ICES REPORT 09-33 - Stochastic Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison

Abstract: Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncer- tainty quantification. An open and relevant research topic is the comparison of these two methods. By introducing a suitable generalization of the classi- cal sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy versus com- putational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyper- bolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method. Finally, numerical results corroborate the optimality of the theoretical estimate of anisotropy ratios introduced by the authors in a previous work for the construction of anisotropic approximation spaces.