A. C. Antoulas. Approximation of Large-Scale Dynamical Systems. Volume 6 of Adv. Des. Control. SIAM Publications, Philadelphia, PA, 2005. ISBN 9780898715293. doi:10.1137/1.9780898718713.


A. C. Antoulas, C. A. Beattie, and S. Gugercin. Interpolatory model reduction of large-scale dynamical systems. In Javad Mohammadpour and Karolos M. Grigoriadis, editors, Efficient Modeling and Control of Large-Scale Systems, pages 3–58. Springer US, 2010. doi:10.1007/978-1-4419-5757-3_1.


S. Barrachina, P. Benner, and E. S. Quintana-Ortí. Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function. Numer. Algorithms, 46(4):351–368, 2007. doi:10.1007/s11075-007-9143-x.


C. A. Beattie and S. Gugercin. Interpolatory projection methods for structure-preserving model reduction. Systems Control Lett., 58(3):225–232, 2009. doi:10.1016/j.sysconle.2008.10.016.


C. A. Beattie and S. Gugercin. Realization-independent $\mathcal H_2$-approximation. In 51st IEEE Conference on Decision and Control (CDC), 4953–4958. 2012. doi:10.1109/CDC.2012.6426344.


P. Benner, M. Köhler, and J. Saak. Sparse-dense Sylvester equations in $H_2$-model order reduction. Preprint MPIMD/11-11, Max Planck Institute Magdeburg, December 2011. URL:


Peter Benner, Zvonimir Bujanović, Patrick Kürschner, and Jens Saak. Radi: a low-rank adi-type algorithm for large scale algebraic riccati equations. Numerische mathematik, 138(2):301–330, 2018.


Peter Binev, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods. SIAM journal on mathematical analysis, 43(3):1457–1472, 2011.


Tobias Breiten, Chris A. Beattie, and Serkan Gugercin. $\mathcal H_2$-gap model reduction for stabilizable and detectable systems. e-prints 1909.13764, arXiv, 2019. math.NA. URL:


Andreas Buhr, Christian Engwer, Mario Ohlberger, and Stephan Rave. A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations. arXiv preprint arXiv:1407.8005, 2014.


Andreas Buhr and Kathrin Smetana. Randomized local model order reduction. SIAM journal on scientific computing, 40(4):A2120–A2151, 2018.


Y. Chahlaoui, D. Lemonnier, A. Vandendorpe, and P. Van Dooren. Second-order balanced truncation. Linear Algebra Appl., 415(2–3):373–384, 2006. doi:10.1016/j.laa.2004.03.032.


Martin A Grepl and Anthony T Patera. A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 39(1):157–181, 2005.


S. Gugercin, A. C. Antoulas, and C. Beattie. $\mathcal H_2$ model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl., 30(2):609–638, 2008. doi:10.1137/060666123.


B. Haasdonk. Reduced basis methods for parametrized PDEs—a tutorial introduction for stationary and instationary problems. In P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, editors, Model Reduction and Approximation: Theory and Algorithms, pages 65–136. SIAM, 2017. doi:10.1137/1.9781611974829.ch2.


B. Haasdonk, M. Dihlmann, and M. Ohlberger. A training set and multiple basis generation approach for parametrized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst., 17(4):423–442, 2011.


B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Math. Model. Numer. Anal., 42(2):277–302, 2008.


Nathan Halko, Per-Gunnar Martinsson, and Joel A Tropp. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM review, 53(2):217–288, 2011.


Jan S Hesthaven and Stefano Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018.


C. Himpe, T. Leibner, and S. Rave. Hierarchical approximate proper orthogonal decomposition. SIAM J. Sci. Comput., 40(5):A3267–A3292, 2018. doi:10.1137/16M1085413.


Michael Hinze, René Pinnau, Michael Ulbrich, and Stefan Ulbrich. Optimization with PDE constraints. Volume 23. Springer Science & Business Media, 2008.


Patrick Kürschner. Efficient low-rank solution of large-scale matrix equations. PhD thesis, Shaker Verlag Aachen, 2016. URL:


Richard Bruno Lehoucq. Analysis and implementation of an implicitly restarted Arnoldi iteration. PhD thesis, Rice University, Houston, USA, 1995. URL:


J.-R. Li and J. White. Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl., 24(1):260–280, 2002. doi:10.1137/S0895479801384937.


D. G. Meyer and S. Srinivasan. Balancing and model reduction for second-order form linear systems. IEEE Trans. Autom. Control, 41(11):1632–1644, 1996. doi:10.1109/9.544000.


D. Mustafa and K. Glover. Controller reduction by $\mathcal H_\infty $-balanced truncation. IEEE Trans. Autom. Control, 36(6):668–682, 1991. doi:10.1109/9.86941.


Jorge Nocedal and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006.


P. C. Opdenacker and E. A. Jonckheere. A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Trans. Circuits Syst., 35(2):184–189, 1988. doi:10.1109/31.1720.


G. Petrova & P. Wojtaszczyk R. DeVore. Greedy algorithms for reduced bases in banach spaces. Constructive Approximation, pages 455–466, 2013.


T. Reis and T. Stykel. Balanced truncation model reduction of second-order systems. Math. Comput. Model. Dyn. Syst., 14(5):391–406, 2008. doi:10.1080/13873950701844170.


Joost Rommes and Nelson Martins. Efficient computation of multivariable transfer function dominant poles using subspace acceleration. IEEE transactions on power systems, 21(4):1471–1483, 2006.


J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton, and J. N. Kutz. On dynamic mode decomposition: theory and applications. Journal of Computational Dynamics, pages 391–421, 2014.


Qian Wang, Jan S Hesthaven, and Deep Ray. Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem. Journal of computational physics, 384:289–307, 2019.


S. Wyatt. Issues in Interpolatory Model Reduction: Inexact Solves, Second-order Systems and DAEs. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA, May 2012. URL:


Y. Xu and T. Zeng. Optimal $\mathcal H_2$ model reduction for large scale MIMO systems via tangential interpolation. Int. J. Numer. Anal. Model., 8(1):174–188, 2011. URL:


K. Zhou, G. Salomon, and E. Wu. Balanced realization and model reduction for unstable systems. Internat. J. Robust Nonlinear Control, 9(3):183–198, 1999. doi:10.1002/(SICI)1099-1239(199903)9:3<183::AID-RNC399>3.0.CO;2-E.