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International Journal for Uncertainty Quantification
IF: 0.967 5-Year IF: 1.301 SJR: 0.531 SNIP: 0.8 CiteScore™: 1.52

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2018026902
pages 495-510

TIME AND FREQUENCY DOMAIN METHODS FOR BASIS SELECTION IN RANDOM LINEAR DYNAMICAL SYSTEMS

John D. Jakeman
Center for Computing Research, Sandia National Laboratories, P.O. Box 5800, MS 1318, Albuquerque, New Mexico, 87185-1320, USA
Roland Pulch
Department of Mathematics and Computer Science, Ernst-Moritz-Arndt-Universitat Greifswald, Walther-Rathenau-Strasse 47, D-17487 Greifswald, Germany

ABSTRACT

Polynomial chaos methods have been extensively used to analyze systems in uncertainty quantification. Furthermore, several approaches exist to determine a low-dimensional approximation (or sparse approximation) for some quantity of interest in a model, where just a few orthogonal basis polynomials are required. We consider linear dynamical systems consisting of ordinary differential equations with random variables. The aim of this paper is to explore methods for producing low-dimensional approximations of the quantity of interest further. We investigate two numerical techniques to compute a low-dimensional 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. We compare the two approaches using a test example from a mechanical application.