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 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.
翻译：我们考虑一种PDE 方法来解决反射天线问题,方法是解决单位领域的最佳运输问题,其成本函数为 $c(x,y) = -2\log\left\Vert x -y\right\Vert$。在球体的每个点,我们用当地正切平面上一个通用的蒙古-安培格方程式取代表面PDE。然后我们使用一个可察觉的可比较的可比较的有限差异方案来接近解决方案并构建反射器。该方法很容易调整,以考虑到非移动的数据和解决方案,从而使它特别适合现实世界的光学问题。计算方法的模型实例表明,在计算一系列具有挑战性的问题时,包括不连续的强度和复杂地貌所支持的强度。
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