2015
DOI: 10.1080/00207179.2015.1081985
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The ADI iteration for Lyapunov equations implicitly performsH2pseudo-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. A link between them is presented by showing that the ADI iteration can always be identified by a Petrov-Galerkin projection with rational block Krylov subspaces. Then a unique Krylov-projected dynamical system can be associated with the ADI iteration, which is proven to be an H 2 pseudo-optimal approximation. This includes… Show more

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Cited by 18 publications
(23 citation statements)
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“…Motivated by the connection of LR-ADI to rational Krylov subspaces [15,17,28,51,52] we can borrow the greedy shift selection strategy from [16] which was developed for the rational Krylov subspace method for (1). Let S ⊂ C − be the convex hull of the set of Ritz values Λ(H ℓ ) and ∂S its boundary.…”
Section: Convex Hull Based Shiftsmentioning
confidence: 99%
“…Motivated by the connection of LR-ADI to rational Krylov subspaces [15,17,28,51,52] we can borrow the greedy shift selection strategy from [16] which was developed for the rational Krylov subspace method for (1). Let S ⊂ C − be the convex hull of the set of Ritz values Λ(H ℓ ) and ∂S its boundary.…”
Section: Convex Hull Based Shiftsmentioning
confidence: 99%
“…In [4] and [16] it was independently shown, that the residual can be factorized as R = B ⊥ B T ⊥ -with real B ⊥ ∈ R n×m , if the set S is closed under conjugation. One way to compute B ⊥ is…”
Section: The Adi Iterationmentioning
confidence: 99%
“…The notation B ⊥ stems from the fact, that a specific projection can be associated to the ADI iteration: B ⊥ is the residual after projecting B, and it fulfills the Petrov-Galerkin condition, see [16] for details. The low-rank formulation R = B ⊥ B T ⊥ allows to compute the induced matrix 2-norm of the residual R 2 by the maximum eigenvalue of the m-by-…”
Section: The Adi Iterationmentioning
confidence: 99%
See 1 more Smart Citation
“…Our method benefits from both Krylov [1] and Lyapunov techniques [2], [3], [4]. The proposed method generates a reduced order models with similar behavior of the original system, minimizes the absolute error, the H ∞ norm error, and preserves the stability of reduced system.…”
Section: Introductionmentioning
confidence: 99%