{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,22]],"date-time":"2025-04-22T15:10:50Z","timestamp":1745334650723,"version":"3.37.3"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T00:00:00Z","timestamp":1721347200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T00:00:00Z","timestamp":1721347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems (MPI Magdeburg)"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2024,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.<\/jats:p>","DOI":"10.1007\/s10444-024-10166-z","type":"journal-article","created":{"date-parts":[[2024,7,19]],"date-time":"2024-07-19T13:02:02Z","timestamp":1721394122000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction"],"prefix":"10.1007","volume":"50","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7942-5703","authenticated-orcid":false,"given":"Quirin","family":"Aumann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Steffen W. R.","family":"Werner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,19]]},"reference":[{"issue":"3","key":"10166_CR1","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/j.sysconle.2008.10.016","volume":"58","author":"CA Beattie","year":"2009","unstructured":"Beattie, C.A., Gugercin, S.: Interpolatory projection methods for structure-preserving model reduction. Syst. Control Lett. 58(3), 225\u2013232 (2009). https:\/\/doi.org\/10.1016\/j.sysconle.2008.10.016","journal-title":"Syst. Control Lett."},{"key":"10166_CR2","unstructured":"Abraham, R., Marsden, J.E.: Foundations of mechanics, 2nd edn. Addison-Wesley Publishing Company, Inc., Redwood City (1987). https:\/\/resolver.caltech.edu\/CaltechBOOK:1987.001"},{"key":"10166_CR3","doi-asserted-by":"publisher","unstructured":"Werner, S.W.R.: Structure-preserving model reduction for mechanical systems. Dissertation, Otto-von-Guericke-Universit\u00e4t, Magdeburg, Germany (2021). https:\/\/doi.org\/10.25673\/38617","DOI":"10.25673\/38617"},{"key":"10166_CR4","doi-asserted-by":"publisher","unstructured":"Wu, K.: Power converters with digital filter feedback control. Academic Press, London (2016). https:\/\/doi.org\/10.1016\/C2015-0-01103-0","DOI":"10.1016\/C2015-0-01103-0"},{"key":"10166_CR5","doi-asserted-by":"publisher","unstructured":"Gao, Q., Karimi, H.R.: Stability, control and application of time-delay systems. Butterworth-Heinemann, Oxford (2019). https:\/\/doi.org\/10.1016\/C2017-0-02175-4","DOI":"10.1016\/C2017-0-02175-4"},{"key":"10166_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115076","volume":"397","author":"Q Aumann","year":"2022","unstructured":"Aumann, Q., Deckers, E., Jonckheere, S., Desmet, W., M\u00fcller, G.: Automatic model order reduction for systems with frequency-dependent material properties. Comput. Methods Appl. Mech. Eng. 397, 115076 (2022). https:\/\/doi.org\/10.1016\/j.cma.2022.115076","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"1","key":"10166_CR7","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1137\/120877556","volume":"35","author":"R Van Beeumen","year":"2013","unstructured":"Van Beeumen, R., Meerbergen, K., Michiels, W.: A rational Krylov method based on Hermite interpolation for nonlinear eigenvalue problems. SIAM J. Sci. Comput. 35(1), 327\u2013350 (2013). https:\/\/doi.org\/10.1137\/120877556","journal-title":"SIAM J. Sci. Comput."