Inexact Linear Solves in Model Reduction of Bilinear Dynamical Systems

The bilinear iterative rational Krylov algorithm (BIRKA) is a very popular, standard, and mathematically sound algorithm for reducing bilinear dynamical systems that arise commonly in science and engineering. This reduction process is termed as a model order reduction (MOR) and leads to a faster simulation of such systems. An efficient variant of the BIRKA, Truncated BIRKA (TBIRKA) has also been recently proposed. Like for any MOR algorithm, these two algorithms also require solving multiple linear systems as part of the model reduction process. For reducing the MOR time, these linear systems are often solved by an iterative solver, which introduces approximation errors (implying inexact solves). Hence, stability analysis of the MOR algorithms with respect to inexact linear solves is important. In our past work, we have shown that under mild conditions, the BIRKA is stable. Here, we look at the stability of the TBIRKA in the same context. Besides deriving the conditions for a stable TBIRKA, our other novel contribution is the more intuitive methodology for achieving this. The stability analysis techniques that we propose here can be extended to many other methods for doing the MOR of bilinear dynamical systems, e.g., using balanced truncation or the ADI methods.