A convergence framework for optimal transport on the sphere

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–Ampère 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–Ampère 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 the 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.