Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients

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.