On the accuracy of the polynomial chaos approximation

Polynomial chaos representations for non-Gaussian random variables and stochastic processes are infinite series of Hermite polynomials of standard Gaussian random variables with deterministic coefficients. Finite truncations of these series are referred to as polynomial chaos (PC) approximations. This paper explores features and limitations of PC approximations. Metrics are developed to assess the accuracy of the PC approximation. A collection of simple, but relevant examples is examined in this paper. The number of terms in the PC approximations used in the examples exceeds the number of terms retained in most current applications. For the examples considered, it is demonstrated that (1) the accuracy of the PC approximation improves in some metrics as additional terms are retained, but does not exhibit this behavior in all metrics considered in the paper, (2) PC approximations for strictly stationary, non-Gaussian stochastic processes are initially nonstationary and gradually may approach weak stationarity as the number of terms retained increases, and (3) the development of PC approximations for certain processes may become computationally demanding, or even prohibitive, because of the large number of coefficients that need to be calculated. However, there have been many applications in which PC approximations have been successful.