Volume 33, October 2011CANUM 2010, 40e Congrès National d'Analyse Numérique
|Page(s)||10 - 21|
|Published online||22 December 2011|
Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients⋆,⋆⋆
1 Applied Mathematics and Computational Science, KAUST Saudi Arabia
2 MOX, Department of Mathematics “F. Brioschi”, Politecnico di Milano, Italy
In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product.
We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids.
Mathematics Subject Classification: 41A10 / 65C20 / 65N12 / 65N35
Key words: Uncertainty Quantification / PDEs with random data / elliptic equations / multivariate polynomial approximation / Best M-Terms approximation / Stochastic Galerkin methods / Smolyak approximation / Sparse grids, Stochastic Collocation methods
The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project ”Effektiva numeriska metoder för stokastiska differentialekvationer med tillämpningar” and King Abdullah University of Science and Technology (KAUST) is also acknowledged.
© EDP Sciences, SMAI 2011
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