A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

In this paper we propose and analyze a Stochastic-Collocation method to solve elliptic 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 provide a rigorous convergence analysis and demonstrate expo...

• Type: Journal
• Year: 2010
• Where: SIAMREV
• Authors: Ivo Babuska, Fabio Nobile, Raúl Tempone
• Keywords: Gauss Points, Input Data, Random Variables, SIAMREV 2010