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Dr. Tim Keil
Institute for Analysis and Numerics
Dr. Tim Keil

Einsteinstr. 62, room 120.009 (Orléansring 10)
48149 Münster

T: +49-(0)251-83-35056
tim.keil@wwu.de

 
  • 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

    2018 – 2022
    PhD in Mathematics
    2015 – 2018
    Master of Science Mathematics with minor Finance, WWU Münster
    2016 – 2017
    year abroad and master thesis with Axel Målqvist, University of Gothenburg
    2012 – 2015
    Bachelor of Science Mathematics with minor Economics, WWU Münster

    Positions

    since 04.2018
    Scientific Assistant, Workgroup Ohlberger, WWU Münster
    2014 – 2018
    Student Assistant, Institute for Computational and Applied Mathematic, WWU Münster
  • Teaching

    Summer Term 2022

    • Practice: Übungen zu Numerical Methods for Partial Differential Equations II [108412]
      (in cooperation with Dr. Stephan Rave)

    Winter Term 2018/19

    • Practice: Übungen zur Vorlesung Numerische Lineare Algebra [104412]
      (in cooperation with Dr. Frank Wübbeling)

    Summer Term 2018

    • Practical: Einführung in die Programmierung zur Numerik mit Python [102387]
      (in cooperation with Tobias Leibner and Prof. Dr. Mario Ohlberger)
  • Project

    • Localized Reduced Basis Methods for PDE-constrained Parameter Optimization – LRB-Opt (2019 – 2023)
      Individual Granted Project: | DFG - Individual Grants Programme | Project Number: OH 98/11-1; SCHI 1493/1-1
  • Publications

    2024

    • Kartmann, Michael, Keil, Tim, Ohlberger, Mario, Volkwein, Stephan, and Kaltenbacher, Barbara. 2024. “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. 2024. “A Relaxed Localized Trust-Region Reduced Basis Approach for Optimization of Multiscale Problems.” Preprint. ESAIM: Mathematical Modelling and Numerical Analysis 58: 79–105. doi: 10.1051/m2an/2023089.
    • Keil, Tim, Ohlberger, Mario, and Schindler, Felix. 2024. “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. 2024. “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.

    2023

    • Keil, Tim, and Rave, Stephan. 2023. “An Online Efficient Two-Scale Reduced Basis Approach for the Localized Orthogonal Decomposition.” SIAM Journal on Scientific Computing 45 (4). doi: 10.1137/21M1460016.

    2022

    • Keil, T, Kleikamp, H, Lorentzen, R, Oguntola, M, and Ohlberger, M. 2022. “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. 2022. “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. 2022. “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. 2022. “A Super-Localized Generalized Finite Element Method.” Preprint. arXiv 2022 doi: 10.48550/arXiv.2211.09461.
    • Keil, Tim. 2022. “Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization.” Dissertation thesis, WWU Münster. doi: 10.48550/arXiv.2211.09607.

    2021

    • Keil, T, Mechelli, L, Ohlberger, M, Schindler, F, and Volkwein, S. 2021. “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.

    2020

    • Hellman, Fredrik, Keil, Tim, and Målqvist, Axel. 2020. “Numerical Upscaling of Perturbed Diffusion Problems.” SIAM Journal on Scientific Computing 2020 (Volume 42, Issue 4): A2014–A2036.. doi: 10.1137/19M1278211.

    2019

    • Hellman, Fredrik, Keil, Tim, and Målqvist, Axel. 2019. “Multiscale methods for perturbed diffusion problems.” Oberwolfach Reports 16: 2099–2181. doi: 10.4171/OWR/2019/35.

    2017

    • Keil, Tim. 2017. Variational crimes in the Localized orthogonal decomposition method (master's thesis)
  • Talks

    2022

    • Keil, Tim 2022: “Adaptive Localized Reduced Basis Methods in Multiscale PDE-Constrained Parameter Optimization”. Model Reduction and Surrogate Modeling -- MORE 2022, contributed talk, Berlin, Germany, Sep 20, 2022.
    • Keil, Tim 2022: “Adaptive Localized Reduced Basis Methods in PDE-constrained Parameter Optimization”. GAMM 2022 - 92nd annual meeting, contributed talk, Aachen, Germany, Aug 17, 2022.
    • Keil, Tim 2022: “Two-scale Reduced Basis Method for Parameterized Multiscale Problems”. Young Mathematicians in Model Order Reduction -- YMMOR 2022, contributed talk, Münster, Germany, Jul 22, 2022.
    • Keil, Tim; Renelt, Lukas 2022: “Introduction to the Reduced Basis Method”. YMMOR - Young Mathematicians in Model Order Reduction, Münster, Jul 18, 2022.
    • Keil, Tim 2022: “Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization”. Forschungsseminar Numerische Mathematik, invited talk workgroup Roland Maier, Jena, Germany, Jul 14, 2022.
    • Keil, Tim 2022: “Adaptive Localized Reduced Basis Methods for Multiscale problems and PDE-Constrained Optimization”. Invited Seminar Talk Workgroup of Daniel Peterseim, Augsburg, Germany, Apr 25, 2022.

    2021

    • Tim Keil 2021: “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, Jul 16, 2021.
    • Tim Keil 2021: “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, Jun 23, 2021.
    • Tim Keil 2021: “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, Mar 18, 2021.

    2020

    • Tim Keil 2020: “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, Sep 14, 2020.

    2019

    • Tim Keil 2019: “Adaptive Trust Region Reduced Basis method for quadratic PDE-constrained Parameter Optimization”. Konstanz Workshop on Optimal Control, invited talk, Konstanz, Germany, Dec 2, 2019.
    • Tim Keil 2019: “The LOD method for perturbed elliptic problems”. Oberwolfach Seminar: Beyond Homogenization, participant talks, Oberwolfach, Germany, Jun 11, 2019.
    • Tim Keil 2019: “Numerical Upscaling of Perturbed Diffusion Problems”. SIAM Conference on Mathematical Computational Issues in the Geosciences 2019, invited talk, Houston, USA, Jun 3, 2019.
    • Tim Keil 2019: “Numerical upscaling of perturbed diffusion problems”. Oberseminar zur Numerik, invited talk, Augsburg, Germany, Jan 22, 2019.

    2018

    • Tim Keil 2018: “Localization of multiscale problems with random defects”. Master- und Oberseminar zu effizienten numerischen Methoden, Münster, Germany, Aug 16, 2018.
  • Contacts
MathematicsMuensterCells in MotionSFB 656 MoBilDEMAIN – Developing Mathematics in InteractionCenter for Nonlinear ScienceCenter for Multiscale Theory and ComputationCompetence for Computing in Science
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Orléans-Ring 10
48149 Münster

Tel: +49 (0) 251 83-35052
Fax: +49 (0) 251 83-32729
seknum@wwu.de
 
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