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.

A framework for global reliability sensitivity analysis in the presence of multi-uncertainty

“…The problem dimension, which can be handled by the approach is guided by the surrogate modelling techniques chosen in both levels. Using an arbitrary surrogate model that is capable of addressing highdimensional problems in level 1 and LRAs or the recently introduced PLS-driven PCEs [54] in level 2, it is applicable up to several thousand input variables. At a fixed accuracy (with respect to Y-model output), the cost of the global surrogate modelling strategy remains constant irrespective of the fraction of reducible and irreducible uncertainty in the model input.…”

Gaussian process metamodeling of functional-input code for coastal flood hazard assessment

“…For a full derivation of B-splines and PCA equations, the interested reader may refer to [50] and [52], respectively. Other projection methods such as PLS [53] or kPCA [22] are valid choices as well, and we encourage the inclusion of multiple projection methods in the analysis for comparison. Furthermore, we impose the projection dimension of every functional input to be the same, for simplicity of exposition.…”

Challenges of order reduction techniques for problems involving polymorphic uncertainty

“…With increasing problem dimension, the efficacy of these approaches deteriorates to varying degrees (drastically for classical PCEs, yet also for LRAs). A possible redemption is provided by PLS‐PCE, which couples a nonlinear partial least squares‐based (PLS) order reduction method with a PCE surrogate. For all of the above approaches, the sensitivity indices can be obtained immediately from postprocessing the model coefficients (see Sudret for classical and sparse PCEs, for LRAs and Ehre et al for PLS‐PCEs).…”