2013/09/16 by Thomas Wolf, Heiko K. F. Panzer, Wolf, Thomas +2 · 1 citation
Decision Sciences · Engineering · Mathematics · Physics and Astronomy · #65F10 #93A15 #93C05 #Dynamical Systems (math.DS) #FOS: Electrical engineering #FOS: Mathematics #Fluid Dynamics and Vibration Analysis #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probabilistic and Robust Engineering Design #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1309.3985
openalex publication_date 2013/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Two approaches for approximating the solution of large-scale Lyapunov\nequations are considered: the alternating direction implicit (ADI) iteration\nand projective methods by Krylov subspaces. A link between them is presented by\nshowing that the ADI iteration can always be identified by a Petrov-Galerkin\nprojection with rational block Krylov subspaces. Then a unique Krylov-projected\ndynamical system can be associated with the ADI iteration, which is proven to\nbe an H2 pseudo-optimal approximation. This includes the generalization of\nprevious results on H2 pseudo-optimality to the multivariable case.\nAdditionally, a low-rank formulation of the residual in the Lyapunov equation\nis presented, which is well-suited for implementation, and which yields a\nmeasure of the "obliqueness" that the ADI iteration is associated with.\n