Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 20 février à 14h30 à l’université Paris-Dauphine (Place du Maréchal de Lattre de Tassigny), en salle P509.
Pour cette édition spéciale, nous aurons le plaisir d’écouter 4 exposés au cours de l’après-midi. Le programme est le suivant:
14h30-15h20: Marc HOFFMANN
15h20-16h10: Olga MULA
16h30-17h20: Laurent COHEN
17h20-18h10: Bruno BOUCHARD
- Exposé de Marc HOFFMANN:
Titre: Estimation statistique pour des modèles structurés en âge dans une limite grande population
Résumé: Motivated by improving mortality tables from human demography databases, we investigate statistical inference of a stochastic age-evolving density of a population alimented by time inhomogeneous mortality and fertility. Asymptotics are taken as the size of the population grows within a limited time horizon: the observation gets closer to the solution of the Von Foerster Mc Kendrick equation, and the difficulty lies in controlling simultaneously the stochastic approximation to the limiting PDE in a suitable sense together with an appropriate parametrisation of the anisotropic solution.
In this setting, we prove new concentration inequalities that enable us to implement the Goldenshluger-Lepski algorithm and derive oracle inequalities. We obtain minimax optimality and adaptation over a wide range of anisotropic Hölder smoothness classes.
- Exposé d’Olga MULA:
Titre: Nonlinear reduced models and state estimation
Abstract: In this talk, we present an overview and some recent results on the problem of reconstructing in real time the state of a physical system from available measurement observations and the knowledge of a physical PDE model. Contrary to classical inverse problem approaches where one seeks for the parameters of the PDE that best satisfy the measurements, we use the PDE models to learn fast reconstruction mappings which satisfy certain optimality properties. The high dimensionality of the problems that arise combined with the very different nature of the potential applications (air pollution, hemodynamics, nuclear safety to name a few) demand the development of compression, optimisation and learning strategies based on sound mathematical grounds. In the talk, we will present recent results on optimal affine algorithms and highlight the prominent role of reduced order modelling of PDEs. However, in its classical formulation, reduced order modelling involves the construction of linear spaces which makes it not suitable to treat hyperbolic problems. We will outline recent results on an approach involving nonlinear mappings to mitigate this obstruction.
- Exposé de Laurent COHEN:
Titre: Méthodes géodésiques pour la segmentation d’images
Résumé: Tubular and tree structures appear very commonly in biomedical images like vessels, microtubules or neuron cells. Minimal paths have been used for long as an interactive tool to segment these structures as cost minimizing curves. The user usually provides start and end points on the image and gets the minimal path as output. These minimal paths correspond to minimal geodesics according to some adapted metric. They are a way to find a (set of) curve(s) globally minimizing the geodesic active contours energy. Finding a geodesic distance can be solved by the Eikonal equation using the fast and efficient Fast Marching method. Introduced first as a way to find the global minimum of a simplified active contour energy, we have recently extended these methods to cover all kinds of active contour energy terms. For example a new way to penalize the curvature in the framework of geodesic minimal paths was introduced, leading to more natural results in vessel extraction for example.
- Exposé de Bruno BOUCHARD:
Titre: Quenched mass transport of particles towards a target
Résumé: We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost-surely transported towards a given target, along the paths of a stochastic differential equation.