Effective error estimation for model reduction with inhomogeneous initial conditions

A priori error bounds have been derived for different balancing-related model reduction methods. The classical result is a bound for balanced truncation and singular perturbation approximation that is applicable for asymptotically stable linear time-invariant systems with zero initial conditions. Recently, there have been a few attempts to generalize the balancing-related reduction methods to the case with inhomogeneous initial conditions. Those strongly rely on the assumption that the space of initial conditions of interest is known a priori In this paper, we show how the exact error representation in terms of the Gramians can be used as a sharp and efficient error bound. In particular, by exploiting an appropriate offline–online decomposition of the computation, this approach is feasible for arbitrary initial conditions. This is in contrast to previous error bounds, which are valid only for an a priori restricted set of initial conditions. Furthermore, our approach can be realized in a large-scale setting, in which case the resulting error estimator is as accurate as the underlying low-rank approximation of the Gramian allows. The effectivity, accuracy and applicability of our bound/estimator for a posteriori estimation and certified model selection are also demonstrated numerically.