Abstract
Given optimal interpolation points σ 1,…,σ r , the \(\mathcal {H}_{2}\)-optimal reduced order model of order r can be obtained for a linear time-invariant system of order n≫r by simple projection (whereas it is not a trivial task to find those interpolation points). Our approach to linear time-invariant systems depending on parameters \(p\in \mathbb {R}^{d}\) is to approximate their parametric dependence as a so-called metamodel, which in turn allows us to set up the corresponding parametrized reduced order models. The construction of the metamodel we suggest involves the coefficients of the characteristic polynomial and radial basis function interpolation, and thus allows for an accurate and efficient approximation of σ 1(p),…,σ r (p). As the computation of the projection still includes large system solves, this metamodel is not sufficient to construct a fast and truly parametric reduced system. Setting up a medium-size model without extra cost, we present a possible answer to this. We illustrate the proposed method with several numerical examples.
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References
Amsallem, D., Farhat, C.: Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46(7), 1803–1813 (2008)
Antoulas, A., Ionita, A, Lefteriu, S: On two-variable rational interpolation. Linear Algebra Appl. 436(8), 2889–2915 (2012)
Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math .Soc. 68(3), 337–404 (1950). doi:10.1090/S0002-9947-1950-0051437-7
Baur, U., Beattie, C.A., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Sci. Comput. 33(5), 2489–2518 (2011a)
Baur, U., Benner, P., Greiner, A., Korvink, J., Lienemann, J., Moosmann, C.: Parameter preserving model order reduction for mems applications. Math. Comp. Model. Dyn. 17(4), 297–317 (2011b). doi:10.1080/13873954.2011.547658
Benner, P., Li, J.R., Penzl, T.: Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems. Numer. Lin. Alg. Appl. 15(9), 755–777 (2008)
Benner, P., Gugercin, S., Willcox, K.: A survey of model reduction methods for parametric systems. Max Planck Institute Magdeburg Preprint MPIMD/13-14, available from http://www.mpi-magdeburg.mpg.de/preprints/ (2013)
Benner, P., Grundel, S., Hornung, N.: Parametric Model Order Reduction with Small \(\mathcal {H}_{2}\)-Error Using Radial Basis Functions. Max Planck Institute Magdeburg Preprint MPIMD/14-01, available from http://www.mpi-magdeburg.mpg.de/preprints/ (2014)
Boyd, J.P.: Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals. J. Sci. Comput. 2(2), 99–109 (1987). doi10.1007/BF01061480
Bunse-Gerstner, A., Kubalińska, D., Vossen, G., Wilczek, D.: h 2-norm optimal model reduction for large scale discrete dynamical MIMO systems. J. Comput. Appl. Math. 233(5), 1202–1216 (2010). doi10.1016/j.cam.2008.12.029
Fasshauer, G.E.: Meshfree Approximation Methods with MATLAB (with CD-ROM). World Scientific (2007)
Grimme, E.J.: Krylov projection methods for model reduction Dissertation. University of Illinois at Urbana-Champaign, USA (1997)
Gugercin, S., Antoulas, A.C., Beattie, C.: \(\mathcal {H}_{2}\) Model Reduction for Large-Scale Linear Dynamical Systems. SIAM. J. Matrix Anal. Appl. 30(2), 609–638 (2008)
Meier, I.L., Luenberger, D.: Approximation of linear constant systems. IEEE Trans. Automat. Control 12(5), 585–588 (1967). doi:10.1109/TAC.1967.1098680
Moosmann, C., Rudnyi, E., Greiner, A., Korvink, J., Hornung, M: Parameter preserving model order reduction of a flow meter. Tech. Proc. Nanotech 2005 (2005)
Panzer, H., Hubele, J., Eid, R, Lohmann, B.: Generating a parametric finite element model of a 3D cantilever Timoshenko beam using MATLAB. Technical reports on automatic control, available from https://mediatum.ub.tum.de (2009)
Panzer, H., Mohring, J., Eid, R., Lohmann, B.: Parametric model order reduction by matrix interpolation. at-Automatisierungstechnik 58(8), 475–484 (2010)
Rieger, C, Schaback, R, Zwicknagl, B: Sampling and stability. In: Dæhlen, M., Floater, M., Lyche, T, Merrien, J.L., Mørken, K., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces, Lect. Notes Comput. Sc., vol. 5862, pp. 347–369 . Springer, Berlin Heidelberg (2010), doi:10.1007/978-3-642-11620-9_23
Schaback, R.: Native Hilbert spaces for radial basis functions I. In: Müller, M.W., Buhmann, M.D., Mache, D.H., Felten, M. (eds.) New Developments in Approximation Theory, ISNM Internat. Ser. Numer. Math., vol 132, Birkhäuser Basel, pp 255–282 (1999), doi:10.1007/978-3-0348-8696-3_16
Positive definite functions and generalizations, an historical survey. Rocky Mountain J. Math. 6(3), 409–434 (1976). doi:10.1216/RMJ-1976-6-3-409
Van Dooren, P., Gallivan, K.A., Absil, P.-A.: \(\mathcal {H}_{2}\)-optimal model reduction of MIMO systems. Appl. Math. Lett. 21(12), 1267–1273 (2008). doi:10.1016/j.aml.2007.09.015
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Communicated by: K. Urban
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Benner, P., Grundel, S. & Hornung, N. Parametric model order reduction with a small \(\mathcal {H}_{2}\)-error using radial basis functions. Adv Comput Math 41, 1231–1253 (2015). https://doi.org/10.1007/s10444-015-9410-7
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DOI: https://doi.org/10.1007/s10444-015-9410-7
Keywords
- Linear time-invariant systems
- Parametric model order reduction
- Rational Krylov \(\mathcal {H}_{2}\) approximation
- Reproducing kernel hilbert spaces
- Radial basis functions