Polynomial meta-models with canonical low-rank approximations: numerical insights and comparison to sparse polynomial chaos expansions

TL;DR

This paper compares polynomial meta-models using canonical low-rank approximations and sparse polynomial chaos expansions, showing LRA's advantages in high-dimensional, data-scarce scenarios and for predicting extreme responses.

Contribution

It provides a detailed numerical comparison between canonical LRA and sparse PCE, highlighting LRA's efficiency and accuracy in specific high-dimensional uncertainty quantification tasks.

Findings

Canonical LRA has smaller errors with limited model evaluations.

LRA outperforms sparse PCE in predicting extreme responses.

LRA is more efficient in high-dimensional problems with scarce data.

Abstract

The growing need for uncertainty analysis of complex computational models has led to an expanding use of meta-models across engineering and sciences. The efficiency of meta-modeling techniques relies on their ability to provide statistically-equivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing meta-models in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the "curse of dimensionality", namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing meta-models with polynomial functions in high-dimensional problems is offered by the newly emerged low-rank approximations (LRA) approach. By exploiting the tensor-product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input dimension. By introducing the conditional generalization error, we further demonstrate that canonical LRA tend to outperform sparse PCE in the prediction of extreme model responses.