Model order reduction and low-dimensional representations for random linear dynamical systems

Abstract

We consider linear dynamical systems of ordinary differential equations or differential algebraic equations. Physical parameters are substituted by random variables for an uncertainty quantification. We expand the state variables as well as a quantity of interest into an orthogonal system of basis functions, which depend on the random variables. For example, polynomial chaos expansions are applicable. The stochastic Galerkin method yields a larger linear dynamical system, whose solution approximates the unknown coefficients in the expansions. The Hardy norms of the transfer function provide information about the input–output behaviour of the Galerkin system. We investigate two approaches to construct a low-dimensional representation of the quantity of interest, which can also be interpreted as a sparse representation. Firstly, a standard basis is reduced by the omission of basis functions, whose accompanying Hardy norms are relatively small. Secondly, a projection-based model order reduction is applied to the Galerkin system and allows for the definition of new basis functions within a low-dimensional representation. In both cases, we prove error bounds on the low-dimensional approximation with respect to Hardy norms. Numerical experiments are demonstrated for two test examples.