A Convergent Finite Difference Method for Optimal Transport on the Sphere

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.