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 $c(x,y) = - 2\log || {x - y} ||$. 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.

中文翻译：

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通过球体上最优输运的反射面天线问题的收敛数值方法

我们考虑采用偏微分方程 (PDE) 方法通过求解单位球面上的最优传输问题来数值求解反射器天线问题，成本函数为 $c(x,y) = - 2\log || {x - y} ||$. 在球体上的每个点，我们用局部切平面上提出的广义 Monge-Ampère 型方程替换表面 PDE。然后我们使用可证明收敛的有限差分格式来近似解并构造反射器。该方法很容易适应考虑高度非光滑的数据和解决方案，这使得它特别适用于现实世界的光学问题。计算示例证明了该方法在计算反射器方面的成功，可以解决一系列具有挑战性的问题，包括不连续强度和复杂几何结构支持的强度。