Dr. Tim Keil
Research Foci
- PDE-constrained Parameter Optimization
- Reduced Basis Methods
- Multiscale Finite Element Methods
- Perturbed problems
Doctoral AbstractThesis
Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization
- Supervisor
- Professor Dr. Mario Ohlberger
- Doctoral Subject
- Mathematik
- Doctoral Degree
- Dr. rer. nat.
- Awarded by
- Department 10 – Mathematics and Computer Science
Model order reduction is an enormously growing field that is particularly suitable for numerical
simulations in real-life applications such as engineering and various natural science disciplines.
Here, partial differential equations are often parameterized towards, e.g., a physical parameter.
Furthermore, it is likely to happen that the repeated utilization of standard numerical methods
like the finite element method (FEM) is considered too costly or even inaccessible.
This thesis presents recent advances in model order reduction methods with the primary aim
to construct online-efficient reduced surrogate models for parameterized multiscale phenomena
and accelerate large-scale PDE-constrained parameter optimization methods. In particular,
we present several different adaptive RB approaches that can be used in an error-aware trustregion
framework for progressive construction of a surrogate model used during a certified
outer optimization loop. In addition, we elaborate on several different enhancements for the
trust-region reduced basis (TR-RB) algorithm and generalize it for parameter constraints.
Thanks to the a posteriori error estimation of the reduced model, the resulting algorithm can be
considered certified with respect to the high-fidelity model. Moreover, we use the first-optimizethen-
discretize approach in order to take maximum advantage of the underlying optimality
system of the problem.
In the first part of this thesis, the theory is based on global RB techniques that use an accurate
FEM discretization as the high-fidelity model. In the second part, we focus on localized model
order reduction methods and develop a novel online efficient reduced model for the localized
orthogonal decomposition (LOD) multiscale method. The reduced model is internally based on
a two-scale formulation of the LOD and, in particular, is independent of the coarse and fine
discretization of the LOD.
The last part of this thesis is devoted to combining both results on TR-RB methods and
localized RB approaches for the LOD. To this end, we present an algorithm that uses adaptive
localized reduced basis methods in the framework of a trust-region localized reduced basis
(TR-LRB) algorithm. The basic ideas from the TR-RB are followed, but FEM evaluations of
the involved systems are entirely avoided.
Throughout this thesis, numerical experiments of well-defined benchmark problems are used
to analyze the proposed methods thoroughly and to show their respective strength compared to
approaches from the literature.
CV
Academic Education
- PhD in Mathematics
- Master of Science Mathematics with minor Finance, WWU Münster
- year abroad and master thesis with Axel Målqvist, University of Gothenburg
- Bachelor of Science Mathematics with minor Economics, WWU Münster
Positions
- Scientific Assistant, Workgroup Ohlberger, WWU Münster
- Student Assistant, Institute for Computational and Applied Mathematic, WWU Münster
Teaching
- Practice: Übungen zu Numerical Methods for Partial Differential Equations II [108412]
(in cooperation with Stephan Rave)
- Practice: Übungen zur Vorlesung Numerische Lineare Algebra [104412]
(in cooperation with Frank Wübbeling)
- Practical: Einführung in die Programmierung zur Numerik mit Python [102387]
(in cooperation with Tobias Leibner and Mario Ohlberger)
Project
- Localized Reduced Basis Methods for PDE-constrained Parameter Optimization – LRB-Opt ()
Individual Granted Project: DFG - Sachbeihilfe/Einzelförderung | Project Number: OH 98/11-1; SCHI 1493/1-1
Publications
- Kartmann, Michael, Keil, Tim, Ohlberger, Mario, Volkwein, Stephan, and Kaltenbacher, Barbara. . “Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems.” Computational Science and Engineering 1 (3): 1–30. doi: 10.1007/s44207-024-00002-z.
- Keil, Tim, and Ohlberger, Mario. . “A Relaxed Localized Trust-Region Reduced Basis Approach for Optimization of Multiscale Problems.” ESAIM: Mathematical Modelling and Numerical Analysis 58: 79–105. doi: 10.1051/m2an/2023089.
- Keil, Tim, Ohlberger, Mario, and Schindler, Felix. . “Adaptive Localized Reduced Basis Methods for Large Scale PDE-constrained Optimization.” Preprint. in Large-Scale Scientific Computations, Vol. 13952 of Lecture Notes in Computer Science, edited by I Lirkov and S Margenov. Cham: Springer Nature. doi: 10.1007/978-3-031-56208-2_10.
- Keil, Tim, Ohlberger, Mario, Schindler, Felix, and Schleuß, Julia. . “Local training and enrichment based on a residual localization strategy.” Preprint. in Proceedings of the Conference Algoritmy 2024, Vol. 8 of Proceedings of the Conference Algoritmy, edited by P Frolkovič, K Mikula and D Ševčovič. Bratislava: Jednota slovenských matematikov a fyzikov.
- Keil, Tim, and Rave, Stephan. . “An Online Efficient Two-Scale Reduced Basis Approach for the Localized Orthogonal Decomposition.” SIAM Journal on Scientific Computing 45 (4). doi: 10.1137/21M1460016.
