# SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

## Abstract

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version ofmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22572343

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 320; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ACCURACY; ALGORITHMS; APPROXIMATIONS; CHAOS THEORY; DATA ANALYSIS; DISTRIBUTION; EFFICIENCY; NONLINEAR PROBLEMS; POLYNOMIALS; PROBABILITY; PROBABILITY DENSITY FUNCTIONS; QUADRATURES; STATISTICAL DATA; STOCHASTIC PROCESSES

### Citation Formats

```
Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk, Belkouchi, B., and Montomoli, F.
```*SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.05.014.

```
Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk, Belkouchi, B., & Montomoli, F.
```*SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos*. United States. doi:10.1016/J.JCP.2016.05.014.

```
Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk, Belkouchi, B., and Montomoli, F. Thu .
"SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos". United States.
doi:10.1016/J.JCP.2016.05.014.
```

```
@article{osti_22572343,
```

title = {SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos},

author = {Ahlfeld, R., E-mail: r.ahlfeld14@imperial.ac.uk and Belkouchi, B. and Montomoli, F.},

abstractNote = {A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.},

doi = {10.1016/J.JCP.2016.05.014},

journal = {Journal of Computational Physics},

number = ,

volume = 320,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 2016},

month = {Thu Sep 01 00:00:00 EDT 2016}

}