Advertisement

Journal of Scientific Computing

, Volume 27, Issue 1–3, pp 455–464 | Cite as

Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs

  • Xiaoliang Wan
  • George Em Karniadakis
Article

Abstract

In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures

Keywords

Uncertainty polynomial chaos stochastic differential equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wiener N. (1938). The homogeneous chaos. Amer. J. Math 60:897–936MathSciNetMATHGoogle Scholar
  2. 2.
    Ghanem R.G., and Spanos P. (1991). Stochastic Finite Eelements: A Spectral Approach. Springer-Verlag, New YorkGoogle Scholar
  3. 3.
    Ghanem R.G. (1999). Stochastic finite elements for heterogeneous media with multiple random non-gaussian properties. ASCE J. Eng. Mech 125(1):26–40CrossRefGoogle Scholar
  4. 4.
    Xiu D., and Karniadakis G.E. (2002). The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput 24(2):619–644CrossRefMathSciNetGoogle Scholar
  5. 5.
    Le Maitre O.P., Njam H.N., Ghanem R.G., and Knio O.M. (2004). Uncertainty propagation using Wiener-Haar expansions. J. Comput. Phys 197:28–57CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Le Maitre O.P., Njam H.N., Ghanem R.G., and Knio O.M. (2004). Multi-resolution analysis of Wiener-type uncertainty propagation schemes. J. Comput. Phys 197:502–531CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Deb M.K., Babuska I., and Oden J.T. (2001). Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng 190:6359–6372CrossRefMathSciNetGoogle Scholar
  8. 8.
    Babuska I., and Chatzipantelidis P. (2002). On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng 191:4093–4122CrossRefMathSciNetGoogle Scholar
  9. 9.
    Babuska I., Tempone R., and Zouraris G.E. (2004). Galerkin finite element approximations of stochastic elliptic differential equations. SIAM J. Numer. Anal 42(2):800–825CrossRefMathSciNetGoogle Scholar
  10. 10.
    Frauenfelder P., Schwab C., and Todor R.A. Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Eng 194:205–228Google Scholar
  11. 11.
    Wan X., and Karniadakis G.E. (2005). An adaptive multi-element generalized polynomial chaos method for stochastic differential equations. J. Comput. Phys 209(2):617–642CrossRefMathSciNetGoogle Scholar
  12. 12.
    Cameron R.H., and Martin W.T. (1947). The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math 48:385CrossRefMathSciNetGoogle Scholar
  13. 13.
    Gautschi W. (1982). On generating orthogonal polynomials. SIAM J. Sci. Stat. Comput 3(3):289–317CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Gautschi W. (1994). Orthpol – a package of routines for generating orthogonal polynomials and gauss-type quadrature rules. ACM Trans. Math. Softw 20(1):21–62CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Schoutens W. (1999). Stochastic Processes in the Askey Scheme. PhD thesis, K.U. Leuven.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

Personalised recommendations