Multi-element flow-driven spectral chaos (ME-FSC) method for uncertainty quantification of dynamical systems

TL;DR

The paper introduces the ME-FSC method, an extension of the flow-driven spectral chaos approach, designed to efficiently handle discontinuities and long-term uncertainty quantification in stochastic dynamical systems.

Contribution

It develops a multi-element spectral chaos method that partitions the probability space to better address discontinuities and nonlinearities in stochastic systems.

Findings

Effective in handling discontinuous stochastic systems

Accurately captures long-time system responses

Achieves low computational cost

Abstract

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.