Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest (1704.02284v2)

Abstract: We examine nonlinear dynamical systems of ordinary differential equations or differential algebraic equations. In an uncertainty quantification, physical parameters are replaced by random variables. The inner variables as well as a quantity of interest are expanded into series with orthogonal basis functions like the polynomial chaos expansions, for example. On the one hand, the stochastic Galerkin method yields a large coupled dynamical system. On the other hand, a stochastic collocation method, which uses a quadrature rule or a sampling scheme, can be written in the form of a large weakly coupled dynamical system. We apply projection-based methods of nonlinear model order reduction to the large systems. A reduced-order model implies a low-dimensional representation of the quantity of interest. We focus on model order reduction by proper orthogonal decomposition. The error of a best approximation located in a low-dimensional subspace is analysed. We illustrate results of numerical computations for test examples.