Convergent numerical method for the reflector antenna problem via optimal transport on the sphere

We consider a partial differential equation (PDE) approach to numerically solve the reflector antenna problem by solving an optimal transport problem on the unit sphere with cost function c(x,y)=-2log⁡||x-y||. At each point on the sphere, we replace the surface PDE with a generalized Monge-Ampère type equation posed on the local tangent plane. We then use a provably convergent finite difference scheme to approximate the solution and construct the reflector. The method is easily adapted to take into account highly nonsmooth data and solutions, which makes it particularly well adapted to real-world optics problems. Computational examples demonstrate the success of this method in computing reflectors for a range of challenging problems including discontinuous intensities and intensities supported on complicated geometries.