Adaptive weighted least-squares polynomial chaos expansion with basis adaptivity and sequential adaptive sampling

Abstract An efficient framework to obtain stochastic models of responses with polynomial chaos expansion (PCE) using an adaptive least-squares approach is presented in this paper. PCE is a high accuracy spectral expansion technique for uncertainty quantification; however, it is hugely affected by the curse of dimensionality with the increase in stochastic dimensions. To alleviate this effect, the basis polynomials are added in an adaptive manner, unlike selecting basis polynomials from a large predefined set in the traditional approach. Also, a refinement strategy is proposed to cull the unnecessary PCE terms based on their contribution to the variance of the response. Furthermore, a sequential optimal sampling is utilized that is capable of adding new samples based on the most recent basis polynomials and also reutilizes the old set of samples. The additional highlights of the algorithm include the implementation of weighted least-squares to reduce the effect of outliers and Kullback–Leibler Divergence to check the convergence of PCE. The algorithm has been implemented to analytical benchmark problems and a composite laminate problem. The substantial computational savings of the proposed framework compared to traditional PCE approaches and a large number of random simulations to achieve similar accuracy were demonstrated by the results.