论文标题
从大地测量到Wasserstein梯度流的变异BDF2方案
From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows
论文作者
论文摘要
我们基于第二阶的经典落后分化公式为Wasserstein梯度流提供了时间离散化。该方案的主要组成部分是在瓦斯坦斯坦空间中的大地测量外推的概念，通常不是唯一的定义。我们为这种操作提出了几个可能的定义，并且在Fokker-Planck方程的情况下，我们证明了所得方案与极限PDE的融合。对于外推的特定选择，我们还证明了更一般的结果，即趋向于EVI流动。最后，我们提出了该方案的各种有限体积离散化，该方案在数值上都可以在时空和时间上达到二阶精度。
We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards EVI flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.
