Abstract
We present a structure-preserving Krylov subspace method for solving large-scale Lyapunov equations where the (approximate) solution is of low rank. This problem arises, e.g., from model order reduction techniques based on Balanced Truncation for large-scale descriptor systems such as those in the simulation of large electrical circuits. The method presented here uses a low-rank approach based on the FGMRES method. For preconditioning the Low Rank Cholesky Factor-Alternating Direct Implicit is applied which turns out to preserve the low-rank structures and allows for the use of inner approximate factorizations.
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References
Benner, P., Li, J.-R., Penzl, T.: Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Lin. Algebra Appl. 15, 755–777 (2008)
Bollhöfer, M., Saad, Y.: Multilevel preconditioners constructed from inverse-based ILUs. SIAM J. Sci. Comput. 27, 1627–1650 (2006)
Golub, G.H., Van Loan, C.F.: Matrix Computations (Johns Hopkins Studies in Mathematical Sciences), 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002)
Penzl, T.: A cyclic low-rank smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21(4), 1401–1418 (2000)
Reis, T., Stykel, T.: PABTEC: Passivity-preserving balanced truncation for electrical circuits. IEEE Trans. Comp. Aided Des. Integr. Circ. Syst. 29(9), 1354–1367 (2010)
Reis, T., Stykel, T.: Positive real and bounded real balancing for model reduction of descriptor systems. Int. J. Contr. 83(1), 74–88 (2010)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)
Stykel, T.: Low-rank iterative methods for projected generalized Lyapunov equations. Electron. Trans. Numer. Anal. 30, 187–202 (2008)
Wachspress, E.L.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1(1), 87–90 (1988)
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© 2012 Springer-Verlag Berlin Heidelberg
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Bollhöfer, M., Eppler, A.K. (2012). A Structure Preserving FGMRES Method for Solving Large Lyapunov Equations. In: Günther, M., Bartel, A., Brunk, M., Schöps, S., Striebel, M. (eds) Progress in Industrial Mathematics at ECMI 2010. Mathematics in Industry(), vol 17. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25100-9_15
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DOI: https://doi.org/10.1007/978-3-642-25100-9_15
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