On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods

On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods

Beck, Joakim; Tempone, Raul; Nobile, Fabio; Tamellini, Lorenzo "On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods." Math. Models Methods Appl. Sci. 22 (2012), no. 9, 1250023, 33 pp.‚Äč

Beck, Joakim; Tempone, Raul; Nobile, Fabio; Tamellini, Lorenzo
Uncertainty quantification; PDEs with random data; elliptic equations; multivariate polynomial approximation; best M-terms polynomial approximation; stochastic Galerkin methods; Smolyak approximation; sparse grids; stochastic collocation methods
2012

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.

DOI: 10.1142/S0218202512500236