},{"key":"10166_CR8","doi-asserted-by":"publisher","unstructured":"Cohen, G., Hauck, A., Kaltenbacher, M., Otsuru, T.: Different types of finite elements. In: Marburg, S., Nolte, B. (eds.) Computational Acoustics of Noise Propagation in Fluids \u2013 Finite and Boundary Element Methods, pp. 57\u201388. Springer, Berlin, Heidelberg (2008). https:\/\/doi.org\/10.1007\/978-3-540-77448-8_3","DOI":"10.1007\/978-3-540-77448-8_3"},{"key":"10166_CR9","doi-asserted-by":"publisher","unstructured":"Antoulas, A.C.: Approximation of large-scale dynamical systems. Adv. Des. Control, vol. 6. SIAM, Philadelphia, PA (2005). https:\/\/doi.org\/10.1137\/1.9780898718713","DOI":"10.1137\/1.9780898718713"},{"key":"10166_CR10","doi-asserted-by":"publisher","unstructured":"Deckers, E., Desmet, W., Meerbergen, K., Naets, F.: Case studies of model order reduction for acoustics and vibrations. In: Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Silveira, L.M. (eds.) Model Order Reduction. Volume 3: Applications, pp. 76\u2013110. De Gruyter, Berlin, Boston (2021). https:\/\/doi.org\/10.1515\/9783110499001-003","DOI":"10.1515\/9783110499001-003"},{"issue":"13","key":"10166_CR11","doi-asserted-by":"publisher","first-page":"1636","DOI":"10.1002\/nme.4271","volume":"90","author":"U Hetmaniuk","year":"2012","unstructured":"Hetmaniuk, U., Tezaur, R., Farhat, C.: Review and assessment of interpolatory model order reduction methods for frequency response structural dynamics and acoustics problems. Int. J. Numer. Methods Eng. 90(13), 1636\u20131662 (2012). https:\/\/doi.org\/10.1002\/nme.4271","journal-title":"Int. J. Numer. Methods Eng."},{"key":"10166_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2022.117363","volume":"543","author":"Q Aumann","year":"2023","unstructured":"Aumann, Q., Werner, S.W.R.: Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods. J. Sound Vib. 543, 117363 (2023). https:\/\/doi.org\/10.1016\/j.jsv.2022.117363","journal-title":"J. Sound Vib."},{"key":"10166_CR13","doi-asserted-by":"publisher","unstructured":"Beddig, R.S., Benner, P., Dorschky, I., Reis, T., Schwerdtner, P., Voigt, M., Werner, S.W.R.: Structure-preserving model reduction for dissipative mechanical systems. In: Eberhard, P. (ed.) Calm, Smooth and Smart. Lect. Notes Appl. Comput. Mech., vol. 102, pp. 209\u2013230. Springer, Cham (2024). https:\/\/doi.org\/10.1007\/978-3-031-36143-2_11","DOI":"10.1007\/978-3-031-36143-2_11"},{"issue":"10","key":"10166_CR14","doi-asserted-by":"publisher","first-page":"5026","DOI":"10.1109\/TAC.2017.2679479","volume":"62","author":"X Cheng","year":"2017","unstructured":"Cheng, X., Kawano, Y., Scherpen, J.M.A.: Reduction of second-order network systems with structure preservation. IEEE Trans. Autom. Control. 62(10), 5026\u20135038 (2017). https:\/\/doi.org\/10.1109\/TAC.2017.2679479","journal-title":"IEEE Trans. Autom. Control."},{"issue":"1","key":"10166_CR15","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1137\/17M1125303","volume":"40","author":"H Egger","year":"2018","unstructured":"Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N., Mehrmann, V.: On structure-preserving model reduction for damped wave propagation in transport networks. SIAM J. Sci. Comput. 40(1), 331\u2013365 (2018). https:\/\/doi.org\/10.1137\/17M1125303","journal-title":"SIAM J. Sci. Comput."},{"key":"10166_CR16","doi-asserted-by":"publisher","unstructured":"Bendokat, T., Zimmermann, R.: Geometric optimization for structure-preserving model reduction of Hamiltonian systems. IFAC-Pap. 55(20), 457\u2013462 (2022). https:\/\/doi.org\/10.1016\/j.ifacol.2022.09.137. 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022","DOI":"10.1016\/j.ifacol.2022.09.137"},{"issue":"9","key":"10166_CR17","doi-asserted-by":"publisher","first-page":"1963","DOI":"10.1016\/j.automatica.2012.05.052","volume":"48","author":"S Gugercin","year":"2012","unstructured":"Gugercin, S., Polyuga, R.V., Beattie, C., Schaft, A.: Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica J. IFAC. 48(9), 1963\u20131974 (2012). https:\/\/doi.org\/10.1016\/j.automatica.2012.05.052","journal-title":"Automatica J. IFAC."