Abstract
Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).
Highlights
Uncertainty is pervasive in science and engineering problems, for which models are widely used to study dynamic behaviors of complex systems [1]
The uncertainty quantification (UQ) algorithm in this work can be summarized as follows, which integrates the modified Gram-Schmidt (GS) orthogonalization, generalized dimension reduction method, and quadrature rules to quickly calculate the Polynomial chaos expansion (PCE) coefficients of model predictions
This paper presents an algorithm for efficient uncertainty quantification (UQ) under many parametric uncertainties that follow arbitrary distributions, which are not considered in the classic polynomial chaos expansion
Summary
Uncertainty is pervasive in science and engineering problems, for which models are widely used to study dynamic behaviors of complex systems [1]. A sparse grid constructed by the Smolyak algorithm [26] was shown to be efficient, but the computational cost and the UQ accuracy are related to how the collocation points are selected [15,28] To address these issues, we have developed an algorithm [29] to combine the generalized dimension reduction method (gDRM), namely S-variate DRM [30], with the PCE. As compared to the non-intrusive methods, attempts to generate sparse collocation points in our algorithm are not required, since the total number of variables in low-dimensional integrals is small This can save modelling efforts for UQ under many parametric uncertainties. Since uncertainty in parameters affects model output, v can be approximated with orthogonal polynomial basis and PCE coefficients as in [24]: M v(ξ) = ∑ vmΨm(ξ). Once the PCE coefficients of v, i.e., {vm} are available, the mean and variance of v can be quickly calculated as discussed in [24]
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