An adaptive dimension-reduction method-based sparse polynomial chaos expansion via sparse Bayesian learning and Bayesian model averaging

Abstract

Polynomial chaos expansion (PCE) is a popular surrogate method for stochastic uncertainty quantification (UQ). Nevertheless, when dealing with high-dimensional problems, the well-known curse of dimensionality and overfitting pose great challenges to the PCE. This study proposes an adaptive sparse learning method based on the automatic relevance determination (ARD) and the Bayesian model averaging (BMA) to solve this problem. Firstly, we construct the sparse PCE model in the framework of sparse Bayesian learning. Secondly, the dimension-reduction method (DRM) is used to reduce the size of candidate PCE bases. Different from the common univariate and bivariate DRMs, this study considers the high-dimensional components in an adaptive way. Thirdly, we propose a novel ARD method based on the analytical B-LASSO to prune the candidate PCE bases. The pruned PCE is regarded as an initial model, and then the BMA is used to refine the high-fidelity sparse PCE. Compared with the direct use of the noninformation prior, the intervention of the pruned PCE model can enhance the rationality of the prior, and improve the accuracy of the posterior estimation. We illustrate the validity of the proposed method from different angles using five classical numerical examples and solve one practical engineering problem. The results indicate that the proposed method is a good choice for UQ under limited design samples.