},{"key":"10166_CR18","doi-asserted-by":"publisher","unstructured":"Hesthaven, J.S., Pagliantini, C., Ripamonti, N.: Rank-adaptive structure-preserving model order reduction of Hamiltonian systems. ESAIM: Math. Model. Numer. Anal. 56(2), 617\u2013650 (2022). https:\/\/doi.org\/10.1051\/m2an\/2022013","DOI":"10.1051\/m2an\/2022013"},{"key":"10166_CR19","doi-asserted-by":"publisher","unstructured":"Chellappa, S., Feng, L., Benner, P.: An adaptive sampling approach for the reduced basis method. In: Beattie, C., Benner, P., Embree, M., Gugercin, S., Lefteriu, S. (eds.) Realization and Model Reduction of Dynamical Systems, pp. 137\u2013155. Springer, Cham (2022). https:\/\/doi.org\/10.1007\/978-3-030-95157-3_8","DOI":"10.1007\/978-3-030-95157-3_8"},{"key":"10166_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.matcom.2015.08.017","volume":"122","author":"T Bonin","year":"2016","unstructured":"Bonin, T., Fa\u00dfbender, H., Soppa, A., Zaeh, M.: A fully adaptive rational global Arnoldi method for the model-order reduction of second-order MIMO systems with proportional damping. Math. Comput. Simul. 122, 1\u201319 (2016). https:\/\/doi.org\/10.1016\/j.matcom.2015.08.017","journal-title":"Math. Comput. Simul."},{"issue":"12","key":"10166_CR21","doi-asserted-by":"publisher","first-page":"1872","DOI":"10.1109\/TCPMT.2015.2491341","volume":"5","author":"L Feng","year":"2015","unstructured":"Feng, L., Korvink, J.G., Benner, P.: A fully adaptive scheme for model order reduction based on moment matching. IEEE Trans. Compon. Packag. Manuf. Technol. 5(12), 1872\u20131884 (2015). https:\/\/doi.org\/10.1109\/TCPMT.2015.2491341","journal-title":"IEEE Trans. Compon. Packag. Manuf. Technol."},{"issue":"10","key":"10166_CR22","doi-asserted-by":"publisher","first-page":"759","DOI":"10.1002\/nme.4609","volume":"97","author":"R Rumpler","year":"2014","unstructured":"Rumpler, R., G\u00f6ransson, P., De\u00fc, J.-F.: A finite element approach combining a reduced-order system, Pad\u00e9 approximants, and an adaptive frequency windowing for fast multi-frequency solution of poro-acoustic problems. Int. J. Numer. Methods Eng. 97(10), 759\u2013784 (2014). https:\/\/doi.org\/10.1002\/nme.4609","journal-title":"Int. J. Numer. Methods Eng."},{"key":"10166_CR23","doi-asserted-by":"publisher","unstructured":"Panzer, H.K.F., Wolf, T., Lohmann, B.: $$H_{2}$$ and $$H_{\\infty }$$ error bounds for model order reduction of second order systems by Krylov subspace methods. In: 2013 European Control Conference (ECC), pp. 4484\u20134489 (2013). https:\/\/doi.org\/10.23919\/ECC.2013.6669657","DOI":"10.23919\/ECC.2013.6669657"},{"key":"10166_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2022.117427","volume":"545","author":"Q Aumann","year":"2023","unstructured":"Aumann, Q., M\u00fcller, G.: Robust error assessment for reduced order vibro-acoustic problems. J. Sound Vib. 545, 117427 (2023). https:\/\/doi.org\/10.1016\/j.jsv.2022.117427","journal-title":"J. Sound Vib."},{"key":"10166_CR25","doi-asserted-by":"publisher","unstructured":"Feng, L., Lombardi, L., Benner, P., Romano, D., Antonini, G.: Model order reduction for delayed PEEC models with guaranteed accuracy and observed stability. IEEE Trans. Circuits Syst. I: Regul. Pap. 69(10), 4177\u20134190 (2022). https:\/\/doi.org\/10.1109\/TCSI.2022.3189389","DOI":"10.1109\/TCSI.2022.3189389"},{"issue":"2","key":"10166_CR26","doi-asserted-by":"publisher","first-page":"928","DOI":"10.1137\/19M125892X","volume":"41","author":"N Aliyev","year":"2020","unstructured":"Aliyev, N., Benner, P., Mengi, E., Voigt, M.: A subspace framework for $${\\cal{H} }_{\\infty }$$-norm minimization. SIAM J. Matrix Anal. Appl. 41(2), 928\u2013956 (2020). https:\/\/doi.org\/10.1137\/19M125892X","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"10166_CR27","doi-asserted-by":"publisher","unstructured":"Schwerdtner, P., Voigt, M.: Adaptive sampling for structure-preserving model order reduction of port-Hamiltonian systems. IFAC-Pap. 54(19), 143\u2013148 (2021). https:\/\/doi.org\/10.1016\/j.ifacol.2021.11.069. 