A reduced-IRKA method for large-scale $\mathcal{H}_2$-optimal model order reduction

The $\mathcal{H}_2$-optimal Model Order Reduction (MOR) is one of the most significant frameworks for reduction methodologies for linear dynamical systems. In this context, the Iterative Rational Krylov Algorithm (\IRKA) is a well established method for computing an optimal projection space of fixed dimension $r$, when the system has small or medium dimensions. However, for large problems the performance of \IRKA\ is not satisfactory. In this paper, we introduce a new rational Krylov subspace projection method with conveniently selected shifts, that can effectively handle large-scale problems. The projection subspace is generated sequentially, and the \IRKA\ procedure is employed on the projected problem to produce a new optimal rational space of dimension $r$ for the reduced problem, and the associated shifts. The latter are then injected to expand the projection space. Truncation of older information of the generated space is performed to limit memory requirements. Numerical experiments on benchmark problems illustrate the effectiveness of the new method.