Polynomial Chaos Expansion for Parametric Problems in Engineering Systems: A Review

In engineering systems with uncertain parameters, it is crucial for system analysis and control to analyze the relationship between these uncertain parameters and system outputs (or states). Nevertheless, the acquisition of this parameter-output function, termed parametric problem in this article, is not easy because it is usually a complicated implicit nonlinear function. Polynomial chaos expansion (PCE), based on the generalized Fourier expansion, is a state-of-the-art method for both parametric problems and uncertainty quantification. Its basic idea is to approximate the implicit parameter-output function with a globally optimal explicit polynomial function, which remains accurate in the case of strong nonlinearity compared with the local Taylor expansion. This article provides a review of PCE and its applications in parametric problems. In terms of PCE theory, a rigorous and complete framework of PCE is established and a detailed review of typical PCE methods is presented. In terms of applications, two kinds of general parametric problems, namely parametric nonlinear algebraic equations and parametric differential equations, as well as their examples in some engineering fields, are introduced. Furthermore, some worthwhile future works are presented at the end of this article, which may facilitate the development of PCE.