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A stochastic collocation method for partial differential equations with random input data

Raul Tempone

Abstract

We present a Stochastic-Collocation method to solve Partial Differential Equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the Stochastic Galerkin method proposed in [Babuska-Tempone-Zouraris, SIAM J. Num. Anal. 42(2004)] and allows one to treat easily a wider range of situations, such as: input data that depend non-linearly on the random variables, diffusivity coefficients with unbounded second moments, random variables that are correlated or have unbounded support.

We give a complete convergence analysis in case of a linear elliptic operator and demonstrate exponential convergence of the "probability error" with respect of the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data.

Numerical examples showing the effectiveness of the method will be presented as well.