Second order hierarchical partial least squares regression-polynomial chaos expansion for global sensitivity and reliability analyses of high-dimensional models

TL;DR

Uncertainty quantification for high-dimensional structural models is challenged by the curse of dimensionality and multicollinearity in polynomial chaos expansions. The authors propose a second-order hierarchical PLSR-PCE (SOHPLSR-PCE) that partitions polynomial terms by interaction and nonlinearity, extracts latent variables with partial least squares regression, and automatically selects optimal orders via cross-validation, enabling Sobol indices to be computed from the expansion coefficients. The approach delivers substantial dimension reduction, memory savings, and dramatic efficiency gains while maintaining accuracy, as demonstrated on three finite-element case studies against the conventional OLSR-PCE. This framework advances practical global sensitivity and reliability analyses for complex, high-dimensional stochastic systems and offers a path toward uncovering latent hierarchical structure in HDMR-like representations.

Abstract

To tackle the curse of dimensionality and multicollinearity problems of polynomial chaos expansion for analyzing global sensitivity and reliability of models with high stochastic dimensions, this paper proposes a novel non-intrusive algorithm called second order hierarchical partial least squares regression-polynomial chaos expansion. The first step of the innovative algorithm is to divide the polynomials into several groups according to their interaction degrees and nonlinearity degrees, which avoids large data sets and reflects the relationship between polynomial chaos expansion and high dimensional model representation. Then a hierarchical regression algorithm based on partial least squares regression is devised for extracting latent variables from each group at different variable levels. The optimal interaction degree and the corresponding nonlinearity degrees are automatically estimated with an improved cross validation scheme. Based on the relationship between variables at two adjacent levels, Sobol' sensitivity indices can be obtained by a simple post-processing of expansion coefficients. Thus, the expansion is greatly simplified through retaining the important inputs, leading to accurate reliability analysis without requirements of additional model evaluations. Finally, finite element models with three different types of structures verified that the proposed method can greatly improve the computational efficiency compared with the ordinary least squares regression-based method.

Figures (14)

• Figure 1: Illustration of first order hierarchical partial least squares regression algorithm
• Figure 2: Configuration and loads of a simply supported beam
• Figure 3: Comparison of the main Sobol indices
• Figure 4: Comparison of the total Sobol indices
• Figure 5: Distributions of order of magnitudes of relative errors of Sobol indices