Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison

Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison

J. Back, F. Nobile, L. Tamellini, and R. Tempone. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In J.S. Hesthaven and E.M. Ronquist, editors, Spectral and High Order Methods for Partial Differential Equations, volume 76 of Lecture Notes in Computational Science and Engineering, pages 43-62. Springer, 2011. Selected papers from the ICOSAHOM '09 conference, June 22-26, Trondheim, Norway.‚Äč
Joakim Back, Fabio Nobile, Lorenzo Tamellini, and Raul Tempone
Elliptic equations Multivariate polynomial approximation PDEs with random data Smolyak approximation Stochastic collocation methods Stochastic Galerkin methods Uncertainty quantification
2011
Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods.
By introducing a suitable generalization of the classical sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy vs. computational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyperbolic 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.
2011