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

TL;DR

This paper introduces a convergent numerical method for the reflector antenna problem by solving an optimal transport problem on the sphere, using a PDE approach with a generalized Monge-Ampère equation and finite difference scheme.

Contribution

It develops a new PDE-based numerical method for the reflector antenna problem that handles nonsmooth data and complex geometries effectively.

Findings

Successfully computes reflectors for discontinuous intensities

Handles highly nonsmooth data and solutions

Demonstrates effectiveness on complex geometries

Abstract

We consider a PDE approach to numerically solving the reflector antenna problem by solving an Optimal Transport problem on the unit sphere with cost function $c(x,y) = -2\log \left\Vert x - y \right\Vert$. At each point on the sphere, we replace the surface PDE with a generalized Monge-Amp\`ere type equation posed on the local tangent plane. We then utilize 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.