Interpolation methods to compute statistics of a stochastic partial differential equation

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.