Time and Frequency Domain Methods for Basis Selection in Random Linear Dynamical Systems

TL;DR

This paper compares frequency and time domain methods for selecting basis functions in polynomial chaos expansions to efficiently approximate quantities of interest in random linear dynamical systems.

Contribution

It introduces and compares two numerical techniques—frequency domain analysis and sparse minimization—for basis selection in uncertainty quantification of dynamical systems.

Findings

Frequency domain approach effectively identifies basis polynomials.

Sparse minimization provides a computationally efficient alternative.

Both methods yield comparable low-dimensional approximations.

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.