Stochastic Galerkin methods and model order reduction for linear dynamical systems

Linear dynamical systems are considered in the form of ordinary differential equations or differential algebraic equations. We change their physical parameters into random variables to represent uncertainties. A stochastic Galerkin method yields a larger linear dynamical system satisfied by an approximation of the random processes. If the original systems own a high dimensionality, then a model order reduction is required to decrease the complexity. We investigate two approaches: the system of the stochastic Galerkin scheme is reduced and, vice versa, the original systems are reduced followed by an application of the stochastic Galerkin method. The properties are analyzed in case of reductions based on moment matching with the Arnoldi algorithm. We present numerical computations for two test examples. Keywords: stochastic modeling, polynomial chaos, stochastic Galerkin method, model order reduction, quadrature, dynamical systems, uncertainty quantificatio