Abstract
Dynamical systems that involve non-linearity of the dynamics is a major challenge encountered in learning these systems. Similarly, the lack of adequate models for phenomena that reflect the governing physics can be an obstacle to an appropriate analysis. Nonetheless, some numerically or experimentally measured data can be found. Based on this data, and using a data-driven method such as the Loewner framework, it is possible to manage this data to derive a high fidelity reduced dynamical system that mimics the behaviour of the original data. In this paper, we tackle the issue of large amount of data presented by samples of transfer functions in a frequency-domain. The main step in this framework consists in computing singular value decomposition (SVD) of the Loewner matrix which provides accurate reduced systems. However, the large amount of data prevents this decomposition from being computed properly. We exploit the fact that the Loewner and shifted Loewner matrices, the key tools of Loewner framework, satisfy certain large scale Sylvester matrix equations. Using an extended block Krylov subspace method, a good approximation in a factored form of the Loewner and shifted Loewner matrices can be obtained and a minimal computation cost of the SVD is ensured. This method facilitates the process of a large amount of data and guarantees a good quality of the inferred model at the end of the process. Accuracy and efficiency of our method are assessed in the final section.
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Notes
The superscript \({^\star }\) stands for the complex transpose conjugate, i.e, \(F^{\star }= {\overline{F}}^T\).
References
Antoulas, A.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)
Antoulas, A., Beattie, C., Gugercine, S.: Interpolatory Methods for Model Reduction. Computer Science and Engineering, vol. 21. SIAM, Philadelphia (2020)
Antoulas, A., Lefteriu, S., Ionita, A.: A tutorial introduction to the Loewner framework for model reduction. Model Reduct. Approx. 8, 0898–1221 (2017)
Abdaoui, I., Elbouyahyaoui, L., Heyouni, M.: An alternative extended block Arnoldi method for solving low-rank Sylvester equations. Comput. Math. Appl. 78(8), 2817–2830 (2019)
Barkouki, H., Bentbib, A.H., Jbilou, K.: An extended nonsymmetric block Lanczos method for model reduction in large scale dynamical systems. Calcolo 55(1), 1–23 (2018)
Benner, P., Kürschner, P.: Computing real low-rank solutions of Sylvester equations by the factored ADI method. Comput. Math. Appl. 67(9), 0898–1221 (2009)
Benner, P., Ohlberger, M., Cohen, A., Willcox, K.: Model Reduction and Approximation. SIAM, Philadelphia (2017)
Cherifi, K., Goyal, P., Benner, P.: A greedy data collection scheme for linear dynamical systems. https://arxiv.org/abs/2107.12950v1 (2021)
Frangos, M., Jaimoukha, I.M.: Adaptive rational Krylov algorithms for model reduction. In: Proceedings of the European Control Conference, pp. 4179–4186 (2007)
Golub, G., Nash, S., Loan, C.: A Hessenberg–Schur method for the problem ax + xb = c. IEEE Trans. Autom. Control 24(6), 909–913 (1979)
Gosea, I., Gugercin, S., Beattie, C.: Data-driven balancing of linear dynamical systems. https://arxiv.org/abs/2104.01006v1 (2021)
Gosea, I., Vassal, C., Antoulas, A.: Data-driven modeling and control of large-scale dynamical systems in the Loewner framework. https://arxiv.org/abs/2108.11870 (2021)
Gosea, I.V., Zhang, Q., Antoulas, A.: Data-driven modeling from noisy measurements. Proc. Appl. Math. Mech. 67(9), 0898–1221 (2021)
Grimme, E.: Krylov projection methods for model reduction. Ph.D. thesis, Coordinated Science Laboratory, University of Illinois at Urbana Champaign (1997)
Gugercin, S., Antoulas, A.C.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2003)
Hamadi, M.A., Jbilou, K., Ratnani, A.: Model reduction method in large scale dynamical systems using an extended-rational block Arnoldi method. J. Appl. Math. Comput. 68(1), 271–293 (2022)
Heyouni, M., Jbilou, K.: An extended block Arnoldi method for large matrix Riccati equations. Electron. Trans. Numer. Anal. 33, 53–62 (2009)
Hochman, A.: Fast singular-value decomposition of Loewner matrices for state-space macromodeling. In: IEEE 24th Electrical Performance of Electronic Packaging and Systems (EPEPS), pp. 177–180 (2015)
Jbilou, K.: Low rank approximate solutions to large Sylvester matrix equations. Appl. Math. Comput. 177(1), 365–376 (2006)
Knizhnerman, L., Druskin, D., Zaslavsky, M.: On optimal convergence rate of the rational Krylov subspace reduction for electromagnetic problems in unbounded domains. SIAM J. Numer. Anal. 47(2), 953–971 (2009)
Lefteriu, S., Antoulas, A.: A new approach to modeling multiport systems from frequency-domain data. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29, 14–27 (2010)
Mayo, A., Antoulas, A.: A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2–3), 634–662 (2007)
Mehrmann, V., Stykel, T.: Balanced truncation model reduction for large-scale systems in descriptor form. In: Lecture Notes in Computational Science and Engineering, vol. 45, pp. 83–115 (2005)
Moore, B.C.: Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Trans. Auto. Control 26, 17–32 (1981)
Palitta, D., Lefteriu, S.: An efficient, memory-saving approach for the Loewner framework. https://arxiv.org/abs/2103.07146 (2021)
Peherstorfer, B., Gugercin, S., Willcox, K.: Data-driven reduced model construction with time-domain Loewner models. SIAM J. Sci. Comput. 39(5), A2152-78 (2017)
Simoncini, V.: A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J. Sci. Comput. 29(3), 1268–1288 (2007)
Simoncini, V., Szyld, D.B., Marlliny, M.: On two numerical methods for the solution of large-scale algebraic Riccati equations. IMA J. Numer. Anal. 34, 904–920 (2014)
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Hamadi, M.A., Jbilou, K. & Ratnani, A. A Data-Driven Krylov Model Order Reduction for Large-Scale Dynamical Systems. J Sci Comput 95, 2 (2023). https://doi.org/10.1007/s10915-023-02122-8
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DOI: https://doi.org/10.1007/s10915-023-02122-8