Projects in Mathematics Muenster

Research area C: Models and Approximations
Unit C2: Multi-scale phenomena and macroscopic structures
Unit C4: Geometry-based modelling, approximation, and reduction

Further Projects
Mathematical Research Data Initiative Driven by the needs and requirements of mathematical research as well as scientific disciplines using mathematics, the NFDI-consortium MaRDI (Mathematical Research Data Initiative) will develop and establish standards and services for mathematical research data. Mathematical research data ranges from databases of special functions and mathematical objects, aspects of scientific computing such as models and algorithms to statistical analysis of data with uncertainties. It is also widely used in other scientific disciplines due to the cross-disciplinary nature of mathematical methods. online
ML-MORE: Machine learning and model order reduction to predict the efficiency of catalytic filters. Subproject 1: Model Order Reduction Reaktiver Stofftransport in porösen Medien in Verbindung mit katalytischen Reaktionen ist die Grundlage für viele industrielle Prozesse und Anlagen, wie z.B. Brennstoffzellen, Photovoltaikzellen, katalytische Filter für Abgase, etc. Die Modellierung und Simulation der Prozesse auf der Porenskala kann bei der Optimierung des Designs von katalytischen Komponenten und der Prozessführung helfen, ist jedoch derzeit dadurch eingeschränkt, dass solche Simulationen zu grossen Datenmengen führen, zeitaufwändig sind und von einer grossen Anzahl von Parametern abhängen. Außerdem werden auf diese Weise die im Laufe der Jahre gesammelten Versuchsdaten nicht wiederverwendet. Die Entwicklung von Lösungsansätzen für die Vorhersage der chemischen Konversionsrate mittels moderner datenbasierter Methoden des Maschinellen Lernens (ML) ist essenziell, um zu schnellen, zuverlässigen prädiktiven Modellen zu gelangen. Hierzu sind verschiedene Methodenklassen erforderlich. Neben den experimentellen Daten sind voll aufgelöste Simulationen auf der Porenskala notwendig. Diese sind jedoch zu teuer, um einen umfangreichen Satz an Trainingsdaten zu generieren. Daher ist die Modellordnungsreduktion (MOR) zur Beschleunigung entscheidend. Es werden reduzierte Modelle fur den betrachteten instationären reaktiven Transport entwickelt, um große Mengen an Trainingsdaten zu simulieren. Als ML-Methodik werden mehrschichtige Kern-basierte Lernverfahren entwickelt, um die heterogenen Daten zu kalibrieren und nichtlineare prädiktive Modelle zur Effizienzvorhersage zu entwickeln.Hierbei werden große Daten (bzgl. Dimensionalität und Sample-Zahl) zu behandeln sein, was Datenkompression und Parallelisierung des Trainings erfordern wird. Das Hauptziel des Projekts ist es, alle oben genannten Entwicklungen in einem prädiktiven ML-Tool zu integrieren, das die Industrie bei der Entwicklung neuer katalytischer Filter unterstützt und auf viele andere vergleichbare Prozesse übertragbar ist. online
Localized Reduced Basis Methods for PDE-constrained Parameter Optimization This projects is concerned with model reduction for parameter optimization of nonlinear elliptic partial differential equations (PDEs). The goal is to develop a new paradigm for PDE-constrained optimization based on adaptive online enrichment. The essential idea is to design a localized version of the reduced basis (RB) method which is called Localized Reduced Basis Method (LRBM). online

Research Interests

Research Interests

$\bullet$ Numerical analysis of partial differential equations.
$\bullet$ Error control and adaptivity for finite element, finite volume, and DG methods.
$\bullet$ Model reduction for parametrised evolution equations, Reduced Basis Methods.
$\bullet$ Development and analysis of numerical multiscale methods.
$\bullet$ Software development and scientific computing.
$\bullet$ Complex applications in life sciences, fluid mechanics and environmental sciences.

Selected Publications

Selected Publications of Mario Ohlberger

$\bullet$ A. Buhr, C. Engwer, M. Ohlberger, and S. Rave. Arbi Lo Mod, a simulation technique designed for arbitrary local modifications. SIAM J. Numer. Anal., 39(4):A1435–A1465, 2017.

$\bullet$ K. Smetana and M. Ohlberger. Hierarchical model reduction of nonlinear partial differential equations based on the adaptive empirical projection method and reduced basis techniques. ESAIM Math. Model. Numer. Anal., 51(2):641–677, 2017.

$\bullet$ P. Henning, M. Ohlberger, and B. Verfürth. A new heterogeneous multiscale method for time-harmonic Maxwell's equations. SIAM J. Numer. Anal., 54(6):3493–3522, 2016.

$\bullet$ M. Ohlberger and F. Schindler. Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput., 37(6):A2865–A2895, 2015.

$\bullet$ P. Henning, M. Ohlberger, and B. Schweizer. An adaptive multiscale finite element method. Multiscale Model. Simul., 12(3):1078–1107, 2014.

$\bullet$ M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34(2):A937–A969, 2012.

$\bullet$ B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM Math. Model. Numer. Anal., 42(2):277–302, 2008.

$\bullet$ P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander. A generic grid interface for parallel and adaptive scientific computing. {II}. Implementation and tests in DUNE. Computing, 82(2-3):121–138, 2008.

$\bullet$ M. Ohlberger. A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model. Simul., 4(1):88–114, 2005.

