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We consider a PDE approach to numerically solving the reflector antennaproblem by solving an Optimal Transport problem on the unit sphere with costfunction $c(x,y) = -2\log \left\Vert x - y \right\Vert$. At each point on thesphere, we replace the surface PDE with a generalized Monge-Amp\`ere typeequation posed on the local tangent plane. We then utilize a provablyconvergent finite difference scheme to approximate the solution and constructthe reflector. The method is easily adapted to take into account highlynonsmooth data and solutions, which makes it particularly well adapted toreal-world optics problems. Computational examples demonstrate the success ofthis method in computing reflectors for a range of challenging problemsincluding discontinuous intensities and intensities supported on complicatedgeometries.

A Convergent Numerical Method for the Reflector Antenna Problem via Optimal Transport on the Sphere