Structured Optimization-Based Model Order Reduction for Parametric Systems

Authors

Paul Schwerdtner, Manuel Schaller

Citation

• Journal: SIAM Journal on Scientific Computing
• Year: 2025
• Volume: 47
• Issue: 1
• Pages: A72–A101
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/22m1524928

BibTeX

@article{Schwerdtner_2025,
  title={{Structured Optimization-Based Model Order Reduction for Parametric Systems}},
  volume={47},
  ISSN={1095-7197},
  DOI={10.1137/22m1524928},
  number={1},
  journal={SIAM Journal on Scientific Computing},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Schwerdtner, Paul and Schaller, Manuel},
  year={2025},
  pages={A72--A101}
}

Download the bib file

References

• Aliyev, N., Benner, P., Mengi, E., Schwerdtner, P. & Voigt, M. Large-Scale Computation of $\mathcal{L}_\infty$-Norms by a Greedy Subspace Method. SIAM Journal on Matrix Analysis and Applications vol. 38 1496–1516 (2017) – 10.1137/16m1086200
• Allaire, D., Kordonowy, D., Lecerf, M., Mainini, L. & Willcox, K. Multifidelity DDDAS Methods with Application to a Self-aware Aerospace Vehicle. Procedia Computer Science vol. 29 1182–1192 (2014) – 10.1016/j.procs.2014.05.106
• Amsallem, D., Cortial, J., Carlberg, K. & Farhat, C. A method for interpolating on manifolds structural dynamics reduced‐order models. International Journal for Numerical Methods in Engineering vol. 80 1241–1258 (2009) – 10.1002/nme.2681
• Amsallem, D. & Farhat, C. An Online Method for Interpolating Linear Parametric Reduced-Order Models. SIAM Journal on Scientific Computing vol. 33 2169–2198 (2011) – 10.1137/100813051
• Antoulas A. C., SIAM (2005)
• Antoulas A. C., SIAM (2020)
• Apkarian, P. & Noll, D. Structured H∞‐control of infinite‐dimensional systems. International Journal of Robust and Nonlinear Control vol. 28 3212–3238 (2018) – 10.1002/rnc.4073
• Barrault, M., Maday, Y., Nguyen, N. C. & Patera, A. T. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus. Mathématique vol. 339 667–672 (2004) – 10.1016/j.crma.2004.08.006
• Baur, U., Beattie, C., Benner, P. & Gugercin, S. Interpolatory Projection Methods for Parameterized Model Reduction. SIAM Journal on Scientific Computing vol. 33 2489–2518 (2011) – 10.1137/090776925
• Baur, U. & Benner, P. Modellreduktion für parametrisierte Systeme durch balanciertes Abschneiden und InterpolationModel Reduction for Parametric Systems Using Balanced Truncation and Interpolation. at - Automatisierungstechnik vol. 57 411–419 (2009) – 10.1524/auto.2009.0787
• Baur U., SIAM (2017)
• Beattie, C., Gugercin, S. & Mehrmann, V. Structure-Preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems. Realization and Model Reduction of Dynamical Systems 235–254 (2022) doi:10.1007/978-3-030-95157-3_13 – 10.1007/978-3-030-95157-3_13
• Benner, P., Gugercin, S. & Willcox, K. A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems. SIAM Review vol. 57 483–531 (2015) – 10.1137/130932715
• Benner P., SIAM (2017)
• Benner, P. & Voigt, M. ℋ∞‐Norm Computation for Large and Sparse Descriptor Systems. PAMM vol. 12 797–800 (2012) – 10.1002/pamm.201210383
• Wijaya, K. P. et al. An epidemic model integrating direct and fomite transmission as well as household structure applied to COVID-19. Journal of Mathematics in Industry vol. 11 (2021) – 10.1186/s13362-020-00097-x
• Breiten, T. & Unger, B. Passivity preserving model reduction via spectral factorization. Automatica vol. 142 110368 (2022) – 10.1016/j.automatica.2022.110368
• Bui-Thanh, T., Willcox, K. & Ghattas, O. Model Reduction for Large-Scale Systems with High-Dimensional Parametric Input Space. SIAM Journal on Scientific Computing vol. 30 3270–3288 (2008) – 10.1137/070694855
• Chaturantabut, S. & Sorensen, D. C. Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM Journal on Scientific Computing vol. 32 2737–2764 (2010) – 10.1137/090766498
• Cohen, A., Dahmen, W., DeVore, R. & Nichols, J. Reduced Basis Greedy Selection Using Random Training Sets. ESAIM: Mathematical Modelling and Numerical Analysis vol. 54 1509–1524 (2020) – 10.1051/m2an/2020004
• Daniel, L., Siong, O. C., Chay, L. S., Lee, K. H. & White, J. A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems vol. 23 678–693 (2004) – 10.1109/tcad.2004.826583
• Degroote, J., Vierendeels, J. & Willcox, K. Interpolation among reduced‐order matrices to obtain parameterized models for design, optimization and probabilistic analysis. International Journal for Numerical Methods in Fluids vol. 63 207–230 (2009) – 10.1002/fld.2089
• Freitag, M. A., Spence, A. & Dooren, P. V. Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method. SIAM Journal on Matrix Analysis and Applications vol. 35 619–635 (2014) – 10.1137/130933228
• Gillis, N. & Sharma, P. On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. Automatica vol. 85 113–121 (2017) – 10.1016/j.automatica.2017.07.047
