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Bilinear dynamical systems are commonly used in science and engineeringbecause they form a bridge between linear and non-linear systems. However,simulating them is still a challenge because of their large size. Hence, a lotof research is currently being done for reducing such bilinear dynamicalsystems (termed as bilinear model order reduction or bilinear MOR). Bilineariterative rational Krylov algorithm (BIRKA) is a very popular, standard andmathematically sound algorithm for bilinear MOR, which is based uponinterpolatory projection technique. An efficient variant of BIRKA, TruncatedBIRKA (or TBIRKA) has also been recently proposed.  Like for any MOR algorithm, these two algorithms also require solvingmultiple linear systems as part of the model reduction process. For reducingvery large dynamical systems, which is now-a-days becoming a norm, scaling ofsuch linear systems with respect to input dynamical system size is abottleneck. For efficiency, these linear systems are often solved by aniterative solver, which introduces approximation errors. Hence, stabilityanalysis of MOR algorithms with respect to inexact linear solves is important.In our past work, we have shown that under mild conditions, BIRKA is stable (inthe sense as discussed above). Here, we look at stability of TBIRKA in the samecontext.  Besides deriving the conditions for a stable TBIRKA, our other novelcontribution is the more intuitive methodology for achieving this. Thisapproach exploits the fact that in TBIRKA a bilinear dynamical system can berepresented by a finite set of functions, which was not possible in BIRKA(because infinite such functions were needed there). The stability analysistechniques that we propose here can be extended to many other methods for doingMOR of bilinear dynamical systems, e.g., using balanced truncation or the ADImethods.

Inexact Linear Solves In Model Reduction of Bilinear Dynamical Systems