Abstract
Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.
Article PDF
Similar content being viewed by others
References
Greif, C., Urban, K.: Decay of the Kolmogorov \(n\)-width for wave problems. Appl. Math. Lett. 96, 216–222 (2019). https://doi.org/10.1016/j.aml.2019.05.013
Ohlberger, M, Rave, S.: Reduced basis methods: success, limitations and future challenges. Proceedings of the Conference Algoritmy,1–12 (2016). http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/389
Pinkus, A.: \(n\)-widths in approximation theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, Heidelberg, Germany, 7 (1985). https://doi.org/10.1007/978-3-642-69894-1
Peherstorfer, B.: Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling. SIAM J. Scientific Comput. 42(5), 2803–2836 (2020). https://doi.org/10.1137/19M1257275
Zimmermann, R., Peherstorfer, B., Willcox, K.: Geometric subspace updates with applications to online adaptive nonlinear model reduction. SIAM J. Matrix Anal. Appl. 39(1), 234–261 (2018). https://doi.org/10.1137/17M1123286
Amsallem, D., Haasdonk, B.: Projection-error based local reduced-order models. Adv. Model. Simulation Eng. Sci. 3(1), 6 (2016). https://doi.org/10.1186/s40323-016-0059-7
Amsallem, D., Zahr, M.J., Farhat, C.: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numerical Methods Eng. 92(10), 891–916 (2012). https://doi.org/10.1002/nme.4371
Dihlmann M., Drohmann M., Haasdonk, B.: Model reduction of parametrized evolution problems using the reduced basis method with adaptive time-partitioning. Proc. of ADMOS 2011, 64 (2011). https://www.ians.uni-stuttgart.de/anm/publications/files_publication_anm/DKH12_pre.pdf
Drohmann M., Haasdonk B., Ohlberger, M.: Adaptive reduced basis methods for nonlinear convection–diffusion equations. In: Finite Volumes for Complex Applications VI Problems & Perspectives, Springer, Berlin, Heidelberg, Germany, pp. 369–377 (2011). https://www.ians.uni-stuttgart.de/anm/publications/files_publication_anm/DOH11.pdf
Eftang, J.L., Patera, A.T., Rønquist, E.M.: An “\(hp\) certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Scientific Comput. 32(6), 3170–3200 (2010). https://doi.org/10.1137/090780122
Koch, O., Lubich, C.: Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29(2), 434–454 (2007). https://doi.org/10.1137/050639703
Kaulmann, S., Haasdonk, B.: Online greedy reduced basis construction using dictionaries. In: Proc. of VI International Conference on Adaptive Modeling and Simulation (ADMOS 2013), pp. 365–376 (2013). http://pnp.mathematik.uni-stuttgart.de/ians/haasdonk/publications/KH13.pdf
Dihlmann, M., Kaulmann, S., Haasdonk, B.: Online reduced basis construction procedure for model reduction of parametrized evolution systems. IFAC Proceedings Volumes 45(2), 112–117 (2012). https://doi.org/10.3182/20120215-3-AT-3016.00020
Maday, Y., Stamm, B.: Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces. SIAM J. Scientific Comput. 35(6), 2417–2441 (2013). https://doi.org/10.1137/120873868
Abgrall, R., Amsallem, D., Crisovan, R.: Robust model reduction by \({L}^{1}\)-norm minimization and approximation via dictionaries: application to nonlinear hyperbolic problems. Adv. Model. Simulation Eng. Sci. 3(1), 1 (2016). https://doi.org/10.1186/s40323-015-0055-3
Abgrall, R., Crisovan, R.: Model reduction using l1-norm minimization as an application to nonlinear hyperbolic problems. Int. J. Numerical Methods Fluids 87(12), 628–651 (2018). https://doi.org/10.1002/fld.4507
Balabanov, O., Nouy, A.: Randomized linear algebra for model reduction-part II: minimal residual methods and dictionary-based approximation. Adv. Computational Math. 47(2), 26 (2021). https://doi.org/10.1007/s10444-020-09836-5
Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.-J.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), 168–192 (2014). https://doi.org/10.1137/130924408
Peherstorfer, B., Willcox, K.: Online adaptive model reduction for nonlinear systems via low-rank updates. SIAM J. Scientific Comput. 37(4), 2123–2150 (2015). https://doi.org/10.1137/140989169
Meyer, K.R., Offin, D.C.: Introduction to Hamiltonian dynamical systems and the N-body problem. Applied Mathematical Sciences, Springer, New York, NY, 90 (2017). https://doi.org/10.1137/1035155
Volkwein, S.: Proper orthogonal decomposition: theory and reduced-order modelling. Lecture Notes, University of Konstanz, 1–29 (2013). https://igdk1754.ma.tum.de/downloads/SummerSchool2013Data/volkwein-slides-1.pdf
Maboudi Afkham, B., Hesthaven, J.: Structure preserving model reduction of parametric Hamiltonian systems. SIAM J. Scientific Comput. 39(6), 2616–2644 (2017). https://doi.org/10.1137/17M1111991
Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Scientific Comput. 38(1), 1–27 (2016). https://doi.org/10.1137/140978922
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Scientific Comput. 32(5), 2737–2764 (2010). https://doi.org/10.1137/090766498
Pagliantini, C.: Dynamical reduced basis methods for Hamiltonian systems. Numerische Mathematik 148(2), 409–448 (2021). https://doi.org/10.1007/s00211-021-01211-w
Benner, P., Ohlberger, M., Cohen, A., Willcox, K.: Model reduction and approximation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2017). https://doi.org/10.1137/1.9781611974829
Benner, P., Grivet-Talocia, S., Quarteroni, A., Rozza, G., Schilders, W., Silveira, L.M.: Model order reduction, snapshot-based methods and algorithms. De Gruyter 2 (2020). https://doi.org/10.1515/9783110671490
Haasdonk, B., Ohlberger, M.: Efficient reduced models and a-posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math. Comput. Model. Dyn. Syst. 17(2), 145–161 (2011). https://doi.org/10.1080/13873954.2010.514703
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 Mathematique 339(9), 667–672 (2004). https://doi.org/10.1016/j.crma.2004.08.006
Da Silva, A.C.: Lectures on symplectic geometry. Springer, Berlin, Heidelberg, Germany (2008). https://doi.org/10.1007/978-3-540-45330-7
Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Scientific Comput. 34(2), 937–969. https://doi.org/10.1137/10081157X
Buchfink, P., Bhatt, A., Haasdonk, B.: Symplectic model order reduction with non-orthonormal bases. Mathematical Computational Appl. 24(2) (2019). https://doi.org/10.3390/mca24020043
Hairer, E., Hochbruck, M., Iserles, A., Lubich, C.: Geometric numerical integration. Oberwolfach Reports 3(1), 805–882 (2006). https://doi.org/10.4171/OWR/2006/14
Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Method for solving the Sine-Gordon equation. Phys Rev Lett 30(25), 1262 (1973). https://doi.org/10.1103/PhysRevLett.30.1262
Funding
Open Access funding enabled and organized by Projekt DEAL. Funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 314733389 and Germany’s Excellence Strategy - EXC 2075 – 390740016. We acknowledge support by the Stuttgart Center for Simulation Science (SimTech).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Ralf Zimmermann
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Herkert, R., Buchfink, P. & Haasdonk, B. Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems. Adv Comput Math 50, 12 (2024). https://doi.org/10.1007/s10444-023-10102-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-023-10102-7