Prof. Dr. Volker Mehrmann

Projects as a project leader

• SE1

  Reduced order modeling for data assimilation

  Prof. Dr. Volker Mehrmann / Dr. Christian Schröder

  Project heads: Prof. Dr. Volker Mehrmann / Dr. Christian Schröder
  Project members: Dr. Matthias Voigt
  Duration: -
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  One of the bottlenecks of current procedures for the generation and distribution of green (wind or solar) energy is the accurate and timely simulation of processes in the ocean and atmosphere that can be used in short term planning and real time control of energy systems. A particular difficulty is the real time construction of physically plausible model initializations and 'controls/inputs' to bring simulations into coherence with available observations when observation locations and observations are coming in at variable times and locations.
  
  The currently best approach for fixed observation times and locations are variational data assimilation techniques. These methods use a four dimensional model that is adapted to the incoming observations using a combination of different filtering techniques and numerical integration of the dynamical system. In order to make these methods efficient in real time data assimilation they have to be combined with appropriate model order reduction methods. A major difficulty in these techniques is the combination of approximate transfer functions and approximate initial and boundary conditions as well as the construction of guaranteed error estimates and the capturing of essential features of the original model. The so-called representer approach formulates the data assimilation problem as the numerical solution of a large-scale nonlinear optimal control problem and incorporates the assimilation of the model to the observations, via an extended ensemble Kalman filter, and the adaptation of the initial data in one approach. Adding further assumptions and linearization this optimization problem usually reduces to a linear quadratic optimal control problem which is solved via the solution of a boundary value problem with Hamiltonian structure.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/SE1/

• SE16

  Numerical solution of dynamic metabolic resource allocation problems for bioenergy production

  Prof. Dr. Alexander Bockmayr / Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Alexander Bockmayr / Prof. Dr. Volker Mehrmann
  Project members: Dr. Markus Arthur Köbis
  Duration: 01.06.2017 - 31.12.2018
  Status: completed
  Located at: Freie Universität Berlin
  

  Description

  In the field of sustainable energies, microbial cell factories such as yeasts and cyanobacteria are receiving increasing interest due to their potential to produce biofuels. A major question is how the metabolism of these microorganisms is coordinated in a dynamic environment such that the correct macromolecules are synthesized at the right time in order to enable growth and survival. Recent mathematical modeling approaches have made it possible to study this problem using an optimal dynamic resource allocation formalism such as dynamic enzyme-cost flux balance analysis (deFBA). The goal of this project is to study the mathematical properties properties of the underlying optimal control problem involving differential-algebraic constraints and to develop efficient numerical solution strategies.
  
  https://www.mi.fu-berlin.de/en/math/groups/mathlife/projects_neu/SE16/index.html

• SE3

  Stability analysis of power networks and power network models

  Prof. Dr. Christian Mehl / Prof. Dr. Volker Mehrmann / Prof. Dr. Caren Tischendorf

  Project heads: Prof. Dr. Christian Mehl / Prof. Dr. Volker Mehrmann / Prof. Dr. Caren Tischendorf
  Project members: Dr. Andreas Steinbrecher
  Duration: -
  Status: completed
  Located at: Humboldt Universität Berlin / Technische Universität Berlin
  

  Description

  In the project the stability of power networks and power network models is analyzed. The classical way of modeling a power network is via a large differential-algebraic system of network equations (DAE). Modifications of the power network by adding extra power lines into the network grid or by removing some power lines can be interpreted as low rank perturbations of matrices and matrix pencils that linearize the DAE system mentioned above. In the project, the influence of these perturbation on the stability of the network is analyzed.
  
