论文标题
多元素流动频谱混乱（ME-FSC）方法用于动态系统的不确定性定量
Multi-element flow-driven spectral chaos (ME-FSC) method for uncertainty quantification of dynamical systems
论文作者
论文摘要
流动频谱混乱（FSC）是一种最近开发的方法，用于使用光谱方法在随机动力学系统的长期响应中跟踪和量化不确定性。该方法使用一种称为“富集随机流图”的新颖概念作为构建不断发展的有限维随机函数空间的手段，该空间既准确又及时地有效。在本文中，我们提出了FSC方法的多元素版本（简称ME-FSC方法）来处理（主要）那些在概率空间上固有不连续的动态系统。在ME-FSC中，随机域被分配为几个元素，然后使用FSC方法在每个随机元素上分别解决问题。随后，将结果汇总以使用总概率定律计算感兴趣的概率力矩。为了证明ME-FSC方法在处理不连续性和随机动力学系统的长期整合方面的有效性，本文介绍了四个代表性的数值示例，包括Van-der-Pol振荡器问题和Kraichnan-Ordzag三模式问题。结果表明，ME-FSC方法能够解决与概率空间相对于概率空间具有强大非线性依赖性的问题，无论是可靠的计算成本和低计算成本。
The flow-driven spectral chaos (FSC) is a recently developed method for tracking and quantifying uncertainties in the long-time response of stochastic dynamical systems using the spectral approach. The method uses a novel concept called 'enriched stochastic flow maps' as a means to construct an evolving finite-dimensional random function space that is both accurate and computationally efficient in time. In this paper, we present a multi-element version of the FSC method (the ME-FSC method for short) to tackle (mainly) those dynamical systems that are inherently discontinuous over the probability space. In ME-FSC, the random domain is partitioned into several elements, and then the problem is solved separately on each random element using the FSC method. Subsequently, results are aggregated to compute the probability moments of interest using the law of total probability. To demonstrate the effectiveness of the ME-FSC method in dealing with discontinuities and long-time integration of stochastic dynamical systems, four representative numerical examples are presented in this paper, including the Van-der-Pol oscillator problem and the Kraichnan-Orszag three-mode problem. Results show that the ME-FSC method is capable of solving problems that have strong nonlinear dependencies over the probability space, both reliably and at low computational cost.
