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

12/02/2020
by   Hugo Esquivel, et al.
0

Uncertainty quantification techniques such as the time-dependent generalized polynomial chaos (TD-gPC) use an adaptive orthogonal basis to better represent the stochastic part of the solution space (aka random function space) in time. However, because the random function space is constructed using tensor products, TD-gPC-based methods are known to suffer from the curse of dimensionality. In this paper, we introduce a new numerical method called the 'flow-driven spectral chaos' (FSC) which overcomes this curse of dimensionality at the random-function-space level. The proposed method is not only computationally more efficient than existing TD-gPC-based methods but is also far more accurate. The FSC method uses the concept of 'enriched stochastic flow maps' to track the evolution of a finite-dimensional random function space efficiently in time. To transfer the probability information from one random function space to another, two approaches are developed and studied herein. In the first approach, the probability information is transferred in the mean-square sense, whereas in the second approach the transfer is done exactly using a new theorem that was developed for this purpose. The FSC method can quantify uncertainties with high fidelity, especially for the long-time response of stochastic dynamical systems governed by ODEs of arbitrary order. Six representative numerical examples, including a nonlinear problem (the Van-der-Pol oscillator), are presented to demonstrate the performance of the FSC method and corroborate the claims of its superior numerical properties. Finally, a parametric, high-dimensional stochastic problem is used to demonstrate that when the FSC method is used in conjunction with Monte Carlo integration, the curse of dimensionality can be overcome altogether.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 27

page 28

05/21/2021

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...
10/15/2019

A neural network approach for uncertainty quantification for time-dependent problems with random parameters

In this work we propose a numerical framework for uncertainty quantifica...
02/22/2021

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...
11/01/2021

Interpolatory tensorial reduced order models for parametric dynamical systems

The paper introduces a reduced order model (ROM) for numerical integrati...
04/11/2020

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

Stochastic dynamical systems arise naturally across nearly all areas of ...
02/06/2019

Toward computing sensitivities of average quantities in turbulent flows

Chaotic dynamical systems such as turbulent flows are characterized by a...
10/17/2018

Path-based measures of expansion rates and Lagrangian transport in stochastic flows

We develop a systematic information-theoretic framework for a probabilis...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.