A Convergent Finite Difference Method for Optimal Transport on the Sphere

TL;DR

This paper presents a new convergent finite difference method for solving optimal transport problems on the sphere, applicable to different cost functions and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a novel finite difference approach adapted for the sphere's geometry and complex cost functions, extending monotone methods for the Monge-Ampère equation.

Findings

Method successfully solves challenging optimal transport problems on the sphere.

Applicable to both squared geodesic and logarithmic costs.

Numerical results confirm convergence and robustness.

Abstract

We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising in the reflector antenna design problem). At each point on the sphere, we replace the surface PDE with a Generated Jacobian equation posed on the local tangent plane using geodesic normal coordinates. The discretization is inspired by recent monotone methods for the Monge-Amp\`ere equation, but requires significant adaptations in order to correctly handle the mix of gradient and Hessian terms appearing inside the nonlinear determinant operator, as well as the singular logarithmic cost function. Numerical results demonstrate the success of this method on a wide range of challenging problems involving both the squared geodesic and the logarithmic cost functions.