A convergence framework for optimal transport on the sphere

TL;DR

This paper develops a PDE-based numerical framework for solving optimal transport problems on the sphere, addressing challenges in convergence and stability for nonlinear Monge-Ampère type equations.

Contribution

It introduces a novel discretization with gradient constraints and adapts the Barles-Souganidis framework to prove convergence for spherical optimal transport.

Findings

Established convergence of the proposed scheme on the sphere

Handled both squared geodesic and logarithmic costs

Provided a stable, monotone discretization method

Abstract

We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge-Amp\`ere type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge-Amp\`ere type equations on the plane. Existing techniques for proving convergence do not immediately apply because the PDE lacks both a comparison principle and a unique solution, which makes it difficult to produce a stable, well-posed scheme. By augmenting this discretization with an additional term that constrains the solution gradient, we obtain a strong form of stability. A modification of the Barles-Souganidis convergence framework then establishes convergence to the mean-zero solution of the original PDE.