Towards a hybrid numerical method using Generalized Polynomial Chaos for Stochastic Differential Equations

Abstract

Generalized polynomial chaos (gPC) is known to fail for problems involving strong nonlinear dependencies on stochastic inputs, especially arising in the context of long term integration. The reason for this is that gPC is a time-independent projection method, not able to capture a dynamic behavior of probability distributions. Recent developments in addressing this problem are represented by decomposing the random space or employing discrete time-dependent basis functionals, both exhibiting promising results but also introducing increasing computational costs. This work focuses on a numerical analysis of these two approaches as well as their hybrid combination with regard to a simple ODE decay problem subject to a uniformly as well as a Gaussian distributed random input. It is observed that depending on the initial probability distribution strong differences occur with respect to the error developments, which efficiently can be reduced when employing local discrete time- dependent basis functionals.