We present a Stochastic-Collocation method to solve Partial Differential Equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables.
The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach.
We will present collocation techniques based on full anisotropic tensor product grids as well as isotropic or anisotropic sparse grids based on the Smolyak construction. The last approach is particularly attractive in the case of input data obtained as truncated expansions of random fields, since the anisotropy can be tuned on the decay properties of the expansion. We will present a priori and a posteriori ways to chose the anisotropy of the sparse grid which are extremely effective in some situations.
We will also present rigorous convergence results in all cases as well as numerical examples where we compare the different approaches with the more traditional Monte Carlo technique. In particular, the sparse grid approach, with a properly chosen anisotropy seems to be very efficient when a moderately large number of input random variables is considered.