Résolution effective de problèmes géométriques non linéaires

Keywords :

• transport optimal numérique
• intégration convexe

In this dissertation, I present my contributions to the field of geometric inference and to several reconstruction problems having connections with various domains such as optimal transport, computational geometry, geometric measure theory and also convex integration theory. The common point of my contribution to these fields  is the development of effective calculations in problems having a geometric flavor. 

In a first part, I show that it is possible to recover robust geometric information of a shape in the euclidean shape from an approximation, such as a finite point set, a triangulation, or a compact set. The general approach consists in using the stability and regularity properties of distance functions and more generally of distance-like functions. I present stability and convergence results on geodesics, normals, Federer's curvature mesures and also on generalized Voronoi covariance measures. 

In a second part, I consider the resolution of certain nonlinear partial differential equations and propose geometric methods to solve them. The far-field reflector problem, arising in non-imaging optics,  is  modeled by a nonlinear PDE on the sphere. The goal is to create a mirror that reflects a given point light source to a prescribed distribution of light at infinity. By combining a variational approach from optimal transport, computational geometry and a geometric construction due to L. A. Caffarelli and V. Oliker, we propose an efficient method to numerically build such a reflector.

Finally, I present the realization of an isometric embedding of the square flat torus in the three dimensional Euclidean space. The existence of such an embedding is due to a famous result of John Nash on isometric embeddings of Riemannian manifolds. In that case, the isometric constraint is expressed by a  nonlinear PDE system involving the metric tensor. Our main theoretical tool is the convex integration theory that has been developed by M. Gromov in the 70s-80s for solving underdeterminated differential systems. We have adapted and implemented this tool so as to get an algorithm for building isometric embeddings of the square flat torus in the ambient space and have discovered a new geometric structure, the smooth fractals.

Raporteurs:

• Mr Joseph Fu (Professeur - Georgia University )
• Mr Bruno Levy (Directeur de Recherche - INRIA Nancy )

Examinators:

• Mr Eric Bonnetier (Professeur - Université Joseph Fourier-Université Grenoble Alpes )
• Mr Hervé Pajot (Professeur - Université Grenoble Alpes )
• Mr Konrad Polthier (Professeur - Freje Universität Berlin )
• Mr Simon Masnou (Professeur - Université Lyon 1 )