Uncertainty propagation in dynamic systems using polynomial chaos based multiresolution approach

A multiresolution approach is presented to study the uncertainty propagation problem in dynamic systems. The proposed approach is an extension of the gener- alized polynomial chaos and involves a separation of random variables from deter- ministic ones in the solution algorithm for a stochastic differential equation. The random variables are expanded in various local polynomial expansions which are further blended together to obtain a global approximation for the random state variables. These polynomials are associated with the assumed probability density functions for the input variables. Galerkin projection is used to generate a system of deterministic differential equations for the expansion coeffIcients. We first present this method to linear systems with parametric uncertainties, and subsequently we generalize it to nonlinear systems with parametric and initial condition uncertain- ties. The effectiveness of the proposed method is demonstrated by considering different numerical examples.