10:30
MS26:New challenges and opportunities for model order reduction (Part 1)
Chair: Katrin Smetana
10:30
25 mins
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Localized Reduced Basis Methods for PDE-constrained Optimization
Tim Keil, Mario Ohlberger, Felix Schindler
Abstract: Model order reduction has become a mature technique for simulation of large classes of parameterized Systems [1]. However, several challenges remain for problems where the solution manifold of the parameterized system cannot be well approximated by linear subspaces [3]. While the online efficiency of these model reduction methods is very convincing for problems with a rapid decay of the Kolmogorov n-width, there are still major drawbacks and limitations. Most importantly, the construction of the reduced system in the offline phase is extremely CPU-time and memory consuming. For practical applications, it is thus necessary to derive model reduction techniques that do not rely on a classical offline/online splitting but allow for more flexibility in the usage of computational resources. A promising approach with this respect is localized model reduction with adaptive enrichment [6, 4].
In this talk we investigate localized model reduction [2] that combines approaches from reduced basis methods, multiscale methods and domain decomposition techniques. We address localized a posteriori error estimation and its use for adaptive enrichment. Finally, we demonstrate the beneficial usage of the localized reduced basis method with adaptive basis enrichment for the solution of PDE-constrained optimization problems [5].
References
[1] P. Benner, A. Cohen, M. Ohlberger, K. Willcox. Model Reduction and Approximation: Theory and Algorithms. Computational Science & Engineering, 15. SIAM, Philadelphia, PA, 2017.
[2] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, K. Smetana. Localized model re- duction for parameterized problems. ArXiv e-prints arXiv:1902.08300 [math.NA], Preprint 2019.
[3] M. Ohlberger, S. Rave. Reduced Basis Methods: Success, Limitations and Future Challenges. In Proceedings of ALGORITMY 2016, Podbanske, Slovakia, March 13-18, 2016, edited by A. Handlovicˇova and D. Sevcˇovicˇ, 1–12. PH Slovak University of Technology, Bratislava, 2016.
[4] M. Ohlberger, S. Rave, F. Schindler. True Error Control for the Localized Reduced Basis Method for Parabolic Problems. In Model Reduction of Parametrized Systems, edited by Benner P., Ohlberger M., Patera A., Rozza G., Urban K., 169–182. Cham: Springer Intern. Publishing, 2017.
[5] M. Ohlberger, M. Schaefer, F. Schindler. Localized Model Reduction in PDE Constrained Optimization. In Shape Optimization, Homogenization and Optimal Control – DFG-AIMS workshop held at the AIMS Center Senegal, March 13-16, 2017, edited by Schulz V, Seck D., 143-163, Basel: Birkha ̈user, 2018.
[6] M. Ohlberger, F. Schindler. Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015.
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10:55
25 mins
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Optimal local approximation spaces for parabolic problems
Julia Schleuß, Kathrin Smetana
Abstract: In this talk we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods that are optimal in the sense of Kolmogorov. The underlying partial differential equation is of parabolic type. [2]
In the last years localized model order reduction has gained importance in various fields such as engineering science. Phenomena of interest are modeled by partial differential equations, but a straightforward numerical simulation, say, via the standard finite element method is often infeasible due to the complexity of the simulated phenomena. One way to address this problem is to decompose the computational domain and to compute basis functions locally from local solutions of the underlying partial differential equation. Hence, localized model order reduction procedures enable an efficient numerical simulation, which is especially useful in real-time and many query contexts.
In this talk we employ the example of the linear heat equation and consider a weak space-time formulation of the initial boundary value problem. To construct optimal local approximation spaces we introduce a compact transfer operator that maps the space of local solutions on an outer subdomain to the space of local solutions on an inner subdomain of the computational domain and covers the full time dimension of the local solutions. Subsequently, the optimal approximation spaces are build by solving the corresponding transfer eigenvalue problem for the composition of the transfer operator and its adjoint. In order to ensure compactness of the transfer operator in the parabolic case, existing elliptic results [1, 3] could not be readily transferred and applied. To apply both a suitable Caccioppoli inequality and the compactness theorem of Aubin-Lions and thus prove compactness of the transfer operator additional regularity results for the weak solution are required.
References
[1] I. Babuska and R. Lipton. Optimal local approximation spaces for generalized finite element methods
with application to multiscale problems. In: Multiscale Modeling & Simulation 9.1 (2011),
pp. 373–406.
[2] J. Schleuß. Optimal local approximation spaces for parabolic problems. Master’s thesis,
Westfälische Wilhelms-Universität Münster, 2019.
[3] K. Smetana and A. T. Patera. Optimal local approximation spaces for componentbased static
condensation procedures. In: SIAM Journal on Scientific Computing 38.5 (2016), A3318–A3356.
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11:20
25 mins
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Model Reduction for Hamilton-Jacobi-Bellman Equations resulting from Intraday Trading of Electricity
Silke Glas, Karsten Urban
Abstract: Due to the growth of renewable energy, the future perspective of energy markets is seen in short-term trading markets. In this talk, we consider the intraday trading of electricity and derive a second-order Hamilton-Jacobi-Bellman (HJB) equation for this setting. As the intraday products are traded hourly, our aim is to find an optimal trading strategy within every hour using the most recent information on the market.
To obtain such an optimal trading strategy, we solve the nonlinear HJB equation for multiple parameters. As no closed-form solution exists for this particular problem, we require a fine discretization such that the full complexity of the problem can be represented. This can result in long computation times, which is why we use the reduced basis method (RBM) to derive a reduced model.
The RBM is a well-known technique to efficiently reduce the numerical efforts for many parametrized problems. We introduce the parametric formulation of our HJB equation, in which the parameter is the incoming data of the market and analyze the reducability of this problem.
We comment on the (to our knowledge) only existing approach for this problem and provide numerical investigations for our method.
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11:45
25 mins
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Greedy controllability for parametrized evolution problems
Laura Iapichino, Giulia Fabrini, Stefan Volkwein
Abstract: Often a dynamical system is characterized by one or more parameters describing physical features of the problem or geometrical configurations of the computational domain. As a consequence, by assuming that the system is controllable, corresponding to different parameter values, a range of optimal controls exists. The goal of the proposed approach is to avoid the computation of a control function for any instance of the parameters. The greedy controllability [M. Lazar, E. Zuazua "Greedy controllability of finite dimensional linear systems", Automatica, (Journal of IFAC) Volume 74 Issue C, Pages 327-340, 2016.] consists in the selection of the most representative values of the parameters that allows a rapid approximation of the control function for any desired new parameter value, ensuring that the system is steered to the target within a certain accuracy. By proposing the Reduced Basis method [J.S.Hesthaven, G. Rozza, B. Stamm, "Certified Reduced Basis Methods for Parametrized Partial Differential Equations", Springer- Briefs in Mathematics, 2016.] in this framework, the computational costs are drastically reduced and the efficiency of the greedy controllability approach is significantly improved.
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