PLS-based adaptation for efficient PCE representation in high dimensions

Abstract

Uncertainty quantification of engineering systems modeled by computationally intensive numerical models remains a challenging task, despite the increase in computer power. Efficient uncertainty propagation of such models can be performed by use of surrogate models, such as polynomial chaos expansions (PCE). A major drawback of standard PCE is that its predictive ability decreases with increase of the problem dimension for a fixed computational budget. This is related to the fact that the number of terms in the expansion increases fast with the input variable dimension. To address this issue, Tipireddy and Ghanem (2014) introduced a sparse PCE representation based on a transformation of the coordinate system in Gaussian input variable spaces. In this contribution, we propose to identify the projection operator underlying this transformation and approximate the coefficients of the resulting PCE through partial least squares (PLS) analysis. The proposed PCE-driven PLS algorithm identifies the directions with the largest predictive significance in the PCE representation based on a set of samples from the input random variables and corresponding response variable. This approach does not require gradient evaluations, which makes it efficient for high dimensional problems with black-box numerical models. We assess the proposed approach with three numerical examples in high-dimensional input spaces, comparing its performance with low-rank tensor approximations. These examples demonstrate that the PLS-based PCE method provides accurate representations even for strongly non-linear problems.