论文标题
Wasserstein-Hamiltonian流量的时间离散
Time Discretizations of Wasserstein-Hamiltonian Flows
论文作者
论文摘要
我们研究了哈密顿系统对配备$ l^2 $ - WASSERSTEIN度量的概率密度歧管的离散化。基于离散的最佳运输理论，得出了不同权重的几个汉密尔顿系统（晶格），可以将其视为原始汉密尔顿系统的空间离散。我们证明了一致性并为这些离散化提供了近似订单。通过使用Fisher信息使系统正规化，我们推导了密度函数的显式下限，该函数可以确保可以使用符号计划来离散。此外，我们显示了这些方案的理想长时间行为，并在几个数字示例中证明了它们的表现。
We study discretizations of Hamiltonian systems on the probability density manifold equipped with the $L^2$-Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on graph (lattice) with different weights are derived, which can be viewed as spatial discretizations to the original Hamiltonian systems. We prove the consistency and provide the approximate orders for those discretizations. By regularizing the system using Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these schemes, and demonstrate their performance on several numerical examples.