7th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC 2021","DOI":"10.1016\/j.ifacol.2021.11.069"},{"issue":"2","key":"10166_CR28","doi-asserted-by":"publisher","first-page":"609","DOI":"10.1137\/060666123","volume":"30","author":"S Gugercin","year":"2008","unstructured":"Gugercin, S., Antoulas, A.C., Beattie, C.: $$\\cal{H} _{2}$$ model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30(2), 609\u2013638 (2008). https:\/\/doi.org\/10.1137\/060666123","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"10166_CR29","unstructured":"Wyatt, S.: Issues in interpolatory model reduction: inexact solves, second-order systems and DAEs. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA (2012). http:\/\/hdl.handle.net\/10919\/27668"},{"key":"10166_CR30","doi-asserted-by":"publisher","unstructured":"Aumann, Q., M\u00fcller, G.: An adaptive method for reducing second-order dynamical systems. IFAC-Pap. 55(20), 337\u2013342 (2022). https:\/\/doi.org\/10.1016\/j.ifacol.2022.09.118. 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022","DOI":"10.1016\/j.ifacol.2022.09.118"},{"key":"10166_CR31","doi-asserted-by":"publisher","unstructured":"Beattie, C.A., Gugercin, S.: Realization-independent $$\\cal{H}_{2}$$-approximation. In: 51st IEEE Conference on Decision and Control (CDC), pp. 4953\u20134958 (2012). https:\/\/doi.org\/10.1109\/CDC.2012.6426344","DOI":"10.1109\/CDC.2012.6426344"},{"key":"10166_CR32","doi-asserted-by":"publisher","unstructured":"Sinani, K., Gugercin, S., Beattie, C.: A structure-preserving model reduction algorithm for dynamical systems with nonlinear frequency dependence. IFAC-Pap. 49(9), 56\u201361 (2016). https:\/\/doi.org\/10.1016\/j.ifacol.2016.07.492. 6th IFAC Symposium on System Structure and Control SSSC 2016","DOI":"10.1016\/j.ifacol.2016.07.492"},{"key":"10166_CR33","doi-asserted-by":"publisher","unstructured":"Mayo, A.J., Antoulas, A.C.: A framework for the solution of the generalized realization problem. Linear Algebra Appl. 425(2\u20133), 634\u2013662 (2007). https:\/\/doi.org\/10.1016\/j.laa.2007.03.008. Special issue in honor of P.\u00a0A. Fuhrmann, Edited by A.\u00a0C. Antoulas, U. Helmke, J. Rosenthal, V. Vinnikov, and E. Zerz","DOI":"10.1016\/j.laa.2007.03.008"},{"key":"10166_CR34","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1016\/j.laa.2017.09.030","volume":"537","author":"P Schulze","year":"2018","unstructured":"Schulze, P., Unger, B., Beattie, C., Gugercin, S.: Data-driven structured realization. Linear Algebra Appl. 537, 250\u2013286 (2018). https:\/\/doi.org\/10.1016\/j.laa.2017.09.030","journal-title":"Linear Algebra Appl."},{"key":"10166_CR35","doi-asserted-by":"publisher","unstructured":"Gosea, I.V., Gugercin, S., Werner, S.W.R.: Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems. e-print 2303.12576, arXiv (2023). https:\/\/doi.org\/10.48550\/arXiv.2303.12576. Numerical Analysis (math.NA)","DOI":"10.48550\/arXiv.2303.12576"},{"key":"10166_CR36","doi-asserted-by":"publisher","unstructured":"Werner, S.W.R., Gosea, I.V., Gugercin, S.: Structured vector fitting framework for mechanical systems. IFAC-Pap. 55(20), 163\u2013168 (2022). https:\/\/doi.org\/10.1016\/j.ifacol.2022.09.089. 