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

Abstract

For decades, uncertainty quantification techniques based on the spectral approach have been demonstrated to be computationally more efficient than the Monte Carlo method for a wide variety of problems, particularly when the dimensionality of the probability space is relatively low. The time-dependent generalized polynomial chaos (TD-gPC) is one such technique that uses an evolving orthogonal basis to better represent the stochastic part of the solution space in time. Here in this paper, we present a new numerical method that uses the concept of enriched stochastic flow maps to track the evolution of the stochastic part of the solution space in time. The computational cost of this proposed flow-driven stochastic chaos (FSC) method is an order of magnitude lower than TD-gPC for comparable solution accuracy. This gain in computational cost is realized because, unlike most existing methods, the number of basis vectors required to track the stochastic part of the solution space does not depend upon the dimensionality of the probability space. Four representative numerical examples are presented to demonstrate the performance of the FSC method for long-time integration of second-order stochastic dynamical systems in the context of stochastic dynamics of structures.

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]
+ Show Author Affiliations
  1. Purdue University, West Lafayette, IN (United States)
Publication Date:
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); Army Research Office (ARO)
OSTI Identifier:
1976899
Alternate Identifier(s):
OSTI ID: 1787714
Grant/Contract Number:  
SC0021142; CNS-1136075; DMS-1555072; DMS-1736364; DMS-205374; CMMI-1634832; CMMI-1560834; 382247; W911NF-15-1-0562
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 398; Journal Issue: C; Journal ID: ISSN 0377-0427
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
uncertainty quantification; long-time integration; stochastic flow map; stochastic dynamics of structures; flow-driven spectral chaos; FSC; TD-gPC

Citation Formats

Esquivel, Hugo, Prakash, Arun, and Lin, Guang. Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems. United States: N. p., 2021. Web. doi:10.1016/j.cam.2021.113674.
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Esquivel, Hugo, Prakash, Arun, & Lin, Guang. Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems. United States. https://doi.org/10.1016/j.cam.2021.113674
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Esquivel, Hugo, Prakash, Arun, and Lin, Guang. Mon . "Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems". United States. https://doi.org/10.1016/j.cam.2021.113674. https://www.osti.gov/servlets/purl/1976899.
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@article{osti_1976899,
title = {Flow-driven spectral chaos (FSC) method for long-time integration of second-order stochastic dynamical systems},
author = {Esquivel, Hugo and Prakash, Arun and Lin, Guang},
abstractNote = {For decades, uncertainty quantification techniques based on the spectral approach have been demonstrated to be computationally more efficient than the Monte Carlo method for a wide variety of problems, particularly when the dimensionality of the probability space is relatively low. The time-dependent generalized polynomial chaos (TD-gPC) is one such technique that uses an evolving orthogonal basis to better represent the stochastic part of the solution space in time. Here in this paper, we present a new numerical method that uses the concept of enriched stochastic flow maps to track the evolution of the stochastic part of the solution space in time. The computational cost of this proposed flow-driven stochastic chaos (FSC) method is an order of magnitude lower than TD-gPC for comparable solution accuracy. This gain in computational cost is realized because, unlike most existing methods, the number of basis vectors required to track the stochastic part of the solution space does not depend upon the dimensionality of the probability space. Four representative numerical examples are presented to demonstrate the performance of the FSC method for long-time integration of second-order stochastic dynamical systems in the context of stochastic dynamics of structures.},
doi = {10.1016/j.cam.2021.113674},
journal = {Journal of Computational and Applied Mathematics},
number = C,
volume = 398,
place = {United States},
year = {Mon May 31 00:00:00 EDT 2021},
month = {Mon May 31 00:00:00 EDT 2021}
}
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https://doi.org/10.1016/j.cam.2021.113674
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