Numerical Optimal Transport on the Sphere

Abstract: The problem of optimal transportation, which involves finding the most cost-efficient mapping between two measures, arises in many different applications. We consider the special case of optimal transport on the sphere, which is of particular importance in the design of optical systems and in mesh generation. This problem can be formulated as a fully nonlinear elliptic partial differential equation on the sphere. However, existing techniques for solving PDEs on surfaces are not well-equipped to handle this challenging equation. We describe a simple approach that reduces this to the problem of discretizing a Monge-Ampere type equation on the plane. We show how existing finite difference methods can be adapted to produce a provably convergent numerical method for optimal transport on the sphere. We conclude with several challenging computational examples, including examples in mesh generation and reflector design.

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