Numerical challenges in the use of polynomial chaos representations for stochastic processes

Bert J. Debusschere*, Habib N. Najm, Philippe P. Pébayt, Omar M. Knio, Roger G. Ghanem, Olivier P. Le Maître

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

362 Scopus citations


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

Original languageEnglish (US)
Pages (from-to)698-719
Number of pages22
JournalSIAM Journal on Scientific Computing
Issue number2
StatePublished - 2005
Externally publishedYes


  • Polynomial chaos
  • Spectral uncertainty quantification
  • Stochastic

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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