Efficient Balance-and-Truncate Model Reduction for Large Scale Systems

We present efficient implement at ions of the balanceand-truncate model reduction technique for large-scale systems. The key observation that distinguishes our approach is that Krylov subspace methods (Arnoldi and Lanczos) directly yield approximate low-rank square roots1 of the system Gramians; the balancing transformation can be then constructed from these square roots, obviating the need for solving any Lyapunov equations. In addition, the order of the reduced model is not fixed a priori as with some existing methods, but is determined from the problem data. Numerical simulations show t,hat our approach performs very well over a range of examples, and offers considerable savings in practice. 'We use the term "square root" to mean the not necessarily symmetric square root of a matrix: If M = M~ = N N T , we say N is the square root of M .