Semi-discrete optimal transport methods for the semi-geostrophic equations

A Newton algorithm for semi-discrete optimal transport

Many problems in geometric optics or convex geometry can be recast as optimal transport problems and a popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton’s algorithm for this type of problems, which is experimentally efficient, and we establish its ...

Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations

In arbitrary dimension, in the discrete setting of finite-differences we prove a Carleman estimate for a semi-discrete parabolic operator, in which the large parameter is connected to the mesh size. This estimate is applied for the derivation of a (relaxed) observability estimate, that yield some controlability results for semi-linear semi-discrete parabilic equations. Sub-linear and super-line...

Chaotic response of the 2D semi-geostrophic and 3D quasi-geostrophic equations to gentle periodic forcing

Symmetries and Hamiltonian structure are combined with Melnikov’s method to show a set of exact solutions to the 2D semi-geostrophic equations in an elliptical tank respond chaotically to gentle periodic forcing of the domain eccentricity (or of the potential vorticity, for that matter) which are sinusoidal in time with nearly any period. A similar approach confirms the chaotic response of the ...

Semi-Lagrangian Particle Methods for Hyperbolic Equations

Particle methods with remeshing of particles at each time-step can be seen as forward semi-lagrangian conservative methods for advection-dominated problems, and must be analyzed as such. In this article we investigate the links between these methods and finite-difference methods and present convergence results as well as techniques to control their oscillations. We emphasize the role of the siz...

Semi-Lagrangian Methods for Level Set Equations