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

Joakim Bäck, Fabio Nobile, Lorenzo Tamellini, Raul Tempone

Research output: Chapter in Book/Report/Conference proceedingConference contribution

125 Scopus citations

Abstract

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 Springer.
Original languageEnglish (US)
Title of host publicationSpectral and High Order Methods for Partial Differential Equations
PublisherSpringer Nature
Pages43-62
Number of pages20
ISBN (Print)9783642153365
DOIs
StatePublished - Sep 17 2010

ASJC Scopus subject areas

  • Engineering(all)
  • Modeling and Simulation
  • Computational Mathematics
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

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