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

TLDR

The method generalizes the stochastic Galerkin approach, enabling treatment of nonlinear random inputs, unbounded second moments, and correlated or unbounded random variables. The paper proposes and analyzes a stochastic collocation method for elliptic PDEs with random coefficients and forcing terms. The method uses a Galerkin spatial discretization combined with Gauss‑point collocation in the probability space, yielding uncoupled deterministic problems. A rigorous convergence analysis shows exponential decay of the probability error with respect to the number of Gauss points, and numerical tests confirm the method’s effectiveness. Reference: Babuška, Tempone, and Zouraris, SIAM J.

Abstract

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 [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to 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.