Recent Advances in Polynomial Chaos Expansion: Theory, Applications and Software

Abstract

he paper is focused on recent advances in uncertainty quantification using polynomial chaos expansion (PCE). PCE is a well-known technique for approximation of costly mathematical models with ran- dom inputs – surrogate model. Although PCE is a widely used technique and it has several advantages over various surrogate models, it has still several lim- itations and research gaps. This paper reviews some of the recent theoretical developments in PCE. Specif- ically a new active learning method optimizing the ex- perimental design and an extension of analytical sta- tistical analysis using PCE will be reviewed. These two topics represent crucial tools for efficient applications: active learning leads generally to a significantly more efficient construction of PCE and improved statistical analysis allows for analytical estimation of higher sta- tistical moments directly from PCE coefficients. Higher statistical moments can be further used for the iden- tification of probability distribution and estimation of design quantiles, which is a crucial task for the proba- bilistic analysis of structures. Selected applications of the theoretical methods are briefly presented in a con- text of civil engineering as well as some preliminary results of further research. A part of the paper also presents UQPy package containing state-of-the-art im- plementation of the PCE theory.