  http://www.math.hu-berlin.de/~numteam1/projects/SE3.php

• SE21

  Data Assimilation for Port-Hamiltonian Power Network Models

  Dr. Raphael Kruse / Prof. Dr. Volker Mehrmann / Dr. Matthias Voigt

  Project heads: Dr. Raphael Kruse / Prof. Dr. Volker Mehrmann / Dr. Matthias Voigt
  Project members: Riccardo Morandin
  Duration: 01.06.2017 - 31.12.2018
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  In this project we will study the modeling of power networks by employing the port-Hamiltonian framework. Energy based modeling with port-Hamiltonian descriptor systems has many advantages, e. g., it accounts for the physical interpretation of its variables, it is best suited for the modular structure of the network, since coupled port-Hamiltonian systems form again a port-Hamiltonian system and it encodes these properties in algebraic and geometric properties that simplify Galerkin type model reduction, stability analysis, and also efficient discretization techniques. To improve the predictions that one obtains from such models we suggest to employ data assimilation and state estimation techniques by incorporating the measurement data. These would allow to take the uncertainty in the measurements and the presence of unmodeled dynamics as well as data and modeling errors into account. The improved predictions can then be used to control the network such that (the expected value of) the load is kept as constant as possible. To control the network we propose to use techniques of model predictive control (MPC) which solve a sequence of finite horizon optimal control problems. The method uses predictions of the state and computes a local optimal control which is then used for the model simulation in the next iteration. This framework is very flexible, since it allows control in real time and the incorporation of nonlinear dynamics and/or inequality constraints. It has already been used successfully within other areas of energy network control. Our new ansatz will also incorporate the stochastic effects into the model predictive control framework using data assimilation. Our vision is to develop numerical methods for network operators that allows the incorporation of model uncertainities for improving simulation and control of power networks.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/SE21/

• SE-AP8

  Entwicklung eines reduzierten Modells eines Pulsed Detonation Combustors

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: -
  Duration: 01.07.2012 - 30.06.2020
  Status: running
  Located at: Technische Universität Berlin
  

  Description

  In this project a model reduction of reactive flows is developed. Model reduction aims to replace complex, high-dimensional models by models of much smaller dimension. Goal of this project is to improve the existing techniques for systems where transport phenomena are dominant. To this end an appropriate error estimator is developed and combined with a model reduction. The small model can then be adaptively improved by adding physically motivated ansatz-functions. By this approach a low order model of a pulsed combustion is derived. This is used for control and design of a pulsed detonation combustor.
  
  The reduced order models shall not only describe the process of the combustion but also show the changes due to specific manipulation. The controllability in the context of mathematical fluid dynamics is determined via adjoint equations. For this the adjoint equations for reactive flows have to be differentiated and implemented.
  
  The reduced models are then used to design the combustor and to control the combustion process.
  
  https://www.sfb1029.tu-berlin.de/menue/teilprojekte/a02/parameter/en/

• SE-AP20

  Analysis, numerical solution and control of delay differential-algebraic equations

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: -
  Duration: 01.01.2011 - 31.12.2022
  Status: running
  Located at: Technische Universität Berlin
  

  Description

  Delay differential-algebraic equations (DDAEs) arise in a variety of applications including flow control, biological systems and electronic networks. We will study existence and uniqueness as well as the development of numerical methods for general nonlinear DDAEs. For this, regularization techniques need to be performed that prepare the DDAE for numerical simulation and control. We will derive such techniques for DDAEs on the basis of a combination of time-differentiations and time-shifts, in particular for systems with multiple delays. We also plan to extend the spectral stability theory, i.e. the concepts of Lyapunov, Bohl and Sacker-Sell spectra, to DDAEs. We will also develop numerical methods for the computation of these spectra using semi-explicit integration methods. Another goal is to study the solution of algebraically constrained partial delay-differential equations arising in flow control and to derive discretization as well as optimal control methods in space and time.
  
  http://www.itp.tu-berlin.de/collaborative_research_center_910/sonderforschungsbereich_910/project_groups/a_theoretical_methods/tp_a2/

• MI-AP7

  Controlled coupling of mixed integer-continuous models with modeled uncertainties

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: -
  Duration: 01.10.2014 - 30.06.2022
  Status: running
  Located at: Technische Universität Berlin
  