- Keil, T, Kleikamp, H, Lorentzen, R, Oguntola, M, and Ohlberger, M. . “Adaptive machine learning based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery.” Advances in Computational Mathematics 2022 (48) 73. doi: 10.1007/s10444-022-09981-z.
- Keil, Tim, and Ohlberger, Mario. . “Model Reduction for Large Scale Systems.” in Large-Scale Scientific Computing, Vol. 13127 of Lecture Notes in Computer Science (LNCS), edited by Ivan Lirkov and Svetozar Margenov. Cham: Springer International Publishing. doi: 10.1007/978-3-030-97549-4_2.
- Banholzer, S, Keil, T, Mechelli, L, Ohlberger, M, Schindler, F, and Volkwein, S. . “An adaptive projected Newton non-conforming dual approach for trust-region reduced basis approximation of PDE-constrained parameter optimization.” Pure and Applied Functional Analysis 7 (5): 1561–1596.
- Freese, P, Hauck, M, Keil, T, and Peterseim, D. . “A Super-Localized Generalized Finite Element Method.” Preprint. arXiv 2022 doi: 10.48550/arXiv.2211.09461.
- Keil, Tim. . “Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization.” Dissertation thesis, WWU Münster. doi: 10.48550/arXiv.2211.09607.
- Keil, T, Mechelli, L, Ohlberger, M, Schindler, F, and Volkwein, S. . “A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization.” ESAIM: Mathematical Modelling and Numerical Analysis 55: 1239–1269.. doi: 10.1051/m2an/2021019.
- Hellman, Fredrik, Keil, Tim, and Målqvist, Axel. . “Numerical Upscaling of Perturbed Diffusion Problems.” SIAM Journal on Scientific Computing 2020 (Volume 42, Issue 4): A2014–A2036.. doi: 10.1137/19M1278211.
- Hellman, Fredrik, Keil, Tim, and Målqvist, Axel. . “Multiscale methods for perturbed diffusion problems.” Oberwolfach Reports 16: 2099–2181. doi: 10.4171/OWR/2019/35.
- Keil, Tim. . Variational crimes in the Localized orthogonal decomposition method (master's thesis)
Talks
- Keil, Tim : “Adaptive Localized Reduced Basis Methods in Multiscale PDE-Constrained Parameter Optimization”. Model Reduction and Surrogate Modeling -- MORE 2022, contributed talk, Berlin, Germany, .
- Keil, Tim : “Adaptive Localized Reduced Basis Methods in PDE-constrained Parameter Optimization”. GAMM 2022 - 92nd annual meeting, contributed talk, Aachen, Germany, .
- Keil, Tim : “Two-scale Reduced Basis Method for Parameterized Multiscale Problems”. Young Mathematicians in Model Order Reduction -- YMMOR 2022, contributed talk, Münster, Germany, .
- Keil, Tim; Renelt, Lukas : “Introduction to the Reduced Basis Method”. YMMOR - Young Mathematicians in Model Order Reduction, Münster, .
- Keil, Tim : “Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization”. Forschungsseminar Numerische Mathematik, invited talk workgroup Roland Maier, Jena, Germany, .
- Keil, Tim : “Adaptive Localized Reduced Basis Methods for Multiscale problems and PDE-Constrained Optimization”. Invited Seminar Talk Workgroup of Daniel Peterseim, Augsburg, Germany, .
- Tim Keil : “Two-scale Reduced Basis Method for Parameterized Multiscale Problems”. New trends in numerical multiscale methods and beyond, invited talk, Institut Mittag-Leffler, Djursholm, Sweden, online, .
- Tim Keil : “Trust-Region Reduced Basis Methods for Large Scale PDE-Constrained Parameter Optimization: A Non-Conforming Dual Approach”. SIAM Conference on Mathematical and Computational Issues in the Geosciences 2021, invited talk, Milano, Italy, online, .
- Tim Keil : “Adaptive Trust Region Reduced Basis Method in PDE-Constrained Parameter Optimization: A Non-Conforming Dual Approach”. GAMM 2021 - 91th Annual Meeting, contributed talk, Kassel, Germany, Online, .
- Tim Keil : “Advances for Reduced Basis methods for PDE-constrained optimization: a non conforming approach”. ALGORITMY 2020, Minisymposium: Advances in Model Order Reduction and its Applications, invited talk, Podbanske, Slovakia, Online, .
- Tim Keil : “Adaptive Trust Region Reduced Basis method for quadratic PDE-constrained Parameter Optimization”. Konstanz Workshop on Optimal Control, invited talk, Konstanz, Germany, .
- Tim Keil : “The LOD method for perturbed elliptic problems”. Oberwolfach Seminar: Beyond Homogenization, participant talks, Oberwolfach, Germany, .
- Tim Keil : “Numerical Upscaling of Perturbed Diffusion Problems”. SIAM Conference on Mathematical Computational Issues in the Geosciences 2019, invited talk, Houston, USA, .
- Tim Keil : “Numerical upscaling of perturbed diffusion problems”. Oberseminar zur Numerik, invited talk, Augsburg, Germany, .
- Tim Keil : “Localization of multiscale problems with random defects”. Master- und Oberseminar zu effizienten numerischen Methoden, Münster, Germany, .