Title:  Stochastic collocation methods for elliptic partial
differential equations with random input data

Abstract:

In this talk we present Stochastic-Collocation methods 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 methods consist in a Galerkin approximation with respect to the space variables and a collocation 
on suitable tensor product grids, utilizing either Clenshaw-Curtis or Gaussian knots, in the 
probability space and naturally lead 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 choose 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 and superior to 
all the others also when a moderately large number of input random variables is considered.