  Description

  The aim of project B03 is the development of a new methodology for the coupling of widely different mathematical models in a network. Moreover, error controllers are developed on the basis of modeled errors and uncertainties using the example of gas networks. These errors and uncertainties in submodels of the complex network are balanced in the overall simulation. Therefore, measures for the errors and uncertainties should be modeled for every submodel and made comparable. This can only succeed on the basis of a detailed model hierarchy, see Figure 1. All the errors and uncertainties in the simulation and optimisation are considered as an error in the finest modeling level using a backward error analysis in the model hierarchy. The estimated error in the finest level forms the mathematical basis for the development of a robust coupling controller. The controller should allow us to control the overall error in such a way that a prescribed simulation or optimisation goal is achieved within a desired tolerance.
  
  http://trr154.fau.de/index.php/en/subprojects/b03e

• MI-AP20

  Eigenvalue Analysis and Model Reduction in the Treatment of Disc Brake Squeal

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: -
  Duration: 01.09.2012 - 31.03.2015
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Disc brake squeal is a frequent and annoying phenomenon. It arises from self-excited vibrations caused by friction forces at the pad-rotor interface for an industrial brake model [1] (see Figure 1a and 1b). In order to satisfy customers, the automotive industry has been trying for decades to reduce squeal by changing the design of the brake and the disc. So far, it has found no satisfactory solutions that can be implemented in a systematic way. To improve the situation, several car manufacturers, suppliers, and software companies initiated a joint project, supported by the German Federal Ministry of Economics and Technology, which included two mechanical engineering groups at Technical University (TU) Berlin and TU Hamburg-Harburg, and the numerical analysis group at TU Berlin [1]. The goal of the project was to develop a mathematical model of a brake system with all effects that may cause squeal, to simulate the brake behavior for many different parameters, and to generate a small-scale reduced-order model that can be used for optimization.
  
  https://sinews.siam.org/DetailsPage/TabId/900/ArtMID/2243/ArticleID/443/Eigenvalue-Analysis-and-Model-Reduction-in-the-Treatment-of-Disc-Brake-Squeal.aspx

• OT3

  Adaptive finite element methods for nonlinear parameter-dependent eigenvalue problems in photonic crystals

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: Dr. Robert Altmann
  Duration: -
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Photonic crystals are periodic materials that affect the propagation of electromagnetic waves. They occur in nature (e.g. on butterfly wings), but they can also be manufactured. They possess certain properties affecting the propagation of electromagnetic waves in the visible spectrum, hence the name photonic crystals. The most interesting (and useful) property of such periodic structures is that for certain geometric and material configurations they have the so-called bandgaps, i.e., intervals of wavelengths that cannot propagate in the periodic structure. Therefore, finding materials and geometries with wide bandgaps is an active research area. Mathematically, finding such bandgaps for different configurations of materials and geometries can be modelled as a PDE eigenvalue problem with the frequency (or wavelength) of the electromagnetic field as the eigenvalue. These eigenvalue problems depend on various parameters describing the material of the structure and typically involve nonlinear functions of the searched frequency. The configuration of the periodic geometry may also be modified and can be considered a parameter. Finally, through the mathematical treatment of the PDE eigenvalue problem another parameter, the quasimomentum, is introduced in order to reduce the problem from an infinite domain to a family of problems, parametrised by the quasimomentum, on a finite domain. These are more accurately solvable. In order to solve the problem of finding a material and geometric structure with an especially wide bandgap, one needs to solve many nonlinear eigenvalue problems during each step of the optimization process. Therefore, the main goal of this project is to find efficient nonlinear eigensolvers. It is well-known that an efficient way of discretizing PDE eigenvalue problems on geometrically complicated domains is an adaptive Finite Element method (AFEM). To investigate the performance of AFEM for the described problems reliable and efficient error estimators for nonlinear parameter dependent eigenvalue problems are needed. Solving the finite dimensional nonlinear problem resulting from the AFEM discretization in general cannot be done directly, as the systems are usually large, and thus produce another error to be considered in the error analysis. Another goal in this research project is therefore to equilibrate the errors and computational work between the discretization and approximation errors of the AFEM and the errors in the solution of the resulting finite dimensional nonlinear eigenvalue problems.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/OT3/

