Numerical Methods for Stochastic Differential Equations in Stiefel Manifolds via the Cayley Transform
Numerical Methods for Stochastic Differential Equations in Stiefel Manifolds via the Cayley Transform
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通过 Cayley 变换求解 Stiefel 流形中随机微分方程的数值方法
DOI：
10.1109/cdc40024.2019.9029436
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Zhichao Wang
中科院分区：
文献类型：
--
作者：
V. Solo;Zhichao Wang
在Stiefel流形上演化的随机微分方程在科学和工程中有许多应用。对于常微分方程在Stiefel流形上的发展，有一个坚实的文献数值实现保证坚持的流形。但对于随机微分方程，数值方法还处于起步阶段。事实上，一些现有的计划未能满足所需的几何约束。基于Cayley变换，我们提出了一种新的有效的方法来模拟Stiefel流形上演化的随机微分方程。特别是，我们展示了如何构建漂移和扩散项服从几何条件，确保Stiefel流形的演变。比较模拟说明了新的计划，表明它是几何保持大量的时间步长。
Stochastic differential equations evolving in a Stiefel manifold occur in several applications in Science and Engineering. For ordinary differential equations evolving in Stiefel manifolds there is a solid literature on numerical implementation guaranteeing adherence to the manifold. But for stochastic differential equations, numerical methods are in their infancy. Indeed some existing schemes fail to satisfy the required geometric constraints. We develop a new and efficient scheme to simulate a stochastic differential equation evolving in a Stiefel manifold, based on the Cayley transform. In particular, we show how to construct drift and diffusion terms to obey geometric conditions, ensuring evolution in the Stiefel manifold. Comparative simulations illustrate the new scheme showing that it is geometry preserving over large numbers of time steps.