The ADI iteration for Lyapunov equations implicitly performs H2 pseudo-optimal model order reduction

ABSTRACT

Two approaches for approximating the solution of large-scale Lyapunov equations are considered: the alternating direction implicit (ADI) iteration and projective methods by Krylov subspaces. We show that they are linked in the way that the ADI iteration can always be identified by a Petrov–Galerkin projection with rational block Krylov subspaces. Therefore, a unique Krylov-projected dynamical system can be associated with the ADI iteration, which is proven to be an ${\mathcal{H}}_2$ pseudo-optimal approximation. This includes the generalisation of previous results on ${\mathcal{H}}_2$ pseudo-optimality to the multivariable case. Additionally, a low-rank formulation of the residual in the Lyapunov equation is presented, which is well-suited for implementation, and which yields a measure of the ‘obliqueness’ that the ADI iteration is associated with.

Keywords:

• Lyapunov equation
• alternating direction implicit method
• model order reduction
• rational Krylov subspace
• ${\mathcal{H}}_2$ optimality

Acknowledgements

The authors thank Prof. Serkan Gugercin for the fruitful discussion at the MODRED 2013 in Magdeburg, and the anonymous reviewers for their valuable remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Available at http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark.