Abstract
In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bernreuther, M.: Nonsmooth PDEs: efficient algorithms, model order reduction, multiobjective PDE-constrained optimization. PhD thesis, Universität Konstanz, Konstanz (2023). http://nbn-resolving.de/urn:nbn:de:bsz:352-2-1jxe38e3x59983
Rappaz, J.: Approximation of a nondifferentiable nonlinear problem related to MHD equilibria. Numer. Math. 45(1), 117–133 (1984)
Xin, J.: An introduction to fronts in random media. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 5. Springer, New York (2009)
Meidner, D., Vexler, B.: A priori error analysis of the Petrov-Galerkin Crank-Nicolson scheme for parabolic optimal control problems. SIAM J. Control and Optimization. 49, 2183–2211 (2011). https://doi.org/10.1137/100809611
Gunzburger, M., Kunoth, A.: Space-time adaptive wavelet methods for optimal control problems constrained by parabolic evolution equations. J. Control Optim. 55, 1150–1170 (2011). https://doi.org/10.1137/100806382
Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numerische Mathematik. 120, 345–386 (2012). https://doi.org/10.1007/s00211-011-0409-9
Langer, U., Steinbach, O.: Space time methods: applications to partial differential equations. Radon Series on Computational and Applied Mathematics, vol. 25. De Gruyter, Berlin (2019)
Steinbach, O.: Space-time finite element methods for parabolic problems. Comput. Methods Appl. Math. 15, 551–566 (2015). https://doi.org/10.1515/cmam-2015-0026
Steinbach, O., Yang, H.: Comparison of algebraic multigrid methods for an adaptive space-time finite element discretization of the heat equation in 3d and 4d. Numer Linear Algebra Appl 25, 2143 (2018). https://doi.org/10.1002/nla.2143
Harbrecht, H., Tausch, J.: A fast sparse grid based space-time boundary element method for the nonstationary heat equation. Numer. Math. 140, 239–264 (2018). https://doi.org/10.1007/s00211-018-0963-5
Hinze, M., Korolev, D.: A space-time certified reduced basis method for quasilinear parabolic partial differential equations. Adv. Comput. Math. 47, 36 (2021). https://doi.org/10.1007/s10444-021-09860-z
Steih, K., Urban, K.: Space-time reduced basis methods for time-periodic partial differential equations. IFAC Proc. Vol. 45, 710–715 (2012). https://doi.org/10.3182/20120215-3-AT-3016.00126
Yano, M., Patera, A.T., Urban, K.: A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation. Math. Models Methods Appl. Sci. 24, 1903–1935 (2014). https://doi.org/10.1142/S0218202514500110
Urban, K., Patera, A.T.: An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83, 1599–1615 (2014). https://doi.org/10.1090/S0025-5718-2013-02782-2
Henning, J., Palitta, D., Simoncini, V., Urban, K.: An ultraweak space-time variational formulation for the wave equation: analysis and efficient numerical solution. ESAIM Math. Model. Numer. Anal. 56, 1173–1198 (2022). https://doi.org/10.1051/m2an/2022035
Beranek, N., Reinhold, A., Urban, K.: A space-time variational method for optimal control problems: well-posedness, stability and numerical solution. Submitted (2022). https://doi.org/10.48550/arXiv.2010.00345
Ballarin, F., Rozza, G., Strazzullo, M.: Chapter 9 – Space-time POD-Galerkin approach for parametric flow control. In: Trélat, E., Zuazua, E. (eds.) Numerical Control: Part A. Handbook of Numerical Analysis, vol. 23, pp. 307–338. Elsevier, ??? (2022). https://doi.org/10.1016/bs.hna.2021.12.009
Strazzullo, M., Ballarin, F., Rozza, G.: A certified reduced basis method for linear parametrized parabolic optimal control problems in space-time formulation. Submitted (2021). https://doi.org/10.48550/arXiv.2103.00460
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, 667–672 (2004). https://doi.org/10.1016/j.crma.2004.08.006
Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010). https://doi.org/10.1137/090766498
Chaturantabut, S., Sorensen, D.C.: A state space estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50, 46–63 (2012). https://doi.org/10.1137/110822724
Betz, L.M.: Second-order sufficient optimality conditions for optimal control of non-smooth, semilinear parabolic equations. J. Control. Optim. 57, 4033–4062 (2019). https://doi.org/10.1137/19M1239106
