Parallel Algorithms for Balanced Truncation of Large-Scale Unstable Systems

We discuss and compare two approaches for model reduction of large-scale unstable systems on parallel computers. The first method proceeds by computing the additive decomposition of the transfer function via a block diagonalization, followed by a reduction of the stable part of the system using techniques based on state-space truncation. The second method employs a representation of the controllability and observability Gramians of an unstable systems in terms of the Gramians of the stabilized system where the particular stabilization is obtained via the solution of dual algebraic Bernoulli equations. Based on these Gramians, balanced truncation is then applied in the usual manner. All core computational steps in these methods can be efficiently solved on parallel computers by means of diverse variants of the Newton iteration for the sign function. Numerical experiments on a cluster of Intel Xeon processors show the numerical and parallel performances of these methods.