Bibliography¶
E. S. Al-Aidarous. Symplectic Gram-Schmidt algorithm with re-orthogonalization. Journal of King Abdulaziz University: Science, 23(1):11–20, 2011. doi:10.4197/Sci.23-1.2.
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 J. Mohammadpour and K. 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.
A. C. Antoulas, S. Lefteriu, and A. C. Ionita. A tutorial introduction to the Loewner framework for model reduction. In P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, editors, Model Reduction and Approximation: Theory and Algorithms, pages 335–376. SIAM, 2017. doi:10.1137/1.9781611974829.ch8.
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. 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. 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, Z. Bujanović, P. Kürschner, and J. Saak. RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations. Numerische Mathematik, 138(2):301–330, 2018. doi:10.1007/s00211-017-0907-5.
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: https://csc.mpi-magdeburg.mpg.de/preprints/2011/11/.
P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods. SIAM Journal on Mathematical Analysis, 43(3):1457–1472, 2011. doi:10.1137/100795772.
T. Breiten, C. Beattie, and S. Gugercin. $\mathcal H_2$-gap model reduction for stabilizable and detectable systems. In C. Beattie, P. Benner, M. Embree, S. Gugercin, and S. Lefteriu, editors, Realization and Model Reduction of Dynamical Systems: A Festschrift in Honor of the 70th Birthday of Thanos Antoulas, pages 317–334. Springer International Publishing, Cham, 2022. doi:10.1007/978-3-030-95157-3\_17.
T. Breiten and B. Unger. Passivity preserving model reduction via spectral factorization. Automatica, 142:110368, 2022. doi:10.1016/j.automatica.2022.110368.
P. Buchfink, A. Bhatt, and B. Haasdonk. Symplectic model order reduction with non-orthonormal bases. Mathematical and Computational Applications, 2019. doi:10.3390/mca24020043.
A. Buhr, C. Engwer, M. Ohlberger, and S. Rave. A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations. arXiv preprint 1407.8005, 2014. doi:10.48550/arXiv.1407.8005.
A. Buhr and K. Smetana. Randomized local model order reduction. SIAM Journal on Scientific Computing, 40(4):A2120–A2151, 2018. doi:10.1137/17M1138480.
A. Carracedo Rodriguez, L. Balicki, and S. Gugercin. The p-AAA algorithm for data-driven modeling of parametric dynamical systems. SIAM Journal on Scientific Computing, 45(3):A1332–A1358, 2023. doi:10.1137/20M1322698.
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.
B. Clapperton, F. Crusca, and M. Aldeen. Bilinear transformation and generalized singular perturbation model reduction. IEEE Transactions on Automatic Control, 41(4):589–593, 1996. doi:10.1109/9.489281.
U. Desai and D. Pal. A transformation approach to stochastic model reduction. IEEE Transactions on Automatic Control, 29(12):1097–1100, 1984. doi:10.1109/TAC.1984.1103438.
R. DeVore, G. Petrova, and P. Wojtaszczyk. Greedy algorithms for reduced bases in Banach spaces. Constructive Approximation, pages 455–466, 2013. doi:10.1007/s00365-013-9186-2.
Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. The Johns Hopkins University Press, fourth edition edition, 2013. ISBN 978-1-4214-0794-4.
M. A. Grepl and A. 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. doi:10.1051/m2an:2005006.
S. Gugercin and A. C. Antoulas. A survey of model reduction by balanced truncation and some new results. International Journal of Control, 77(8):748–766, 2004. doi:10.1080/00207170410001713448.
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.
S. Gugercin, R. V. Polyuga, C. Beattie, and A. van der Schaft. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica, 48(9):1963–1974, 2012. doi:10.1016/j.automatica.2012.05.052.
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. doi:10.1080/13873954.2011.547674.
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. doi:10.1051/m2an:2008001.
P. Harshavardhana, E.A. Jonckheere, and L.M. Silverman. Stochastic balancing and approximation-stability and minimality. IEEE Transactions on Automatic Control, 29(8):744–746, 1984. doi:10.1109/TAC.1984.1103631.
J. S. Hesthaven and S. Ubbiali. Non-intrusive reduced order modeling of nonlinear problems using neural networks. Journal of Computational Physics, 363:55–78, 2018. doi:10.1016/j.jcp.2018.02.037.
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.
M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE constraints. Volume 23. Springer Science & Business Media, 2008. doi:10.1007/978-1-4020-8839-1.
S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–1780, November 1997. doi:10.1162/neco.1997.9.8.1735.
T. Keil, L. Mechelli, M. Ohlberger, F. Schindler, and S. Volkwein. A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization. ESAIM: M2AN, 55(3):1239–1269, 2021. doi:10.1051/m2an/2021019.
B. Kramer and S. Gugercin. Tangential interpolation-based eigensystem realization algorithm for MIMO systems. Mathematical and Computer Modelling of Dynamical Systems, 22(4):282–306, 2016. doi:10.1080/13873954.2016.1198389.
S. Kung. A new identification and model reduction algorithm via singular value decomposition. In Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers, 705–714. 1978.
P. Kürschner. Efficient low-rank solution of large-scale matrix equations. PhD thesis, Shaker Verlag Aachen, 2016. URL: http://pubman.mpdl.mpg.de/pubman/.
R. B. Lehoucq. Analysis and implementation of an implicitly restarted Arnoldi iteration. PhD thesis, Rice University, Houston, USA, 1995. URL: https://scholarship.rice.edu/handle/1911/16844.
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.
V. Mehrmann and B. Unger. Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numerica, 32:395–515, 2023. doi:10.1017/S0962492922000083.
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.
R. Minster, A. K. Saibaba, J. Kar, and A. Chakrabortty. Efficient algorithms for eigensystem realization using randomized SVD. SIAM Journal on Matrix Analysis and Applications, 42(2):1045–1072, 2021. doi:10.1137/20M1327616.
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.
Y. Nakatsukasa, O. Sète, and L. N. Trefethen. The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing, 40(3):A1494–A1522, 2018. doi:10.1137/16M1106122.
Jorge Nocedal and Stephen Wright. Numerical optimization. Springer Science & Business Media, 2006. doi:10.1007/978-0-387-40065-5.
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.
L. Peng and K. Mohseni. Symplectic model reduction of Hamiltonian systems. SIAM Journal on Scientific Computing, 38(1):A1–A27, 2016. doi:10.1137/140978922.
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.
J. Rommes and N. Martins. Efficient computation of multivariable transfer function dominant poles using subspace acceleration. IEEE Transactions on Power Systems, 21(4):1471–1483, 2006. doi:10.1109/TPWRS.2006.881154.
A. K. Saibaba, J. Hart, and B. van Bloemen Waanders. Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis. Numerical Linear Algebra with Applications, 2021. doi:10.1002/nla.2364.
A. K. Saibaba, J. Lee, and P. K. Kitanidis. Randomized algorithms for generalized Hermitian eigenvalue problems with application to computing Karhunen-Loève expansion. Numerical Linear Algebra with Applications, 2016. doi:10.1002/nla.2026.
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, 1:391–421, 2014. doi:10.3934/jcd.2014.1.391.
Q. Wang, J. S. Hesthaven, and D. 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. doi:10.1016/j.jcp.2019.01.031.
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: http://hdl.handle.net/10919/27668.
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: http://www.math.ualberta.ca/ijnam/Volume-8-2011/No-1-11/2011-01-10.pdf.
Y. Yue and K. Meerbergen. Accelerating optimization of parametric linear systems by model order reduction. SIAM Journal on Optimization, 23(2):1344–1370, 2013. doi:10.1137/120869171.
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.