10th Vienna International Conference on Mathematical Modelling MATHMOD 2022","DOI":"10.1016\/j.ifacol.2022.09.089"},{"key":"10166_CR37","doi-asserted-by":"publisher","unstructured":"Antoulas, A.C., Lefteriu, S., Ionita, A.C.: A tutorial introduction to the Loewner framework for model reduction. In: Benner, P., Ohlberger, M., Cohen, A., Willcox, K. (eds.) Model Reduction and Approximation: Theory and Algorithms. Computational Science & Engineering, pp. 335\u2013376. SIAM, Philadelphia, PA (2017). https:\/\/doi.org\/10.1137\/1.9781611974829.ch8","DOI":"10.1137\/1.9781611974829.ch8"},{"issue":"3","key":"10166_CR38","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1080\/13873954.2018.1464030","volume":"24","author":"A Castagnotto","year":"2018","unstructured":"Castagnotto, A., Lohmann, B.: A new framework for $$\\cal{H} _{2}$$-optimal model reduction. Math. Comput. Model. Dyn. Syst. 24(3), 236\u2013257 (2018). https:\/\/doi.org\/10.1080\/13873954.2018.1464030","journal-title":"Math. Comput. Model. Dyn. Syst."},{"issue":"5","key":"10166_CR39","doi-asserted-by":"publisher","first-page":"1231","DOI":"10.1007\/s10444-015-9410-7","volume":"41","author":"P Benner","year":"2015","unstructured":"Benner, P., Grundel, S., Hornung, N.: Parametric model order reduction with a small $$\\cal{H} _{2}$$-error using radial basis functions. Adv. Comput. Math. 41(5), 1231\u20131253 (2015). https:\/\/doi.org\/10.1007\/s10444-015-9410-7","journal-title":"Adv. Comput. Math."},{"key":"10166_CR40","doi-asserted-by":"publisher","unstructured":"Vuillemin, P., Poussot-Vassal, C., Alazard, D.: $$\\cal{H}_{2}$$ optimal and frequency limited approximation methods for large-scale LTI dynamical systems. IFAC Proc. Vol. 46(2), 719\u2013724 (2013). https:\/\/doi.org\/10.3182\/20130204-3-FR-2033.00061. 5th IFAC Symposium on System Structure and Control","DOI":"10.3182\/20130204-3-FR-2033.00061"},{"key":"10166_CR41","doi-asserted-by":"publisher","unstructured":"Karachalios, D.S., Gosea, I.V., Antoulas, A.C.: The Loewner framework for system identification and reduction. In: Benner, P., Schilders, W., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Silveira, L.M. (eds.) Model Order Reduction. Volume 1: System- and Data-Driven Methods and Algorithms, pp. 181\u2013228. De Gruyter, Berlin, Boston (2021). https:\/\/doi.org\/10.1515\/9783110498967-006","DOI":"10.1515\/9783110498967-006"},{"issue":"3","key":"10166_CR42","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1109\/61.772353","volume":"14","author":"B Gustavsen","year":"1999","unstructured":"Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Del. 14(3), 1052\u20131061 (1999). https:\/\/doi.org\/10.1109\/61.772353","journal-title":"IEEE Trans. Power Del."},{"issue":"2","key":"10166_CR43","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1137\/140961511","volume":"37","author":"Z Drma\u010d","year":"2015","unstructured":"Drma\u010d, Z., Gugercin, S., Beattie, C.: Quadrature-based vector fitting for discretized $$\\cal{H} _{2}$$ approximation. SIAM J. Sci. Comput. 37(2), 625\u2013652 (2015). https:\/\/doi.org\/10.1137\/140961511","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"10166_CR44","doi-asserted-by":"publisher","first-page":"2049","DOI":"10.1137\/15M1025426","volume":"39","author":"M Berljafa","year":"2017","unstructured":"Berljafa, M., G\u00fcttel, S.: The RKFIT algorithm for nonlinear rational approximation. SIAM J. Sci. Comput. 39(5), 2049\u20132071 (2017). https:\/\/doi.org\/10.1137\/15M1025426","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"10166_CR45","doi-asserted-by":"publisher","first-page":"1494","DOI":"10.1137\/16M1106122","volume":"40","author":"Y Nakatsukasa","year":"2018","unstructured":"Nakatsukasa, Y., S\u00e8te, O., Trefethen, L.N.: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40(3), 1494\u20131522 (2018). https:\/\/doi.org\/10.1137\/16M1106122","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"10166_CR46","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1109\/59.486093","volume":"11","author":"N Martins","year":"1996","unstructured":"Martins, N., Lima, L.T.G., Pinto, H.J.C.P.: Computing dominant poles of power system transfer functions. IEEE Trans. Power Syst. 11(1), 162\u2013170 (1996). https:\/\/doi.org\/10.1109\/59.486093","journal-title":"IEEE Trans. Power Syst."