A convergent finite difference method for optimal transport on the sphere

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