A Minimum Action Method for Dynamical Systems with Constant Time Delays
A Minimum Action Method for Dynamical Systems with Constant Time Delays
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DOI：
10.1137/20m1349163
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
X. Wan;Jiayu Zhai
中科院分区：
文献类型：
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作者：
X. Wan;Jiayu Zhai
在这项工作中，我们构建了具有恒定时滞的动力系统的最小作用方法。最小作用方法（MAM）在寻找小噪声引起的最可能的转移路径方面发挥着重要作用。最小作用量法存在两种表述：一种是基于莫佩尔蒂原理的几何表述，另一种是时间表述。几何公式依赖于与 Freidlin-Wentzell 作用泛函相对应的哈密顿量守恒。对于具有时滞的系统，由于对时滞的显式依赖，哈密顿量不守恒，这意味着几何 MAM 不适用。我们使用 MAM 的时间公式来解决时间延迟问题。通过定义辅助路径，我们通过最优线性时间缩放消除了对时间的优化。辅助路径和延迟转移路径之间的逐点对应关系通过包含在动作泛函中的惩罚项来处理。然后通过有限元方法对作用函数进行离散化，并开发了 h 自适应网格细化策略。数值例子证明了我们算法的有效性。
In this work, we construct a minimum action method for dynamical systems with constant time delays. The minimum action method (MAM) plays an important role in seeking the most probable transition pathway induced by small noise. There exist two formulations of the minimum action method: one is the geometric formulation based on the Maupertuis principle, and the other one is the temporal formulation. The geometric formulation relies on the conservation of Hamiltonian corresponding to the Freidlin–Wentzell action functional. For systems with time delays, the Hamiltonian does not conserve due to the explicit dependence on the time delay, which implies that the geometric MAM is not applicable. We work with the temporal formulation of MAM for problems with time delays. By defining an auxiliary path, we remove the optimization with respect to time through the optimal linear time scaling. The pointwise correspondence between the auxiliary path and the delayed transition path is dealt with by a penalty term included into the action functional. The action functional is then discretized by the finite element method, and strategies for h-adaptive mesh refinement have been developed. Numerical examples have been presented to demonstrate the effectiveness of our algorithm.