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