$\bullet$ M. Ohlberger. A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. ESAIM Math. Model. Numer. Anal., 35(2):355–387, 2001.

Current Publications

$\bullet $ Bernard Haasdonk, Hendrik Kleikamp, Mario Ohlberger, Felix Schindler, and Tizian Wenzel. A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs. arXiv e-prints, April 2022. arXiv:2204.13454.

$\bullet $ Tim Keil and Mario Ohlberger. A relaxed localized trust-region reduced basis approach for optimization of multiscale problems. arXiv e-prints, March 2022. arXiv:2203.09964.

$\bullet $ Hendrik Kleikamp, Mario Ohlberger, and Stephan Rave. Nonlinear model order reduction using diffeomorphic transformations of a space-time domain. arXiv e-prints, March 2022. arXiv:2203.05833.

$\bullet $ Tim Keil, Hendrik Kleikamp, Rolf J Lorentzen, Micheal B Oguntola, and Mario Ohlberger. Adaptive machine learning based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery. arXiv e-prints, March 2022. arXiv:2203.01674.

$\bullet $ Bernard Haasdonk, Mario Ohlberger, and Felix Schindler. An adaptive model hierarchy for data-augmented training of kernel models for reactive flow. arXiv e-prints, October 2021. arXiv:2110.12388.

$\bullet $ Manuel Landstorfer, Mario Ohlberger, Stephan Rave, and Marie Tacke. A modeling framework for efficient reduced order simulations of parametrized lithium-ion battery cells. arXiv e-prints, October 2021. arXiv:2110.06011.

$\bullet $ Tim Keil, Luca Mechelli, Mario Ohlberger, Felix Schindler, and Stefan Volkwein. A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization. ESAIM: M2AN, 55(3):1239–1269, May 2021. doi:10.1051/m2an/2021019.

$\bullet $ Tim Keil and Mario Ohlberger. Model reduction for large scale systems. arXiv e-prints, May 2021. arXiv:2105.01433.

$\bullet $ Pavel Gavrilenko, Bernard Haasdonk, Oleg Iliev, Mario Ohlberger, Felix Schindler, Pavel Toktaliev, Tizian Wenzel, and Maha Youssef. A full order, reduced order and machine learning model pipeline for efficient prediction of reactive flows. arXiv e-prints, April 2021. arXiv:2104.02800.

$\bullet $ Peter Bastian, Markus Blatt, Andreas Dedner, Nils-Arne Dreier, Christian Engwer, René Fritze, Carsten Gräser, Christoph Grüninger, Dominic Kempf, Robert Klöfkorn, Mario Ohlberger, and Oliver Sander. The DUNE framework: Basic concepts and recent developments. Computers and Mathematics with Applications, 81:75–112, January 2021. doi:10.1016/j.camwa.2020.06.007.

$\bullet $ Andreas Buhr, Laura Iapichino, Mario Ohlberger, Stephan Rave, Felix Schindler, and Kathrin Smetana. Localized model reduction for parameterized problems. In Model order reduction. Volume 2: Snapshot-based methods and algorithms, pages 245–305. January 2021. doi:10.1515/9783110671490-006.

$\bullet $ Peter Bastian, Mirco Altenbernd, Nils-Arne Dreier, Christian Engwer, Jorrit Fahlke, René Fritze, Markus Geveler, Dominik Göddeke, Oleg Iliev, Olaf Ippisch, Jan Mohring, Steffen Müthing, Mario Ohlberger, Dirk Ribbrock, Nikolay Shegunov, and Stefan Turek. Exa-duneflexible PDE solvers, numerical methods and applications. In Hans-Joachim Bungartz, Severin Reiz, Benjamin Uekermann, Philipp Neumann, and Wolfgang E. Nagel, editors, Software for Exascale Computing - SPPEXA 2016-2019, pages 225–269. Cham, July 2020. doi:10.1007/978-3-030-47956-5_9.

$\bullet $ Tobias Leibner and Mario Ohlberger. A new coordinate-transformed discretization method for minimum entropy moment approximations of linear kinetic equations. arXiv e-prints, July 2020. arXiv:2007.04467.

$\bullet $ Mario Ohlberger, Ben Schweizer, Maik Urban, and Barbara Verfürth. Mathematical analysis of transmission properties of electromagnetic meta-materials. Networks & Heterogeneous Media, 15(1):29–56, January 2020. doi:10.3934/nhm.2020002.

$\bullet $ Mario Ohlberger, Andreas Buhr, Dennis Eikhorn, Christian Engwer, and Stephan Rave. Advances in model order reduction for large scale or multi-scale problems. Oberwolfach Reports, 2019:38–40, September 2019.

$\bullet $ Julian Feinauer, Simon Hein, Stephan Rave, Sebastian Schmidt, Daniel Westhoff, Jochen Zausch, Oleg Iliev, Arnulf Latz, Mario Ohlberger, and Volker Schmidt. MULTIBAT: Unified workflow for fast electrochemical 3D simulations of lithium-ion cells combining virtual stochastic microstructures, electrochemical degradation models and model order reduction. Journal of Computational Science, 31:172–184, February 2019. doi:10.1016/j.jocs.2018.03.006.

$\bullet $ Stefan Hain, Mario Ohlberger, Mladjan Radic, and Karsten Urban. A hierarchical a posteriori error estimator for the Reduced Basis Method. Advances in Computational Mathematics, 45(5-6):2191–2214, February 2019. doi:10.1007/s10444-019-09675-z.