Uncertainty quantification in kinetic and hyperbolic problems

Local polynomial chaos expansion for high dimensional stochastic PDE

Dongbin Xiu

University of Utah


We present a localized polynomial chaos expansion for PDE with random inputs. In particular, we focus on time independent linear stochastic problems with high dimensional random inputs, where the traditional polynomial chaos methods, and most of the existing methods, incur prohibitively high simulation cost. The local polynomial chaos method employs a domain decomposition technique to approximate the stochastic solution locally. In each subdomain, a subdomain problem is solved independently and more importantly, in a much lower dimensional random space. In a post-procesing stage, accurate samples of the original stochastic problems are obtained from the samples of the local solutions, by enforcing the correct stochastic structure of the random inputs and the coupling conditions at the interfaces of the subdomains. Overall, the method is able to solve stochastic PDEs in very large dimensions by solving a collection of low dimensional local problems and can be highly efficient. We present the mathematical framework of the method and use numerical examples to demonstrate its efficiency.