SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

JOURNAL OF COMPUTATIONAL PHYSICS

卷:320

Article

Ahlfeld, R.¹  Belkouchi, B.¹  Montomoli, F.¹ 

[1] Univ London Imperial Coll Sci Technol & Med, Dept Aeronaut Engn, London SW7 2AZ, England

关键词: Uncertainty Quantification;    Non-Intrusive Polynomial Chaos;    Arbitrary Polynomial Chaos;    Sparse Gaussian Quadrature;    Anisotropic Smolyak Grid;    SAMBA;   

DOI  :  10.1016/j.jcp.2016.05.014

来源: Elsevier

【 摘 要 】

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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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