Abstract
The solution of large-scale Lyapunov equations is an important tool for the solution of several engineering problems arising in optimal control and model order reduction. In this work, we investigate the case when the coefficient matrix of the equations presents a band structure. Exploiting the structure of this matrix, we can achive relevant reductions in the memory requirements and the number of floating-point operations. Additionally, the new solver efficiently leverages the parallelism of CPU–GPU platforms. Furthermore, it is integrated in the lyapack library to facilitate its use. The new codes are evaluated on the solution of several benchmarks, exposing significant runtime reductions with respect to the original CPU version in lyapack.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For numerical reasons, pivoting is required in the factorization of \(A\). Although, for simplicity, it is not included in our discussion, the lapack implementations and all the implementations presented include pivoting.
References
Antoulas A (2005) Approximation of large-scale dynamical systems. Philadelphia
Benner P (1997) Contributions to the numerical solution of algebraic Riccati equations and related eigenvalue problems. Logos-Verlag, Berlin
Benner P, Dufrechou E, Ezzatti P, Igounet P, Quintana-Ortí ES, Remón A (2014) Accelerating band linear algebra operations on GPUs with application in model reduction. In: Lecture notes in computer science. Springer
Benner P, Ezzatti P, Kressner D, Quintana-Ortí ES, Remón A (2010) Accelerating model reduction of large linear systems with graphics processors. In: Lecture notes in computer science. Springer, pp 88–97
Benner P, Ezzatti P, Kressner D, Quintana-Ortí ES, Remón A (2011) A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms. Parallel Comput 37(8):439–450
Benner P, Mehrmann V, Sorensen D (2005) Dimension reduction of large-scale systems. In: Lecture notes in computational science and engineering. Springer, Berlin
Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th national conference, ACM ’69ACM, New York, pp 157–172
Datta B (2004) Numerical methods for linear control systems. Elsevier Science, San Diego, CA, USA, London, UK
Dufrechou E, Ezzatti P, Quintana-Ortí ES, Remón A (2013) Accelerating the Lyapack library using GPUs. J Supercomput 65:1114–1124
Green M, Limebeer D (1995) Linear robust control. In: Prentice-Hall information and system sciences series. Prentice Hall, Englewood Cliffs, NJ
IMTEK: Oberwolfach model reduction benchmark collection. http://portal.uni-freiburg.de/imteksimulation/downloads/benchmark. Accessed 3 Nov 2014
Penzl T (1999) A cyclic low-rank smith method for large sparse Lyapunov equations. SIAM J Sci Comput 21(4):1401–1418
Penzl T (2000) LYAPACK: A Matlab toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear-quadratic optimal control problems. Users guide (version 1.0) .
Petkov P, Christov N, Konstantinov M (1991) Computational methods for linear control systems. Hertfordshire, UK
Wachspress E (2013) The ADI model problem. Springer, New York
Acknowledgments
Ernesto Dufrechou and Pablo Ezzatti acknowledge the support from Programa de Desarrollo de las Ciencias Básicas, and Agencia Nacional de Investigación e Innovación of Uruguay. Enrique S. Quintana-Ortí was supported by project TIN2011-23283 of the Ministry of Science and Competitiveness (MINECO) and EU FEDER, and project P1-1B2013-20 of the Fundació Caixa Castelló-Bancaixa and UJI.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Benner, P., Remón, A., Dufrechou, E. et al. Extending lyapack for the solution of band Lyapunov equations on hybrid CPU–GPU platforms. J Supercomput 71, 740–750 (2015). https://doi.org/10.1007/s11227-014-1322-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-014-1322-7