Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest
  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.
