Balanced Truncation Model Reduction of Large and Sparse Generalized Linear Systems

We investigate model reduction of large-scale linear time-invariant systems in generalized state-space form. We consider sparse state matrix pencils, including pencils with banded structure. The balancing-based methods employed here are composed of well-known linear algebra operations and have been recently shown to be applicable to large models by exploiting the structure of the matrices defining the dynamics of the system. In this paper we propose a modification of the LR-ADI iteration to solve large-scale generalized Lyapunov equations together with a practical convergence criterion, and several other implementation refinements. Using kernels from several serial and parallel linear algebra libraries, we have developed a parallel package for model reduction, SpaRed, extending the applicability of balanced truncation to sparse systems with up to $O(10^5)$ states. Experiments on an SMP parallel architecture consisting of Intel Itanium 2 processors illustrate the numerical performance of this approach and the potential of the parallel algorithms for model reduction of large-scale sparse systems.