Search

In this paper we show that the maximal viscosity solution of a class of quasiconvex Hamilton-Jacobi equations, coupled with inequality constraints on the boundary, can be recovered by taking the limit as p → ∞ in a family of Finsler p-Laplace problems. The approach also enables us to provide an optimal solution to a Beckmann-type problem in general Finslerian setting and allows recovering a bench of known results based on the Evans-Gangbo technique.

We suggest a new approach to solve a class of degenerate Hamilton-Jacobi equations without any assumptions on the emptiness of the Aubry set. It is based on the characterization of the maximal subsolution by means of the Fenchel-Rockafellar duality. This approach enables us to use augmented Lagrangian methods as alternatives to the commonly used methods for numerical approximation of the solution, based on finite difference approximation or on optimal control interpretation of the solution.

In this thesis we propose some variational methods for the mathematical and numerical analysis of a class of HJ equations. Thanks to the metric character of these equations, the set of subsolution corresponds to the set of 1-Lipschitz functions with respect to the Finsler metric associated to the Hamiltonian. Equivalently, it corresponds to the set of functions whose gradient belongs to a Finsler ball. The solution we are looking for is the maximal one, which can be described via a Hopf-Lax formula, solves a maximization problem under gradient constraint. We derive the associated dual problem which involves the Finsler total variation of vector measures under a divergence constraint. We take advantage of this saddle-point structure to use the augmented Lagrangian method for the numerical approximation of HJ equation. This characterization of the HJ equation allows making the link with some optimal transport problems. This link with optimal transport leads us to generalize the Evans-Gangbo approach. In fact, we show that the maximal viscosity subsolution of the HJ equation can be recovered by taking p→ ∞ in a class of Finslerp-Laplace problems with boundary obstacles. In addition, this allows us to construct the optimal flow for the associated Beckmann problem. As an application, we use our variational approach for the Shape from Shading problem., L’objectif de cette thèse est de proposer des méthodes variationnelles pour l’analyse mathématiques et numérique d’une classe d’équations d’HJ. Le caractère métrique de ces équations permet de caractériser l’ensemble des sous-solutions, à savoir, elles sont 1-Lipschitz par rapport à la distance Finslerienne associée au Hamiltonien. De manière équivalente, cela revient à dire que le gradient de ces fonctions appartient à une certaine boule Finslerienne. La solution recherchée est la sous-solution maximale, qui peut être décrite par une formule du type Hopf-Lax, qui résout un problème de maximisation avec contrainte sur le gradient. Nous dérivons un problème dual associé faisant intervenir la variation totale Finslerienne de mesures vectorielles avec contrainte divergente. Nous exploitons la structure de point-selle pour proposer une résolution numérique avec la méthode du Lagrangien augmenté. Cette caractérisation de l’équation d’HJ montre aussi le lien avec des problèmes de transport optimal vers/depuis le bord. Ce lien avec le transport optimal de masse nous amène à généraliser l’approche d’Evans-Gangbo. En effet, nous montrons que la sous-solution maximale de l’équation d’HJ s’obtient en faisant tendre p→∞ dans une classe de p-Laplaciens de type Finsler avec des obstacles sur le bord. Cela nous permet aussi de construire le flux optimal pour le problème de Beckmann associé. Parmi les applications que l’on regarde, le problème du Shape from Shading qui consiste à reconstruire la surface d’un objet en 3D à partir d’une image en nuances de gris de cet objet.

The aim of this note is to give a Beckmann-type problem as well as the corresponding optimal mass transportation problem associated with a degenerate Hamilton-Jacobi equation H ( x , ∇ u ) = 0 , H(x,\nabla u)=0, coupled with non-zero Dirichlet condition u = g u=g on ∂ Ω \partial \Omega . In this article, the Hamiltonian H H is continuous in both arguments, coercive and convex in the second, but not enjoying any property of existence of a smooth strict sub-solution. We also provide numerical examples to validate the correctness of theoretical formulations.

The continuous Lambertian shape from shading is studied using a PDE approach in terms of Hamilton–Jacobi equations. The latter will then be characterized by a maximization problem. In this paper we show the convergence of discretization and propose to use the well-known Chambolle–Pock primal-dual algorithm to solve numerically the shape from shading problem. The saddle-point structure of the problem makes the Chambolle–Pock algorithm suitable to approximate solutions of the discretized problems.