Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs

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

[1]  W. T. Martin,et al.  The Orthogonal Development of Non-Linear Functionals in Series of Fourier-Hermite Functionals , 1947 .

[2]  P. Frauenfelder,et al.  Finite elements for elliptic problems with stochastic coefficients , 2005 .

[3]  W. Gautschi.  On Generating Orthogonal Polynomials , 1982 .

[4]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[5]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[6]  N. Cutland,et al.  On homogeneous chaos , 1991, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  I. Babuska,et al.  Solution of stochastic partial differential equations using Galerkin finite element techniques , 2001 .

[8]  R. Ghanem.  Stochastic Finite Elements For Heterogeneous Media with Multiple Random Non-Gaussian Properties , 1997 .

[9]  R. Ghanem,et al.  Uncertainty propagation using Wiener-Haar expansions , 2004 .

[10]  Roger Ghanem,et al.  Stochastic Finite Elements with Multiple Random Non-Gaussian Properties , 1999 .

[11]  Raúl Tempone,et al.  Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations , 2004, SIAM J. Numer. Anal..

[12]  WALTER GAUTSCHI.  Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules , 1994, TOMS.

[13]  R. Ghanem,et al.  Multi-resolution analysis of wiener-type uncertainty propagation schemes , 2004 .

[14]  Ivo Babuška,et al.  On solving elliptic stochastic partial differential equations , 2002 .

[15]  G. Karniadakis,et al.  An adaptive multi-element generalized polynomial chaos method for stochastic differential equations , 2005 .