SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos
File(s)
Accepted version
Author(s)
Ahlfeld, R
Belkouchi, B
Montomoli, F
Type
Journal Article
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 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.
Date Issued
2016-09-01
Date Acceptance
2016-05-05
Citation
Journal of Computational Physics, 2016, 320 (1), pp.1-16
ISSN
0021-9991
Publisher
Elsevier
Start Page
1
End Page
16
Journal / Book Title
Journal of Computational Physics
Volume
320
Issue
1
Copyright Statement
© 2016, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Identifier
https://www.sciencedirect.com/science/article/pii/S0021999116301516
Subjects
Science & Technology
Technology
Physical Sciences
Computer Science, Interdisciplinary Applications
Physics, Mathematical
Computer Science
Physics
Uncertainty Quantification
Non-Intrusive Polynomial Chaos
Arbitrary Polynomial Chaos
Sparse Gaussian Quadrature
Anisotropic Smolyak Grid
SAMBA
DRIVEN UNCERTAINTY QUANTIFICATION
INTERPOLATION
SIMULATIONS
SYSTEMS
STORAGE
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Applied Mathematics
Publication Status
Published
Date Publish Online
2016-05-11