A convergent finite difference method for optimal transport on the sphere
A convergent finite difference method for optimal transport on the sphere
复制标题
球体最优输运的收敛有限差分法
DOI：
10.1016/j.jcp.2021.110621
复制
发表时间：
2021
影响因子：
4.1
通讯作者：
Turnquist, Axel G.R.
中科院分区：
文献类型：
--
作者：
Hamfeldt, Brittany Froese;Turnquist, Axel G.R.
给出了求解球面上最优运输问题的收敛有限差分法。该方法既适用于传统的平方测地线成本（网格生成），也适用于对数成本（反射器天线设计问题）。在球上的每一点，我们用一个生成的雅可比方程替换表面的PDE，这个雅可比方程是用测地线法坐标在局部切平面上构成的。离散化的灵感来自最近monge - amp<e:1>方程的单调方法，但需要进行重大调整，以便正确处理非线性行列式算子中出现的梯度和Hessian项的混合，以及奇异对数代价函数。数值结果表明，该方法在涉及平方测地线和对数代价函数的一系列具有挑战性的问题上取得了成功。
We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising in the reflector antenna design problem). At each point on the sphere, we replace the surface PDE with a Generated Jacobian equation posed on the local tangent plane using geodesic normal coordinates. The discretization is inspired by recent monotone methods for the Monge-Ampère equation, but requires significant adaptations in order to correctly handle the mix of gradient and Hessian terms appearing inside the nonlinear determinant operator, as well as the singular logarithmic cost function. Numerical results demonstrate the success of this method on a wide range of challenging problems involving both the squared geodesic and the logarithmic cost functions.