The ADI iteration for Lyapunov equations implicitly performsH2pseudo-optimal model order reduction

2015/08/20 by T. Wolf, Thomas Wolf, H.K.F. Panzer +1 · 2 citations
Physics and Astronomy · Decision Sciences · Mathematics · #Model Reduction and Neural Networks #Probabilistic and Robust Engineering Design #Numerical methods for differential equations

paper · doi:10.1080/00207179.2015.1081985

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 H2 pseudo-optimal approximation. This includes the generalisation of previous results on H2 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.

Cited by

• Riccati ADI: Existence, uniqueness and new iterative methods
• Towards a mixed-precision ADI method for Lyapunov equations

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