Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems
Abstract
Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a lowdimensional approximation (or sparse approximation) for some quantity of interest in a model, where just a few orthogonal basis polynomials are required. In this work, we consider linear dynamical systems consisting of ordinary differential equations with random variables. The aim of this paper is to explore methods for producing lowdimensional approximations of the quantity of interest further. We investigate two numerical techniques to compute a lowdimensional representation, which both fit the approximation to a set of samples in the time domain. On the one hand, a frequency domain analysis of a stochastic Galerkin system yields the selection of the basis polynomials. It follows a linear least squares problem. On the other hand, a sparse minimization yields the choice of the basis polynomials by information from the time domain only. An orthogonal matching pursuit produces an approximate solution of the minimization problem. Finally, we compare the two approaches using a test example from a mechanical application.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 University of Greifswald (Germany)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1496998
 Report Number(s):
 SAND201810420J
Journal ID: ISSN 21525080; 672091
 Grant/Contract Number:
 AC0494AL85000; NA0003525
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 International Journal for Uncertainty Quantification
 Additional Journal Information:
 Journal Volume: 8; Journal Issue: 6; Journal ID: ISSN 21525080
 Publisher:
 Begell House
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; linear dynamical system; random variable; orthogonal basis; polynomial chaos; stochastic Galerkin method; least squares problem; orthogonal matching pursuit; uncertainty quantification
Citation Formats
Jakeman, John Davis, and Pulch, Roland. Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems. United States: N. p., 2018.
Web. doi:10.1615/Int.J.UncertaintyQuantification.2018026902.
Jakeman, John Davis, & Pulch, Roland. Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems. United States. doi:10.1615/Int.J.UncertaintyQuantification.2018026902.
Jakeman, John Davis, and Pulch, Roland. Mon .
"Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems". United States. doi:10.1615/Int.J.UncertaintyQuantification.2018026902. https://www.osti.gov/servlets/purl/1496998.
@article{osti_1496998,
title = {Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems},
author = {Jakeman, John Davis and Pulch, Roland},
abstractNote = {Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a lowdimensional approximation (or sparse approximation) for some quantity of interest in a model, where just a few orthogonal basis polynomials are required. In this work, we consider linear dynamical systems consisting of ordinary differential equations with random variables. The aim of this paper is to explore methods for producing lowdimensional approximations of the quantity of interest further. We investigate two numerical techniques to compute a lowdimensional representation, which both fit the approximation to a set of samples in the time domain. On the one hand, a frequency domain analysis of a stochastic Galerkin system yields the selection of the basis polynomials. It follows a linear least squares problem. On the other hand, a sparse minimization yields the choice of the basis polynomials by information from the time domain only. An orthogonal matching pursuit produces an approximate solution of the minimization problem. Finally, we compare the two approaches using a test example from a mechanical application.},
doi = {10.1615/Int.J.UncertaintyQuantification.2018026902},
journal = {International Journal for Uncertainty Quantification},
issn = {21525080},
number = 6,
volume = 8,
place = {United States},
year = {2018},
month = {1}
}
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