Raul Tempone
NADA
KTH
Stockholm
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.