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

MOX 71

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

Thursday 24th November 2005

Babuska, I.; Nobile, Fabio; Tempone, Raul

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 fi nite 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 prob¬lems as in the Monte Carlo approach. It can be seen as a generalization of the Stochastic Galerkin method proposed in [Babuˇ ska -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 exponential con¬vergence 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 show the effectiveness of the method.

This report, or a modified version of it, has been also submitted to, or published on
I. Babuska, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal., 2007, vol. 45/3, pp. 1005--1034