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.

MSC codes

  1. 60G99
  2. 65C20
  3. 33C45
  4. 37H10

MSC codes

  1. polynomial chaos
  2. stochastic
  3. spectral uncertainty quantification

Get full access to this article

View all available purchase options and get full access to this article.


Serial Fortran Solvers for ODE Initial Value Problems, http://www.llnl.gov/CASC/odepack/ (2002).
Slatec Common Mathematical Library, Version 4.1, http://www.netlib.org/slatec/ (1993).
G. Andrews and R. Askey, Classical orthogonal polynomials, in Polyn⊚mes Orthogonaux et Applications, C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and P. Ronveaux, eds., Lecture Notes in Math. 1171, Springer, Berlin, 1985, pp. 36–62.
Richard Askey, James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc., 54 (1985), 0–0iv+55
R. Cameron, W. Martin, The orthogonal development of non‐linear functionals in series of Fourier‐Hermite functionals, Ann. of Math. (2), 48 (1947), 385–392
Alexandre Chorin, Hermite expansions in Monte‐Carlo computation, J. Computational Phys., 8 (1971), 472–482
Alexandre Chorin, Gaussian fields and random flow, J. Fluid Mech., 63 (1974), 21–32
B. Debusschere, H. Najm, A. Matta, O. Knio, R. Ghanem, and O. Le Maître, Protein labeling reactions in electrochemical microchannel flow: Numerical simulation and uncertainty propagation, Phys. Fluids, 15 (2003), pp. 2238–2250.
B. Debusschere, H. Najm, A. Matta, O. Knio, R. Ghanem, and O. Le Maître, Study of sample dispersion mechanisms in an electroosmotically pumped microchannel, in Technical Proceedings of the 2003 Nanotechnology Conference and Trade Show, Vol. 1, Computational Publications, Boston, 2003, pp. 154–157.
B. Debusschere, H. Najm, A. Matta, T. Shu, O. Knio, R. Ghanem, and O. Le Maître, Uncertainty quantification in a reacting electrochemical microchannel flow model, in Proceedings of the 5th International Conference on Modeling and Simulation of Microsystems, Computational Publications, Boston, 2002, pp. 384–387.
R. Field, Jr and M. Grigoriu, On the accuracy of the polynomial chaos approximation, Probab. Engrg. Mech., 19 (2004), pp. 65–80.
R. Ghanem, Probabilistic characterization of transport in heterogeneous media, Comput. Methods Appl. Mech. Engrg., 158 (1998), pp. 199–220.
Roger Ghanem, Ingredients for a general purpose stochastic finite elements implementation, Comput. Methods Appl. Mech. Engrg., 168 (1999), 19–34
R. Ghanem, The nonlinear Gaussian spectrum of log‐normal stochastic processes and variables, ASME J. Appl. Mech., 66 (1999), pp. 964–973.
R. Ghanem, Stochastic finite elements for heterogeneous media with multiple random non‐Gaussian properties, ASCE J. Engrg. Mech., 125 (1999), pp. 26–40.
R. Ghanem, J. Red‐Horse, and A. Sarkar, Modal properties of a space‐frame with localized system uncertainties, in Proceedings of the 8th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, A. Kareem, A. Haldar, B. F. Spencer, Jr., and E. A. Johnson, eds., American Society of Civil Engineers, Reston, VA, 2000, paper PMC200‐269.
Roger Ghanem, Pol Spanos, Stochastic finite elements: a spectral approach, Springer‐Verlag, 1991x+214
Alan Hindmarsh, ODEPACK, a systematized collection of ODE solvers, IMACS Trans. Sci. Comput., I, IMACS, New Brunswick, NJ, 1983, 55–64
Olivier Le Maître, Omar Knio, Habib Najm, Roger Ghanem, A stochastic projection method for fluid flow. I. Basic formulation, J. Comput. Phys., 173 (2001), 481–511
Olivier Le Maître, Matthew Reagan, Habib Najm, Roger Ghanem, Omar Knio, A stochastic projection method for fluid flow. II. Random process, J. Comput. Phys., 181 (2002), 9–44
F. Maltz, D. Hitzl, Variance reduction in Monte Carlo computations using multidimensional Hermite polynomials, J. Comput. Phys., 32 (1979), 345–376
W. Meecham and D. Jeng, Use of the Wiener‐Hermite expansion for nearly normal turbulence, J. Fluid Mech., 32 (1968), pp. 225–249.
David Nualart, The Malliavin calculus and related topics, Probability and its Applications (New York), Springer‐Verlag, 1995xii+266
M. Reagan, H. Najm, R. Ghanem, and O. Knio, Uncertainty quantification in reacting flow simulations through non‐intrusive spectral projection, Combustion and Flame, 132 (2003), pp. 545–555.
S. Sakamoto and R. Ghanem, Polynomial chaos decomposition for the simulation of non‐Gaussian non‐stationary stochastic processes, ASCE J. Engrg. Mech., 128 (2002), pp. 190–201.
Wim Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statistics, Vol. 146, Springer‐Verlag, 2000xiv+163
W. Vandevender and K. Haskell, The SLATEC mathematical subroutine library, SIGNUM Newsletter, 17 (1982), pp. 16–21.
N. Wiener, The homogenous chaos, Amer. J. Math., 60 (1938), pp. 897–936.
Dongbin Xiu, George Karniadakis, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), 4927–4948
Dongbin Xiu, George Karniadakis, The Wiener‐Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), 619–644
D. Xiu, D. Lucor, C.‐H. Su, and G. Karniadakis, Stochastic modeling of flow‐structure interactions using generalized polynomial chaos, ASME J. Fluids Engrg., 124 (2002), pp. 51–59.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 698 - 719
ISSN (online): 1095-7197


Published online: 26 July 2006

MSC codes

  1. 60G99
  2. 65C20
  3. 33C45
  4. 37H10

MSC codes

  1. polynomial chaos
  2. stochastic
  3. spectral uncertainty quantification



Olivier P. Le Maı⁁tre

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.







Copy the content Link

Share with email

Email a colleague

Share on social media