Meyer, C., Susu, L.M.: Optimal control of nonsmooth, semilinear parabolic equations. J. Control. Optim. 55, 2206–2234 (2017). https://doi.org/10.1137/15M1040426
Drohmann, M., Haasdonk, B., Ohlberger, M.: Adaptive reduced basis methods for nonlinear convection-diffusion equations. In: Finite Volumes for Complex Applications VI Problems & Perspectives (2010). https://doi.org/10.1007/978-3-642-20671-9_39
Bernreuther, M., Müller, G., Volkwein, S.: Reduced basis model order reduction in optimal control of a nonsmooth semilinear elliptic PDE. In: Herzog, R., Heinkenschloss, M., Kalise, D., Stadler, G., Trélat, E. (eds.) Optimization and Control for Partial Differential Equations, pp. 1–32. De Gruyter, Berlin, Boston (2022). https://doi.org/10.1515/9783110695984-001
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island (2010)
Zeidler, E.: Nonlinear functional analysis and its applications. Linear Monotone Operators vol. II/A. Springer, New York (1989)
Zeidler, E.: Nonlinear functional analysis and its applications. Nonlinear Monotone Operators vol. II/B. Springer, New York (1989)
Thomée, V.: Galerkin finite element methods for parabolic problems. Springer, Berlin (1997)
Hintermüller, M.: Semismooth Newton methods and applications. Oberwolfach-Seminar on Mathematics of PDE-Constrained Optimization at Mathematisches Forschungsinstitut in Oberwolfach (2010). https://www.math.uni-hamburg.de/home/hinze/Psfiles/ Hintermueller_OWNotes.pdf
Bernreuther, M.: RB-based PDE-constrained non-smooth optimization. Master’s thesis, Universität Konstanz (2019). http://nbn-resolving.de/urn:nbn:de:bsz:352-2-t4k1djyj77yn3
Braun, J.: Space-time reduced basis method for solving parameterized heat equations. Master’s thesis, Universität Konstanz (2023)
Gubisch, M., Volkwein, S.: Chapter 1: POD for linear-quadratic optimal control. In: Model Reduction and Approximation - Theory and Algorithms. Computational Science & Engineering, pp. 3–63. SIAM, Philadelphia (2017). https://doi.org/10.1137/1.9781611974829.ch1
Hesthaven, J.S., Rozza, G., Stamm, B.: Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer, Cham (2016)
Zeng, J.-p., Yu, H.-x.: Error estimates of the lumped mass finite element method for semilinear elliptic problems. Journal of Computational and Applied Mathematics. 236, 1993–2004 (2012). https://doi.org/10.1016/j.cam.2011.11.009
Chellappa, S., Feng, L., Benner, P.: Adaptive basis construction and improved error estimation for parametric nonlinear dynamical systems. Int. J. Numer. Methods Eng. 121, 5320–5349 (2020). https://doi.org/10.1002/nme.6462
Peherstorfer, B., Willcox, K.: Online adaptive model reduction for nonlinear systems via low-rank updates. SIAM J. Sci. Comput. 37, 2123–2150 (2015). https://doi.org/10.1137/140989169
Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Archive of Numerical Software. 3 (2015). https://doi.org/10.11588/ans.2015.100.20553
Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Jarrod Millman, K., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, İ., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P., Contributors, S...: SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods. 17, 261–272 (2020). https://doi.org/10.1038/s41592-019-0686-2
Acknowledgements
We are grateful to Denis Korolev (Berlin), Dominik Meidner (Munich) and Karsten Urban (Ulm) for fruitful discussions and very helpful remarks.
Funding
Open Access funding enabled and organized by Projekt DEAL. This research was supported by the German Research Foundation (DFG) under grant number VO 1658/5-2 within the DFG Priority Program “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization” (SPP 1962).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Additional information
Communicated by: Gianluigi Rozza
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
Bernreuther, M., Volkwein, S. An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations. Adv Comput Math 50, 48 (2024). https://doi.org/10.1007/s10444-024-10137-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-024-10137-4
Keywords
- Nonsmooth parabolic equations
- Space-time discretization
- Reduced basis
- Discrete empirical interpolation
- A-posteriori error estimation
- Semismooth Newton