Research of the Working Group Lebiedz

Dynamical Systems and Mathematical Physics

Skizze von langsamen invarianten Mannigfaltigkeiten als Symbolbild für Dynamische Systeme und Mathematische Physik/Chemie

Multi-scale differential equation models from physics, chemistry and biology place high demands on analysis simulation. Due to a multitude of state variables and parameters, a dimensional reduction of the models can be helpful. For this purpose we develop analytical and numerical methods. Interdisciplinary idea transfer, e.g. from classical mechanics in variable Lagrange-Hamilton-Jacobi form, statistical physics and relativity theory, plays a central role.

Differential Geometry and Symmetry

Skizze einer Mannigfaltigkeit als Symbolbild für Riemannsche Flächen und algebraische Geometrie

We apply coordinate independent, differential geometric and function theoretical methods for the analytical and numerical characterization of phase space geometry and topology of dynamic systems (ordinary and parabolic partial differential equations). In particular, geometric approaches for model reduction and approximation of dynamic systems are developed and tools for complex analysis are made available through analytical continuations.

Optimization and Optimal Control

Skizze eines Rührkessels (CSTR) als Symbolbild für Optimierung und Optimale Steuerung

Optimization and optimum control of process sequences play an important role in flow-through reactors in chemistry and biotechnology, for example. We use powerful numerical methods based on theory, especially with regard to robustness and efficiency. We also use model reduction and real-time algorithms (collaboration with working group of Prof. Moritz Diehl, IMTEK Freiburg), which are used in the field of control engineering for autonomous driving and wind power generation.

Working Group Lebiedz 2019

Current research assistants of working group of Dirk Lebiedz (from left to right): Jörn Dietrich, Marcus Heitel, Johannes Poppe, missing: Stephan Scholz


Research Areas and Applications

  • finite- and infinite-dimensional (real and complex) dynamic systems
  • applications in physics, chemistry and biology/biomedicine
  • geometric manifold-based model reduction and approximation
  • optimization and optimal control
  • differential geometry and complex analysis
  • symmetry concepts

Completed dissertations of the working group: see Mathematical Genealogy

Possible topics for theses

Former research assistants, year and current profession:

  • Dr. Oliver Inderwildi (Physikalische Chenmie, Universität Heidelberg 2005) - Senior Consultant, mm1, Zürich
  • Dr. Julia Kammerer (Mathematik, Universität Heidelberg 2007) - Aktuarin, Frankfurt
  • Dr. Qingyun Su (Physikalische Chemie, Universität Heidelberg 2007) - Dalian University of Technology, China
  • Dr. Mario Mommer (Modellierung und Systemoptimierung Mommer GmbH)
  • Dr. Oliver Slaby (Physikalische Chemie, Universität Heidelberg 2008) - Linde AG, München
  • Dr. Volkmar Reinhardt (Mathematik, Universität Heidelberg 2008) - SEW Eurodrive, Bruchsal
  • Dr. Osman Shahi Shaik (Physikalische Chemie, Universität Heidelberg 2008) - L'Oreal, Bangalore, Indien
  • Dr. Nikita Vladimirov (Systembiologie, Universität Heidelberg 2009) - Max-Delbrück Zentrum Berlin
  • Jun.-Prof. Dr. Johannes Stegmaier (Systembiologie, Universität Freiburg 2011) - Juniorprofessor RWTH Aachen
  • Dr. Dominik Skanda (Mathematik, Universität Freiburg 2012) - Vector Informatik GmbH, Stuttgart
  • Dr. Jochen Siehr (Mathematik, Universität Heidelberg 2013) - Deutsche Accumotive GmbH, Daimler AG
  • Dr. Marcel Rehberg (Mathematik, Universität Ulm 2013) - DECOIT, Karlsruhe
  • Dr. Marc Fein (Mathematik, Universität Ulm 2014) - ZF Friedrichshafen AG
  • Dr. Jonas Unger (Mathematik, Universität Ulm 2016) - Horaios, Blaustein
  • Dr. Pascal Heiter (Mathematik, Universität Ulm 2017) - Continental, Ulm

Selection of completed Bachelor's and Master's theses

  • C. Schlosser: A functional analytic approach to slow invariant manifolds.
  • C. Ott: Physikalische und mathematische Modelltheorie: Revolution oder Evolution wissenschaftlicher Tatsachen – eine Analyse zweier Fallbeispiele.
  • S. Rist: Laufzeitoptimierung einer mannigfaltigkeitsbasierten Modellreduktionssoftware mittels CUDA.
  • A. Dürr: Robuste Geometrieoptimierung elektrischer Maschinen.
  • M. Hermann: Die Bestimmung der optimalen Bestellmenge im Einzelhandel – Modellierung und Optimierung.
  • A. Mayer: Die Berechnung von invarianten Mannigfaltigkeiten in holomorphen Flüssen mittels SIM Methoden.
  • J. Dietrich: Symmetries of slow invariant manifolds.
  • J. Späth: Python Interface für eine mannigfaltigkeitsbasierte Modellreduktionssoftware.
  • F. Hof: Investigation of a pharmocokinetic multi-transit-compartment model: analytic solution and numerical modeling.
  • M. Brüche: Numerische Simulation und Analyse von Reaktions-Diffusionssystemen zur Untersuchung von Strukturbildungsphänomenen des H2O2-NaOHSCN-Cu2+ Oszillators.
  • M. Kreuzer: Flexible energy balance climate models for teaching and research.
  • M. Heitel: Comparison of numerical optimization techniques for a variational problem formulation of manifold-based model reduction.
  • C. Winter-Emden: Mathematische Modellierung und Fehleranalyse eines Patientenpositionier-Roboters.
  • C. Fitzer: Topologieoptimierung von Bauteilen bei Metallgußprozessen in Bezug auf Fliessdynamik und Strömungsgeschwindigkeiten.
  • J. Dietrich: Trajectory based model reduction of dynamical systems using methods of optimal control.
  • J. Gabriel: Modellierung und Simulation einer nicht-vorgemischten Gleichstrom-Wasserstoff-Verbrennung.
  • P. Heiter: On numerical methods for stiff ordinary differential equation systems.
  • A. Erbach: The mammalian circadian clock: an application for numerical optimal control.

Selected Publications

List of all publications


Heitel, Marcus; Lebiedz, Dirk
A View on Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamic Systems and Complex-Time Newton Flows
December 2019
Heitel, Marcus; Verschueren, Robin; Diehl, Moritz; Lebiedz, Dirk
Considering slow manifold based model reduction for multiscale chemical optimal control problems


Heiter, Pascal; Lebiedz, Dirk
Towards Differential Geometric Characterization of Slow Invariant Manifolds in Extended Phase Space: Sectional Curvature and Flow Invariance
SIAM Journal on Applied Dynamical Systems, 17 :732-753
March 2018


Lebiedz, Dirk; Unger, Jonas
On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models
Mathematical and Computer Modelling of Dynamical Systems, 22 (2) :87--112
March 2016


Skanda, Dominik; Lebiedz, Dirk
A robust optimization approach to experimental design for model discrimination of dynamical systems
Mathematical Programming, 141 (1-2) :405-433
October 2013