High-order Numerical Methods for Differential Equations 
with Random Inputs

Dongbin Xiu
Department of Chemical Engineering
Princeton University

Recently there has been a growing interest in designing efficient methods 
for solutions of ordinary/partial differential equations with random 
inputs. Unlike in deterministic simulations where relatively mature 
techniques exist for monitoring computational accuracy and accessing 
discretizational errors, design of high-order methods and analysis of 
their accuracy and efficiency for random differential equations 
are still at an early stage of development. To this end, Stochastic 
Galerkin (SG) methods appear to be superior to other non-sampling methods, 
and in many cases, to several sampling methods. This type of methods take 
advantage of the smoothness assumption of the solutions in random spaces 
and can achieve fast convergence. In this talk, I will discuss recent 
advances of these methods and their applications to various stochastic 
problems, ranging from simple model equations to large-scale complicated 
systems.