Model order reduction for random nonlinear dynamical systems and low-dimensional representations for their quantities of interest
For researchers in uncertainty quantification and model reduction, this work provides a theoretical error analysis for reduced-order models of random nonlinear dynamical systems, but it is incremental as it extends existing POD methods to this specific setting.
The paper applies projection-based model order reduction, specifically proper orthogonal decomposition, to large coupled dynamical systems arising from uncertainty quantification in nonlinear ODEs/DAEs, achieving low-dimensional representations for quantities of interest with analyzed approximation error.
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.