We first summarize the main abstract result presented one year ago. We then discuss its applications to Schrödinger and Kirchhoff systems. We end up by stating several open questions.
Initial data for Minkowski stability with arbitrary decay
In this talk, I will present a construction of arbitrarily decaying initial data for the stability of Minkowski spacetime as solutions to the Einstein equations. Initial data on a spacelike hypersurface need to solve the so-called constraint equations, i.e a geometric nonlinear underdetermined elliptic system. I will show how one can parametrize solutions in a neighborhood of Minkowski spacetime and address linear obstructions coming from conservation laws in general relativity. This is a joint work with Allen Juntao Fang and Jérémie Szeftel.
Models and algorithms for configuring and testing prototype cars
In this presentation, we consider a new industrial problem, which occurs in the context of the automobile industry. This problem occurs during the testing phase of a new vehicle. It involves determining all the variants of the vehicle to be manufactured in order to carry out these tests, and scheduling these tests over time. We model this problem as a new scheduling scheduling problem. Given a set of machines, and a set of jobs, we seek a fixed configuration for each machine (i.e. a set of values for various parameters), and an assignment of jobs to machines along the time horizon that respects compatibility constraints between jobs and machine configurations. Two objectives are lexicographically optimized: the number of late jobs, and the number of machines used. This problem involves a notion of configuration that is not addressed in the literature. First we prove that even finding a feasible solution for the problem is NP-hard, and characterize the cases where compatibility constraints amount to ensuring that only pairwise compatible jobs are assigned to each machine. We then propose a mathematical model for this problem, and a reformulation into a path-flow formulation. To deal with the new notion of configuration, we propose a refined labelling algorithm embedded in a column-and-row generation algorithm to generate primal and dual bounds from this formulation. We conducted computational experiments on industrial data from Renault, and compared our results with those obtained by solving a constraint programming model provided by the company. Our approach finds better solutions than those obtained by the company, and proves the optimality for all instances of our benchmark for the first objective function. We also obtain small optimality gaps for the second objective function.
Second-order algorithms for large-scale optimization and deep learning
Non-convex non-smooth optimization has gained a lot of interest due to the efficiency of neural networks in many practical applications and the need to "train" them. Training amounts to solving very large-scale optimization problems. In this context, standard algorithms almost exclusively rely on inexact (sub-)gradients through automatic differentiation and mini-batch sub-sampling. As a result, first-order methods (SGD, ADAM, etc.) remain the most used ones to train neural networks. Driven by a dynamical system approach, we build INNA, an inertial and Newtonian algorithm, exploiting second-order information on the function only by means of first-order automatic differentiation and mini-batch sub-sampling. By analyzing together the dynamical system and INNA, we prove the almost-sure convergence of the algorithm. We discuss practical considerations and empirical results on deep learning experiments. We finally depart from non-smooth optimization and provide insights into recent results that pave the way for designing faster second-order methods.
Titre de la thèse :"Méthodes numériques pour la tomographie par impédance électrique dans le cadre de l'électrocardiographie". Directrice de thèse : Lisl Weynans
Un système dynamique (topologique) est la donnée d’un homéomorphisme T sur un espace métrique compact X. Un problème classique consiste à étudier le centralisateur de T i.e., le groupe des transformations commutant avec T. Nous nous concentrerons sur les systèmes symboliques engendrés par des sous-shifts, ou sous-décalage, qui forment une classe riche de systèmes dynamiques offrant une diversité de comportements. Après avoir évoqué les motivations, nous présenterons un survol de résultats récents obtenus dans ce contexte, mettant en lumière comment les propriétés topologiques du système (complexité, minimalité, etc) influent sur ce groupe.
Redressement des champs d'ellipses : une nouvelle preuve du théorème d'Ahlfors-Bers
Le théorème d'Ahlfors-Bers énonce la dépendance holomorphe du redressement d'un champ d'ellipses quand le champ dépend holomorphiquement d'un paramètre et est un point crucial de plusieurs preuves en dynamique holomorphe et chez les groupes Kleiniens. Sa preuve classique repose sur l'étude d'un opérateur de convolution à noyau singulier. En considérant une suite de surfaces de similitude, on peut en donner une preuve plus géométrique. Si le temps le permet je décrirai un objet limite inattendu dans le cas où le champ d'ellipses est suffisamment lisse.
