An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties

In (Augustin et al. in European J. Appl. Math. 19:149-190, 2008) we considered the Polynomial Chaos Expansion for the treatment of uncertainties in industrial applications. For many applications the method has been proven to be a computationally superior alternative to Monte Carlo evaluations. In the current overview we compare the accuracy of Polynomial Chaos type methods for the propagation of uncertainties in nonlinear problems and verify them on two examples relevant for industry. For weakly nonlinear time-dependent models, the generalized Kalman filter equations define an efficient method, yielding good approximations if the quantities of interest are restricted to the first two moments of the solution. Secondly, stochastic collocation is discussed. The method is applied to delay differential equations and random ordinary differential equations. Finally, a generalized PC method is discussed which is based on a subdivision of the random space. This approach is even suitable for highly nonlinear models.