# Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

## Abstract

Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar ormore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22382160

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 280; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CHAOS THEORY; COMPARATIVE EVALUATIONS; CONVERGENCE; EXPANSION; HERMITE POLYNOMIALS; LEGENDRE POLYNOMIALS; MARKOV PROCESS; MATHEMATICAL SOLUTIONS; MINIMIZATION; MONTE CARLO METHOD; PARTIAL DIFFERENTIAL EQUATIONS; PROBABILISTIC ESTIMATION; RANDOMNESS

### Citation Formats

```
Hampton, Jerrad, and Doostan, Alireza, E-mail: alireza.doostan@colorado.edu.
```*Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.09.019.

```
Hampton, Jerrad, & Doostan, Alireza, E-mail: alireza.doostan@colorado.edu.
```*Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies*. United States. doi:10.1016/J.JCP.2014.09.019.

```
Hampton, Jerrad, and Doostan, Alireza, E-mail: alireza.doostan@colorado.edu. Thu .
"Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies". United States. doi:10.1016/J.JCP.2014.09.019.
```

```
@article{osti_22382160,
```

title = {Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies},

author = {Hampton, Jerrad and Doostan, Alireza, E-mail: alireza.doostan@colorado.edu},

abstractNote = {Sampling orthogonal polynomial bases via Monte Carlo is of interest for uncertainty quantification of models with random inputs, using Polynomial Chaos (PC) expansions. It is known that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion via solution to an ℓ{sub 1}-minimization problem. Utilizing results for orthogonal polynomials, we bound the coherence parameter for polynomials of Hermite and Legendre type under their respective natural sampling distribution. In both polynomial bases we identify an importance sampling distribution which yields a bound with weaker dependence on the order of the approximation. For more general orthonormal bases, we propose the coherence-optimal sampling: a Markov Chain Monte Carlo sampling, which directly uses the basis functions under consideration to achieve a statistical optimality among all sampling schemes with identical support. We demonstrate these different sampling strategies numerically in both high-order and high-dimensional, manufactured PC expansions. In addition, the quality of each sampling method is compared in the identification of solutions to two differential equations, one with a high-dimensional random input and the other with a high-order PC expansion. In both cases, the coherence-optimal sampling scheme leads to similar or considerably improved accuracy.},

doi = {10.1016/J.JCP.2014.09.019},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = ,

volume = 280,

place = {United States},

year = {2015},

month = {1}

}