Sensitivity analysis of random linear differential–algebraic equations using system norms

Abstract

We consider linear dynamical systems composed of differential–algebraic equations (DAEs), where a quantity of interest (QoI) is assigned as output. Physical parameters of a system are modelled as random variables to quantify uncertainty, and we investigate a variance-based sensitivity analysis of the random QoI. Based on expansions via generalised polynomial chaos, the stochastic Galerkin method yields a new deterministic system of DAEs of high dimension. We define sensitivity measures by system norms, i.e., the H∞-norm of the transfer function associated with the Galerkin system for different combinations of outputs. To ameliorate the enormous computational effort required to compute norms of high-dimensional systems, we apply balanced truncation, a particular method of model order reduction (MOR), to obtain a low-dimensional linear dynamical system that produces approximations of system norms. MOR of DAEs is more sophisticated in comparison to systems of ordinary differential equations. We show an a priori error bound for the sensitivity measures satisfied by the MOR method. Numerical results are presented for two stochastic models given by DAEs.