ce manuscrit, nous décrivons un système de particules actives sur réseau interagissant localement pour aligner leurs vitesses. L'objet de cette thèse est l'obtention rigoureuse, à l'aide du formalisme des limites hydrodynamiques pour les gaz sur réseau, de la limite macroscopique de ce système hors-équilibre, qui pose de nombreuses difficultés techniques et théoriques.
• An explicit martingale version of the one-dimensional Brenier theorem
  
  • Henry-Labordère Pierre
  • Touzi Nizar
  Finance and Stochastics, Springer Verlag (Germany), 2016, 20 (3), pp.635-668. (10.1007/s00780-016-0299-x)
  DOI : 10.1007/s00780-016-0299-x