• OT10

  Model Reduction for Nonlinear Parameter-Dependent Eigenvalue Problems in Photonic Crystals

  Dr. Robert Altmann / Prof. Dr. Volker Mehrmann

  Project heads: Dr. Robert Altmann / Prof. Dr. Volker Mehrmann
  Project members: Marine Froidevaux
  Duration: 01.06.2017 - 30.09.2019
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Photonic crystals are special materials having a periodic structure that can be used for trapping, filtering and guiding light. The key property of such materials is their ability to prevent light waves with specific frequencies from propagating in any direction. Because it is very challenging to build photonic crystals featuring a spectrum that can comply with the specific requirements of applications, a mathematical description and analysis of the electromagnetic properties of photonic crystals is needed, in order to support engineers in finding suitable components as well as optimal crystal geometries for new promising applications. The goal of the project is to develop an efficient solver for parameter-dependent non-linear eigenvalue problems arising in the search of photonic band-gaps. This solver should combine, in a computing-time optimal way, adaptive finite element methods (AFEM) for PDE eigenvalue problems, numerical methods for nonlinear eigenvalue problems, and low-dimensional approximations for a parameter space. The free parameters needed for the design of photonic crystals describe, e. g., the geometry of the crystal or the electromagnetic properties of the material. In order to optimize the properties of the photonic crystals over a given parameter set, we need to apply techniques from model order reduction. We plan to use approximations of the eigenfunctions, obtained by AFEM for several parameters in order to construct a reduced basis. These computations may be performed in parallel and, ideally, result in a set of eigenfunctions that contains good approximations of the eigenfunctions for all parameter values. We want to approximate the set of locally-expressed eigenfunctions with a low-dimensional non-local basis. Moreover, having efficient computations in mind, we need rigorous error bounds in order to equilibrate the different kinds of errors introduced at every level of approximation. Indeed, the total numerical error includes the discretization error arising from the AFEM, the algebraic error arising from the (iterative) solution of the nonlinear eigenvalue problems, and the model reduction error arising from the discretization of the parameter set. Since all these errors are normally measured in different norms, a unifying setting has to be developed in order to be able to compare all types of errors.
  
  http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/OT10/

• OT-AP5

  MODSIMCONMP - Modeling, simulation and control of multiphysics systems

  Prof. Dr. Volker Mehrmann

  Project heads: Prof. Dr. Volker Mehrmann
  Project members: -
  Duration: 01.04.2011 - 31.03.2016
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  The project aims at developing and analyzing a fundamentally new interdisciplinary approach for the modeling, simulation, control and optimization of multi-physics and multi-scale dynamical systems.
  
  The innovative feature is to generate models via a network of modularized uni-physics components, where each component incorporates a mathematical model for the dynamical behavior as well as a model for the uncertainties, arising, e.g., by modeling, discretization or finite precision computation errors.
  
  Based on this new modeling concept also new numerical simulation,control, and optimization techniques will be developed and incorporated, that allow a systematic adaptive error control - including the appropriate treatment of different scales, and the uncertainties - for the components as well as for the whole multi-physics model.
  
  The new remodeled systems will be designed such that they allow an efficient and accurate dynamical simulation with high order numerical integration techniques as well as the application of efficient methods for model reduction and open and closed loop control.
  
  In order to cope with the differential-algebraic and multi-scale character of the systems we plan to develop and analyze remodeling techniques for the components as well as for the whole network including the uncertainties as well as special structures of the system.
  
  In an interdisciplinary corporation with colleagues from engineering and computer science we plan to extend the modeling language Modelica to incorporate the new features - in particular the uncertainties and modeling errors - and to implement the complete approach as a new software platform.
  
  http://www3.math.tu-berlin.de/multiphysics/Description/