Computational aspects for constructing realizations of polynomial chaos in high dimension

Abstract : This paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension. The method proposed consists of (1) constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and (2) performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the Gram-Schmidt orthogonalization algorithm which is not stable in high dimension. A brief review of polynomial chaos expansion with arbitrary measure is given. The statistically independent realizations of multivariate monomials are introduced. The centered statistically independent realizations of orthonormal multivariate polynomials are developed. Finally, a quantification of the errors induced by the usual methods is given.