In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new 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. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids. © 2012 World Scientific Publishing Company.
|Original language||English (US)|
|Journal||Mathematical Models and Methods in Applied Sciences|
|State||Published - Sep 2012|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The authors would like to recognize the support of the PECOS center at ICES, University of Texas at Austin (Project No. 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and uncertainty quantification for models of porous media" is also acknowledged. The second and third authors have been supported by the Italian grant FIRB-IDEAS (Project No. RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems".
ASJC Scopus subject areas
- Modeling and Simulation
- Applied Mathematics