A novel polynomial dimension decomposition method based on sparse Bayesian learning and Bayesian model averaging

Abstract

The polynomial dimensional decomposition (PDD), an orthogonal polynomial-based metamodel, has received increasing attention in uncertainty quantification (UQ). Nevertheless, for complex high-dimensional problems, its computational burden may become unaffordable, which is usually called curse of dimensionality. To solve this problem, sparse regression methods can be considered to establish a sparse PDD model. However, when the design samples are limited, their computational accuracy may be low due to the enormous size of the polynomial bases. Aimed at this issue, we proposed a novel sparse PDD metamodel based on the Bayesian LASSO (least absolute shrinkage and selection operator) method and an adaptive candidate basis selection and model updating method (CBSMU). Firstly, to improve the CPU efficiency, an analytical Bayesian LASSO based on the sparse Bayesian learning is used for the regression analysis, which replaces the time-consuming Markov chain Monte Carlo sampling of the traditional method with the efficient iteration algorithm for calculating the posterior estimations. Then, to reduce the size of the polynomial bases, this study proposes the adaptive CBSMU for screening the significant candidate polynomial bases and updating the metamodel sequentially. The CBSMU can find out the candidate bases that contributes to improve the prediction accuracy in the view of Bayesian model averaging. Thus, during the process of the sparse PDD modeling, the size of candidate bases is relatively small, which facilitates to improve the final computational accuracy when the design samples are limited. We verify the proposed method using three high-dimensional numerical examples, and apply it to solve one complex high-dimensional engineering problem. The results show that the proposed method is more accurate for UQ than the two common methods with the same computational costs, and is well-suited for solving complex high-dimensional structural dynamic problem.