Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

This paper gives an overview of the use of Polynomial Chaos expansions to represent stochastic processes in numerical simulations. Several methods are presented to perform arithmetic on, as well as to evaluate polynomial and non-polynomial functions of variables respresented with Polynomial Chaos expansions. These methods include Taylor series, a newly developed integration method as well as a sampling based spectral projection method for non-polynomial function evaluations. A detailed analysis of the accuracy of the Polynomial Chaos representations, and of the different methods for non-polynomial function evaluations is performed. It is found that the integration method offers a robust and accurate approach to evaluate non-polynomial functions, even when very high order information is present in the PC expansions.