  This paper considers the analysis of partial differential equations (PDE)
containing multiple random variables. Recently developed collocation methods
enable the construction of high-order stochastic solutions by converting a
stochastic PDE into a system of deterministic PDEs. This interpolation method
requires that the probability distribution of all random input variables is
known a priori, which is often not the case in industrially relevant
applications. Additionally, this method suffers from a curse of dimensionality,
i.e., the number of deterministic PDEs to be solved grows exponentially with
respect to the number of random variables. This paper presents an alternative
interpolation method, based on a radial basis function (RBF) metamodel, to
compute statistics of the stochastic PDE. The RBF metamodel can be constructed
even if the probability distribution of all random variables is not known.
Then, a lot of statistic scenarios with different probability distributions of
the random variables can be computed with this single metamodel. In order to
reduce the model complexity, we present a parameter screening technique which
can be combined with an interpolation method to solve a reduced stochastic
model. Numerical results of a model problem demonstrate that the RBF metamodel
is as fast as a low order collocation approach and achieves a good accuracy.
The parameter screening is able to reduce the dimension and, thus, to
accelerate the computation of the stochastic solution.
