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

by   Hugo Esquivel, et al.

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.


page 16

page 19

page 22


Flow-driven spectral chaos (FSC) method for simulating long-time dynamics of arbitrary-order non-linear stochastic dynamical systems

Uncertainty quantification techniques such as the time-dependent general...

Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems

For decades, uncertainty quantification techniques based on the spectral...

Sequential Bayesian experimental design for estimation of extreme-event probability in stochastic dynamical systems

We consider a dynamical system with two sources of uncertainties: (1) pa...

Probabilistic Evolution of Stochastic Dynamical Systems: A Meso-scale Perspective

Stochastic dynamical systems arise naturally across nearly all areas of ...

Toward computing sensitivities of average quantities in turbulent flows

Chaotic dynamical systems such as turbulent flows are characterized by a...

Long-time predictive modeling of nonlinear dynamical systems using neural networks

We study the use of feedforward neural networks (FNN) to develop models ...