## 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 low-dimensional 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 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. Finally, we compare the two approaches using a test example from a mechanical application.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- University of Greifswald (Germany)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1496998

- Report Number(s):
- SAND-2018-10420J

Journal ID: ISSN 2152-5080; 672091

- Grant/Contract Number:
- AC04-94AL85000; NA0003525

- Resource Type:
- Accepted Manuscript

- Journal Name:
- International Journal for Uncertainty Quantification

- Additional Journal Information:
- Journal Volume: 8; Journal Issue: 6; Journal ID: ISSN 2152-5080

- 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 low-dimensional 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 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. 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},

number = 6,

volume = 8,

place = {United States},

year = {2018},

month = {1}

}