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Low-Rank Cholesky Factor Krylov Subspace Methods for Generalized Projected Lyapunov Equations

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System Reduction for Nanoscale IC Design

Part of the book series: Mathematics in Industry ((MATHINDUSTRY,volume 20))

Abstract

Large-scale descriptor systems arising from circuit simulation often require model reduction techniques. Among many methods, Balanced Truncation is a popular method for constructing a reduced order model. In the heart of Balanced Truncation methods, a sequence of projected generalized Lyapunov equations has to be solved. In this article we present a general framework for the numerical solution of projected generalized Lyapunov equations using preconditioned Krylov subspace methods based on iterates with a low-rank Cholesky factor representation. This approach can be viewed as alternative to the LRCF-ADI method, a well established method for solving Lyapunov equations. We will show that many well-known Krylov subspace methods such as (F)GMRES, QMR, BICGSTAB and CG can be easily modified to reveal the underlying low-rank structures.

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Notes

  1. 1.

    Matthias Bollhöfer and Yousef Saad. ILUPACK - preconditioning software package. Available online at http://ilupack.tu-bs.de/.ReleaseV2.4,June2011.

  2. 2.

    http://sites.google.com/site/rionutiu2/research/software.

  3. 3.

    http://www.pardiso-project.org.

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Acknowledgements

The work reported in this paper was supported by the German Federal Ministry of Education and Research (BMBF), grant no. 03BOPAE4. Responsibility for the contents of this publication rests with the authors.

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Authors and Affiliations

  1. TU Braunschweig, Carl-Friedrich-Gauß-Fakultät, Institut Computational Mathematics, Universitätspl. 2, 38106, Braunschweig, Germany

    Matthias Bollhöfer & André K. Eppler

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  1. Matthias Bollhöfer
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  2. André K. Eppler
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Correspondence to Matthias Bollhöfer .

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  1. Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

    Peter Benner

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Bollhöfer, M., Eppler, A.K. (2017). Low-Rank Cholesky Factor Krylov Subspace Methods for Generalized Projected Lyapunov Equations. In: Benner, P. (eds) System Reduction for Nanoscale IC Design. Mathematics in Industry, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-07236-4_5

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