Implicitly Restarted Krylov Subspace Methods for Stable Partial Realizations

This paper considers an implicitly restarted Krylov subspace method that approximates a stable, linear transfer function $f(s)$ of order $n$ by one of order $m$, where $n\gg m$. It is well known that oblique projections onto a Krylov subspace may generate unstable partial realizations. To remedy this situation, the oblique projectors obtained via classical Krylov subspace methods are supplemented with further projectors which enable the formation of stable partial realizations directly from $f(s)$. A key feature of this process is that it may be incorporated into an implicit restart scheme. A second difficulty arises from the fact that Krylov subspace methods often generate partial realizations that contain nonessential modes. To this end, balanced truncation may be employed to discard the unwanted part of the reduced-order model. This paper proposes oblique projection methods for large-scale model reduction that simultaneously compute stable reduced-order models while discarding all nonessential modes. It is shown that both of these tasks may be effected by a single oblique projection process. Furthermore, the process is shown to naturally fit into an implicit restart framework. The theoretical properties of these methods are thoroughly investigated, and exact low-dimensional expressions for the $\L^{\infty}$-norm of the residual errors are derived. Finally, the behavior of the algorithm is illustrated on two large-scale examples.