• Gugercin, S., Antoulas, A. C. & Beattie, C. $\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems. SIAM Journal on Matrix Analysis and Applications vol. 30 609–638 (2008) – 10.1137/060666123
• Gugercin, S., Polyuga, R. V., Beattie, C. & van der Schaft, A. Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica vol. 48 1963–1974 (2012) – 10.1016/j.automatica.2012.05.052
• Guglielmi, N., Gürbüzbalaban, M. & Overton, M. L. Fast Approximation of the $H_\infty$ Norm via Optimization over Spectral Value Sets. SIAM Journal on Matrix Analysis and Applications vol. 34 709–737 (2013) – 10.1137/120875752
• Hauschild S.-A., Control Cybernet. (2019)
• Hesthaven, J. S., Rozza, G. & Stamm, B. Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics (Springer International Publishing, 2016). doi:10.1007/978-3-319-22470-1 – 10.1007/978-3-319-22470-1
• Hund, M., Mitchell, T., Mlinarić, P. & Saak, J. Optimization-based Parametric Model Order Reduction via ${ {\mathcal{H}_2} \otimes {\mathcal{L}_2} }$ First-order Necessary Conditions. SIAM Journal on Scientific Computing vol. 44 A1554–A1578 (2022) – 10.1137/21m140290x
• Kleyman, V. et al. Modeling and parameter identification for real-time temperature controlled retinal laser therapies. at - Automatisierungstechnik vol. 68 953–966 (2020) – 10.1515/auto-2020-0074
• Lasance, C. J. M. Two benchmarks to facilitate the study of compact thermal modeling phenomena. IEEE Transactions on Components and Packaging Technologies vol. 24 559–565 (2001) – 10.1109/6144.974943
• Lassila, T., Quarteroni, A. & Rozza, G. A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems. SIAM Journal on Scientific Computing vol. 34 A1187–A1213 (2012) – 10.1137/110819950
• Liu, D. C. & Nocedal, J. On the limited memory BFGS method for large scale optimization. Mathematical Programming vol. 45 503–528 (1989) – 10.1007/bf01589116
• Lohmann B., Methoden und Anwendungen der Regelungstechnik (2007)
• Mainini, L. & Willcox, K. Surrogate Modeling Approach to Support Real-Time Structural Assessment and Decision Making. AIAA Journal vol. 53 1612–1626 (2015) – 10.2514/1.j053464
• Mitchell, T. & Overton, M. L. Hybrid expansion–contraction: a robust scaleable method for approximating theH∞norm. IMA Journal of Numerical Analysis vol. 36 985–1014 (2015) – 10.1093/imanum/drv046
• K Mogensen, P. & N Riseth, A. Optim: A mathematical optimization package for Julia. Journal of Open Source Software vol. 3 615 (2018) – 10.21105/joss.00615
• Moore, B. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control vol. 26 17–32 (1981) – 10.1109/tac.1981.1102568
• Mullis, C. & Roberts, R. Synthesis of minimum roundoff noise fixed point digital filters. IEEE Transactions on Circuits and Systems vol. 23 551–562 (1976) – 10.1109/tcs.1976.1084254
• Niederreiter, H. Coding Theory and Cryptology. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (2002) doi:10.1142/5078 – 10.1142/5078
• Panzer, H., Mohring, J., Eid, R. & Lohmann, B. Parametric Model Order Reduction by Matrix Interpolation. auto vol. 58 475–484 (2010) – 10.1524/auto.2010.0863
• Polyuga, R. V. & van der Schaft, A. Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica vol. 46 665–672 (2010) – 10.1016/j.automatica.2010.01.018
• Prud’homme, C. et al. Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods. Journal of Fluids Engineering vol. 124 70–80 (2001) – 10.1115/1.1448332
• Sato, K. Riemannian optimal model reduction of linear port-Hamiltonian systems. Automatica vol. 93 428–434 (2018) – 10.1016/j.automatica.2018.03.051
• Schaller, M. et al. Parameter estimation and model reduction for model predictive control in retinal laser treatment. Control Engineering Practice vol. 128 105320 (2022) – 10.1016/j.conengprac.2022.105320
• Schwerdtner, P. & Voigt, M. Adaptive Sampling for Structure-Preserving Model Order Reduction of Port-Hamiltonian Systems. IFAC-PapersOnLine vol. 54 143–148 (2021) – 10.1016/j.ifacol.2021.11.069
• Son, N. T. A real time procedure for affinely dependent parametric model order reduction using interpolation on Grassmann manifolds. International Journal for Numerical Methods in Engineering vol. 93 818–833 (2012) – 10.1002/nme.4408
• Unger, B. & Gugercin, S. Kolmogorov n-widths for linear dynamical systems. Advances in Computational Mathematics vol. 45 2273–2286 (2019) – 10.1007/s10444-019-09701-0
• Weile, D. S., Michielssen, E., Grimme, E. & Gallivan, K. A method for generating rational interpolant reduced order models of two-parameter linear systems. Applied Mathematics Letters vol. 12 93–102 (1999) – 10.1016/s0893-9659(99)00063-4
• Wittmuess, P., Tarin, C., Keck, A., Arnold, E. & Sawodny, O. Parametric Model Order Reduction via Balanced Truncation with Taylor Series Representation. IEEE Transactions on Automatic Control vol. 61 3438–3451 (2016) – 10.1109/tac.2016.2521361