A convergence framework for optimal transport on the sphere
A convergence framework for optimal transport on the sphere
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DOI：
10.1007/s00211-022-01292-1
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发表时间：
2021-03
影响因子：
2.1
通讯作者：
Brittany Froese Hamfeldt;Axel G. R. Turnquist
中科院分区：
文献类型：
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作者：
Brittany Froese Hamfeldt;Axel G. R. Turnquist
我们考虑一个偏微分方程的方法来数值求解最优运输问题的球。我们专注于传统的平方测地线成本和对数成本，这是在反射器天线的设计问题。在球面上的每一点上，我们用一个广义的Monge-Ampère型方程代替表面PDE，该方程使用法向坐标在切平面上提出。 由此产生的非线性偏微分方程，然后可以近似为任何一致的，单调计划的广义Monge-Ampère型方程在平面上。现有的技术证明收敛不立即适用，因为PDE缺乏比较原则和唯一的解决方案，这使得它很难产生一个稳定的，适定的计划。通过增加一个额外的项，约束解的梯度的离散化，我们得到了一个强形式的稳定性。一个修改的Barles-Souganwestern收敛框架，然后建立收敛到原来的偏微分方程的均值零解。
We consider a PDE approach to numerically solving the optimal transportation problem on the sphere. We focus on both the traditional squared geodesic cost and a logarithmic cost, which arises in the reflector antenna design problem. At each point on the sphere, we replace the surface PDE with a generalized Monge–Ampère type equation posed on the tangent plane using normal coordinates. The resulting nonlinear PDE can then be approximated by any consistent, monotone scheme for generalized Monge–Ampère type equations on the plane. Existing techniques for proving convergence do not immediately apply because the PDE lacks both a comparison principle and a unique solution, which makes it difficult to produce a stable, well-posed scheme. By augmenting the discretization with an additional term that constrains the solution gradient, we obtain a strong form of stability. A modification of the Barles–Souganidis convergence framework then establishes convergence to the mean-zero solution of the original PDE.