Abstract
In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the \(d\)-dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krylov-Plus-Inverted-Krylov (K-PIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.
Similar content being viewed by others
References
Antoulas, A., Sorensen, D., Zhou, Y.: On the decay rate of Hankel singular values and related issues. Syst. Control Lett. 46(5), 323–342 (2002)
Beckermann, B.: An error analysis for rational Galerkin projection applied to the Sylvester equation. SIAM J. Numer. Anal. 49, 2430 (2011)
Benner, P., Breiten, T.: On \({H}_{2}\)-model reduction of linear parameter-varying systems. Proc. Appl. Math. Mech. 11(1), 805–806 (2011). doi: 10.1002/pamm.201110391
Benner, P., Breiten, T.: On optimality of interpolation-based low-rank approximations of large-scale matrix equations. MPI Magdeburg Preprints MPIMD/11-10 (2011). Available from http://www.mpi-magdeburg.mpg.de/preprints/abstract.php?nr=11-10&year=2011
Benner, P., Breiten, T.: Interpolation-based \({H}_{2}\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33(3), 859–885 (2012)
Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49(2), 686–711 (2011)
Benner, P., Li, J.R., Penzl, T.: Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer. Linear Algebra Appl. 15(9), 755–777 (2008)
Benner, P., Li, R., Truhar, N.: On the ADI method for Sylvester equations. J. Comput. Appl. Math. 233(4), 1035–1045 (2009)
Benner, P., Saak, J.: Linear-quadratic regulator design for optimal cooling of steel profiles. Tech. Rep. SFB393/05-05, Sonderforschungsbereich 393 Parallele Numerische Simulation für Physik und Kontinuumsmechanik, TU Chemnitz, 09107 Chemnitz, FRG (2005). Available from http://www.tu-chemnitz.de/sfb393
Condon, M., Ivanov, R.: Krylov subspaces from bilinear representations of nonlinear systems. COMPEL 26, 11–26 (2007)
D’Alessandro, P., Isidori, A., Ruberti, A.: Realization and structure theory of bilinear dynamical systems. SIAM J. Control Optim. 12(3), 517–535 (1974)
Damm, T.: Rational Matrix Equations in Stochastic Control, vol. 297. Springer, Berlin (2004)
Damm, T.: Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations. Numer. Linear Algebra Appl. 15(9), 853–871 (2008)
Dorissen, H.: Canonical forms for bilinear systems. Syst. Control Lett. 13(2), 153–160 (1989)
Druskin, V., Knizhnerman, L., Simoncini, V.: Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J. Numer. Anal. 49, 1875–1898 (2011)
Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60, 546–560 (2011)
Eppler, A.K., Bollhöfer, M.: An alternative way of solving large Lyapunov equations. Proc. Appl. Math. Mech. 10(1), 547–548. doi:10.1002/pamm.201010266
Eppler, A., Bollhöfer, M.: 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, pp. 131–136. Springer, Berlin (2012). doi:10.1007/978-3-642-25100-9-15
Flagg, G.M., Gugercin, S.: On the ADI method for the Sylvester equation and the optimal-H2 points. Appl. Numer. Math. 64, 50–58 (2013). doi:10.1016/j.apnum.2012.10.001
Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72(3–4), 247–265 (2004)
Grasedyck, L.: Existence of a low rank or \(\cal {H}\)-matrix approximant to the solution of a sylvester equation. Numer. Linear Algebra Appl. 11(4), 371–389 (2004). doi:10.1002/nla.366
Gray, W., Mesko, J.: Energy functions and algebraic Gramians for bilinear systems. In: Preprints of the 4th IFAC Nonlinear Control Systems Design, Symposium, pp. 103–108 (1998)
Gugercin, S., Antoulas, A., Beattie, S.: \({H}_{2}\) model reduction for large-scale dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609–638 (2008)
Hartmann, C., Zueva, A., Schäfer-Bung, B.: Balanced model reduction of bilinear systems with applications to positive systems (2010) (Submitted)
Horn, R., Johnson, C.: Matrix Analysis. Cambridge University Press, Cambridge (1990)
Jaimoukha, I., Kasenally, E.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227–251 (1994)
Jbilou, K., Riquet, A.J.: Projection methods for large Lyapunov matrix equations. Linear Algebra Appl. 415(2–3), 344–358 (2006)
Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009). doi:10.1137/07070111X
Kressner, D., Tobler, C.: Krylov subspace methods for linear systems with tensor product structure. SIAM J. Matrix Anal. Appl. 31(4), 1688–1714 (2010)
Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM J. Matrix Anal. Appl. 32(4), 1288–1316 (2011)
Kressner, D., Tobler, C.: Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems. CMAM 11(3), 363–381 (2011)
Li, J.R., White, J.: Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl. 24(1), 260–280 (2002)
Penzl, T.: Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Syst. Control Lett. 40(2), 139–144 (2000)
Rugh, W.: Nonlinear System Theory. The Johns Hopkins University Press, Baltimore (1982)
Saad, Y.: Numerical solution of large Lyapunov equation. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds.) Signal Processing, Scattering, Operator Theory and Numerical Methods, pp. 503–511. Birkhauser, Basel (1990)
Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268–1288 (2007)
Sorensen, D., Zhou, Y.: Bounds on eigenvalue decay rates and sensitivity of solutions to Lyapunov equations. Tech. Rep. TR02-07, Dept. of Comp. Appl. Math., Rice University, Houston, TX (2002). Available online from http://www.caam.rice.edu/caam/trs/tr02.html#TR02
Stenger, F.: Numerical methods based on Sinc and analytic functions. In: Springer Series in Computational Mathematics, vol. 20. Springer, New York (1993)
Wachspress, E.: Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 107, 87–90 (1988)
Wachspress, E.: The ADI model problem (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benner, P., Breiten, T. Low rank methods for a class of generalized Lyapunov equations and related issues. Numer. Math. 124, 441–470 (2013). https://doi.org/10.1007/s00211-013-0521-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-013-0521-0