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

Abstract

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. Here, 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,more » 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.« less

Authors:

Esquivel, Hugo  ^[1];

• Search DOE PAGES for author "Esquivel, Hugo"
• Search DOE PAGES for ORCID "0000-0001-7596-4849"

Prakash, Arun ^[1]; Lin, Guang  ^[2]

• Search DOE PAGES for author "Lin, Guang"
• Search DOE PAGES for ORCID "0000-0002-0976-1987"

---

+ Show Author Affiliations

1. Purdue Univ., West Lafayette, IN (United States). Lyles School of Civil Engineering
2. Purdue Univ., West Lafayette, IN (United States). School of Mechanical Engineering. Dept. of Mathematics

Publication Date:

Fri Feb 19 00:00:00 EST 2021

Research Org.:

Purdue Univ., West Lafayette, IN (United States)

Sponsoring Org.:

USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); National Science Foundation (NSF); US Army Research Office (ARO)

OSTI Identifier:

1853725

Alternate Identifier(s):

OSTI ID: 1775925

Grant/Contract Number:  

SC0021142; CNS-1136075; DMS-1555072; DMS-1736364; CMMI-1634832; CMMI-1560834; W911NF-15-1-0562

Resource Type:

Accepted Manuscript

Journal Name:

Journal of Computational Physics

Additional Journal Information:

Journal Volume: 430; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:

Elsevier

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; computer science; physics; uncertainty quantification; long-time integration; stochastic flow map; (nonlinear) stochastic dynamical systems; flow-driven spectral chaos (FSC); TD-gPC