Hierarchical adaptive polynomial chaos expansions

TL;DR

This paper introduces a hierarchical adaptive polynomial chaos expansion method that leverages model structure and heredity principles to improve accuracy and reduce computational costs in high-dimensional uncertainty quantification.

Contribution

It proposes a novel hierarchical adaptive sparse PCE approach that exploits model structure and heredity principles for more efficient uncertainty quantification.

Findings

Reduces computational burden in high-dimensional problems

Achieves higher accuracy with the same computational budget

Effectively exploits hierarchical model structures

Abstract

Polynomial chaos expansions (PCE) are widely used in the framework of uncertainty quantification. However, when dealing with high dimensional complex problems, challenging issues need to be faced. For instance, high-order polynomials may be required, which leads to a large polynomial basis whereas usually only a few of the basis functions are in fact significant. Taking into account the sparse structure of the model, advanced techniques such as sparse PCE (SPCE), have been recently proposed to alleviate the computational issue. In this paper, we propose a novel approach to SPCE, which allows one to exploit the model's hierarchical structure. The proposed approach is based on the adaptive enrichment of the polynomial basis using the so-called principle of heredity. As a result, one can reduce the computational burden related to a large pre-defined candidate set while obtaining higher accuracy with the same computational budget.