},{"issue":"5","key":"10166_CR47","doi-asserted-by":"publisher","first-page":"1010","DOI":"10.1137\/130906635","volume":"35","author":"S Gugercin","year":"2013","unstructured":"Gugercin, S., Stykel, T., Wyatt, S.: Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput. 35(5), 1010\u20131033 (2013). https:\/\/doi.org\/10.1137\/130906635","journal-title":"SIAM J. Sci. Comput."},{"key":"10166_CR48","doi-asserted-by":"publisher","unstructured":"Borggaard, J.T., Gugercin, S.: Model reduction for DAEs with an application to flow control. In: King, R. (ed.) Active Flow and Combustion Control 2014. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 127, pp. 381\u2013396. Springer, Cham (2015). https:\/\/doi.org\/10.1007\/978-3-319-11967-0_23","DOI":"10.1007\/978-3-319-11967-0_23"},{"key":"10166_CR49","doi-asserted-by":"publisher","unstructured":"Saak, J., K\u00f6hler, M., Benner, P.: M-M.E.S.S. \u2013 The matrix equations sparse solvers library (version 2.2). see also:\u00a0https:\/\/www.mpi-magdeburg.mpg.de\/projects\/mess (2022). https:\/\/doi.org\/10.5281\/zenodo.5938237","DOI":"10.5281\/zenodo.5938237"},{"key":"10166_CR50","doi-asserted-by":"publisher","unstructured":"Aumann, Q., Werner, S.W.R.: Code, data and results for the numerical experiments in \u201cAdaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction\u201d (version 1.2) (2024). https:\/\/doi.org\/10.5281\/zenodo.10945494","DOI":"10.5281\/zenodo.10945494"},{"key":"10166_CR51","doi-asserted-by":"publisher","unstructured":"Gugercin, S., Antoulas, A.C., Bedrossian, M.: Approximation of the international space station 1R and 12A models. In: Proceedings of the 40th IEEE Conference on Decision and Control, pp. 1515\u20131516 (2001). https:\/\/doi.org\/10.1109\/CDC.2001.981109","DOI":"10.1109\/CDC.2001.981109"},{"issue":"3\u20134","key":"10166_CR52","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1007\/s00211-023-01380-w","volume":"155","author":"P Benner","year":"2023","unstructured":"Benner, P., Gugercin, S., Werner, S.W.R.: A unifying framework for tangential interpolation of structured bilinear control systems. Numer. Math. 155(3\u20134), 445\u2013483 (2023). https:\/\/doi.org\/10.1007\/s00211-023-01380-w","journal-title":"Numer. Math."},{"key":"10166_CR53","unstructured":"Higham, N.J., Negri\u00a0Porzio, G.M., Tisseur, F.: An updated set of nonlinear eigenvalue problems. e-print 2019.5, MIMS (2019). http:\/\/eprints.maths.manchester.ac.uk\/id\/eprint\/2699"},{"key":"10166_CR54","unstructured":"Liao, B.-S.: Subspace projection methods for model order reduction and nonlinear eigenvalue computation. Dissertation, University of California, Davis, California, USA (2007)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-024-10166-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-024-10166-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-024-10166-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T07:55:01Z","timestamp":1724745301000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-024-10166-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,19]]},"references-count":54,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["10166"],"URL":"https:\/\/doi.org\/10.1007\/s10444-024-10166-z","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2024,7,19]]},"assertion":[{"value":"30 April 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 June 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 July 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"79"}}