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

Abstract

We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

A new implementation of the geometric method for solving the Eady slice equations

“…The most computationally intensive part of solving the ODE (32) numerically is evaluating the function c. In addition, c is continuously differentiable (on the set of distinct seed positions), but not twice continuously differentiable in general: see [4]. As such, we can only expect convergence of the ODE solver up to second order in the time step size.…”

“…In this appendix we give a formal derivation of the ODE (31) from the Lagrangian equation ( 16) by assuming that P is piecewise affine and satisfies the stability principle. For a rigorous derivation, see [4]. By equations ( 15), ( 22), ( 27), (28), we have…”

“…One of the main differences between our implementation of the geometric method and the original implementation in [10] is that we restate it in the language of semi-discrete optimal transport theory [28, Chapter 4], which allows us to exploit recent results from this field, such as the damped Newton method of Kitagawa, Mérigot & Thibert (2019) [23]. In our companion paper [4], the pioneering design of the geometric method is put on a rigorous footing for the closely-related threedimensional semi-geostrophic equations. The original implementation of the geometric method in [10] is now regarded as the first numerical method for solving the semi-discrete optimal transport problem [28,Section 4.1].…”

“…The semi-geostrophic Eady Slice equation, and related semi-geostrophic systems, have been studied extensively in their potential vorticity formulation in the mathematical analysis literature: see, for example,[1,3,16,18,26]. The relation of this viewpoint to the discrete formulation of the dynamics used in the geometric method is the subject of[4].…”

Entropic optimal transport solutions of the semigeostrophic equations

“…Numerical solutions were obtained in [11] under a Lagrangian discretization but available OT solvers at that time were at best of cubic complexity and the resolution was coarse/costly. Later advances in Semi-Discrete Optimal transport [22], leading to linear cost OT solvers have generated a renewed interest in solving numerically the Semigeostrophic equations [5] and higher resolution solutions have been obtained in [14]. Another efficient OT numerical resolution approach based on an Entropic penalization [13] [20] is developing rapidly, see [25] and also [23].…”