Sparse Polynomial Chaos Expansion based on Fully Adaptive Forward-Backward Selection Method

Uncertainties consequentially exist in almost all of engineering and physical problems. These uncertainties may cause the system performance to change or fluctuate, or even cause severe deviation and result in unanticipated or even unprecedented function fault and mission failure. As an efficient uncertainty quantification (UQ) methodology for moment propagation and probability analysis of quantities of interest (QoI's), polynomial chaos (PC) expansions have received broad and sustained attentions. However, the exponentially increasing cost of building PC representations with increasing dimension of uncertainty, i.e., the curse of dimensionality, seriously restricts the practical application of PC at the industrial level. Some efficient strategies applying adaptive basis selection algorithm for sparse optimization (or l_1-minimization) of PC show great potential compared to the classical full PC. However, these strategies mainly focus on forward selection algorithms, which are incapable of correcting any error made by these algorithms. Hence, a novel adaptive forward-backward selection (AFBS) algorithm has been developed for reconstructing sparse PC. The proposed algorithm by a reasonable combination of forward selection and adaptively backward elimination technique can efficiently correct mistakes made by earlier forward selection steps, which retains the most significant PC terms and discards redundant or insignificant ones. The proposed algorithm was first proposed in reference (Zhao, H., Gao, Z., et al. "An efficient adaptive forward-backward selection method for sparse polynomial chaos expansion," Computer Methods in Applied Mechanics and Engineering Vol. 355, 2019, pp. 456-491.). In this paper a fully adaptive forward-backward selection (FAFBS) algorithm is proposed by involving an efficient optimization search algorithm for adaptively selecting the optimal sparse PC. The developed FAFBS method is investigated by several analytical functions and a challenging aerodynamic analysis application. The results demonstrate that the proposed FAFBS method can efficiently identify the significant PC contributions describing the problems, and reproduce sparser PC metamodel and more accurate UQ results than classical orthogonal matching pursuit (OMP) and full PC methods.