Convergent approaches for the Dirichlet Monge amp\`ere problem

In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite difference discretization and monotone schemes are obtained. Hence, we apply the Barles-Souganidis theory to prove the convergence of the schemes and the Damped Newtons method is used to compute the solutions of the schemes. Finally, some numerical results are illustrated.Comment: 14pages, 7figure

Multi-physics Optimal Transportation and Image Interpolation

International audienceOptimal transportation theory is a powerful tool to deal with image interpolation. This was first investigated by Benamou and Brenier \cite{BB00} where an algorithm based on the minimization of a kinetic energy under a conservation of mass constraint was devised. By structure, this algorithm does not preserve image regions along the optimal interpolation path, and it is actually not very difficult to exhibit test cases where the algorithm produces a path of images where high density regions split at the beginning before merging back at its end. However, in some applications to image interpolation this behaviour is not physically realistic. Hence, this paper aims at studying how some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Amp\ue8re equations

We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Amp\ue8re type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited

Eigenvalue problems for fully nonlinear elliptic partial differential equations with transport boundary conditions

Fully nonlinear elliptic partial differential equations (PDEs) arise in a number of applications. From mathematical finance to astrophysics, there is a great deal of interest in solving them. Eigenvalue problems for fully nonlinear PDEs with transport boundary conditions are of particular interest as alternative formulations of PDEs that require data to satisfy a solvability condition, which may not be known explicitly or may be polluted by noisy data. Nevertheless, these have not yet been well-explored in the literature. In this dissertation, a convergence framework for numerically solving eigenvalue problems for fully nonlinear PDEs is introduced. In addition, existing two-dimensional methods for nonlinear equations are extended to handle transport boundary conditions and eigenvalue problems. Finally, new techniques are designed to enable appropriate discretization of a large range of fully nonlinear three-dimensional equations