Parabolic optimal transport equations on manifolds

TL;DR

This paper introduces a parabolic flow approach to solve optimal transport problems on compact Riemannian manifolds, proving convergence under specific curvature and singularity conditions, with applications to spheres and antenna design.

Contribution

It establishes conditions under which a parabolic equation converges to the optimal transport solution on manifolds, extending previous methods to more general cost functions.

Findings

Solution exists for all time under strong MTW and stay-away conditions.

Flow converges exponentially to the optimal transport solution.

Applicable to sphere and antenna cost functions.

Abstract

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condition exists for all time and it converges exponentially to the solution to the optimal transportation problem. Such results hold in particular, on the sphere for the distance squared cost of the round metric and for the far-field reflector antenna cost, among others.