Overview

• Application Fields
  • Clinical Research and Health Care
  • Metropolitan Infrastructure
  • Optical Technologies
  • Sustainable Energies
  • Geometric Design and Visualization
  • Education and Outreach
• Projects
  • Running projects
  • Successfully completed projects
• Publications
• Scientists
• Preprints
• Institutional Cooperation
• Archiv 02-14

Successfully completed projects

• OT1

  Mathematical modeling, analysis, and optimization of strained Germanium-microbridges

  Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas

  Project heads: Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas
  Project members: Dr. Lukas Adam / Dr. Dirk Peschka
  Duration: -
  Status: completed
  Located at: Humboldt Universität Berlin / Weierstraß-Institut
  

  Description

  The goal of the project Mathematical Modeling, Analysis, and Optimization of Strained Germanium-Microbridges is to optimize the design of a strained Germanium microbridge with respect to the light emission. It is a joint project with the Humboldt-University Berlin (M. Hintermüller, T. Surowiec) and the Weierstrass Institute (A. Mielke, M. Thomas), that also involves the close collaboration with the Department for Materials Research at IHP (Leibniz-Institute for Innovative High Performance Microelectronics, Frankfurt Oder).
  
  http://www.wias-berlin.de/projects/ECMath-OT1/

• OT2

  Turbulence and extreme events in non-linear optics

  PD Dr. Uwe Bandelow / Dr. M. Wolfrum

  Project heads: PD Dr. Uwe Bandelow / Dr. M. Wolfrum
  Project members: Dr Shalva Amiranashvili
  Duration: -
  Status: completed
  Located at: Weierstraß-Institut
  

  Description

  Many modern photonic devices show complex dynamical features in space and time resulting from optical nonlinearities in active, often nanostructured materials. The project is focussed specifically on high-dimensional dynamical regimes in optoelectronic systems. Such a complex spatio-temporal behavior, in which nearly all modes are excited, is characterized by the fact that, in contrast to e.g. solitons or pulsations, it cannot be reduced to a low-dimensional description in terms of classical bifurcation theory. This so-called optical turbulence can be observed both in a Hamiltonian and in a dissipative context. A mathematical treatment of the resulting multi-scale and multi-physics problems presents major challenges for modelling, numerical, and analytical investigations. A simulation of the mostly 2+1 dimensional PDE-systems requires efficient parallelization strategies, instability mechanisms can be described only in terms of amplitude equations, and multi-scale effects in complex device structures can lead to singularly perturbed dynamical problems.
  
  http://www.wias-berlin.de/projects/ECMath-OT2/project_OT2.jsp

• 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/

• OT5

  Reduced basis computation of highly complex geometries

  Prof. Dr. Frank Schmidt

  Project heads: Prof. Dr. Frank Schmidt
  Project members: Sven Herrmann
  Duration: -
  Status: completed
  Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Description

  A typical trend in nanotechnology is to extend technology from basically 2D structures to 3D structures, from simple 2D layouts to complex 3D layouts. This has mainly two reasons: (i) There are fundamental physical effects bound to 3D structures, e.g., manifold properties in reciprocal space, and (ii) economic reasons as in semiconductor industry which enforce denser packaging and ever more complex functionalities.
  
  The automatic optimization of nano-photonic device geometries is becoming increasingly important and, due to enhanced complexity, increasingly difficult. Typical one-way simulations become unfeasible in many-query and real-time contexts. Model reduction techniques could be a way out. Potentially they offer online speed ups in the order of magnitudes. The reported success, however, is often linked to relatively simple structured objects. Slightly more complex examples fail immediately due to geometric and mesh constrains. To show the potential in real-world examples, however, complex 3D objects including comprehensive parametrizations have to be assembled.
  
  The project aims to establish a link from 3D solid models obtained by CAD techniques, including full parametrizations, to reduced basis models. Establishing this critical link would facilitate systematic device geometry optimizations to be carried out using rigorous 3D electromagnetic field simulations. The main question is, how we can realize a large scale parametrization maintaining topologically equivalent meshes.
  
  http://www.zib.de/projects/reduced-basis-computation-highly-complex-geometries

• OT6

  Optimization and Control of Electrowetting on Dielectric for Digital Microfluidics in Emerging Technologies

  Prof. Dr. Michael Hintermüller

  Project heads: Prof. Dr. Michael Hintermüller
  Project members: Dr. Soheil Hajian
  Duration: 01.06.2017 - 31.12.2018
  Status: completed
  Located at: Humboldt Universität Berlin
  

  Description

  A number of emerging key technologies in microbiology, medical diagnostic devices, personal genomics, as well as next-generation low-energy OLED displays and liquid lenses make use of a phenomenon known as electrowetting on dielectric (EWOD). Electrowetting involves the manipulation of small (microscopic) droplets on a dielectric surface by the actuation of the underlying current. In fact, droplets in a typical EWOD device are situated between two separated hydrophobic surfaces, one of which contains an array of controllable electrodes. The air-liquid-solid contact angle can then by changed by varying the voltages on separate electrodes, which causes the droplets to move. Thus, the voltages are a natural choice for influencing (controlling) the motion of a droplet. The project pursues both sharp interface and phase field models, respectively, for the movement of droplets in an EWOD device. Both models make use of a macroscopic description for contact line pinning, which is due to contact angle hysteresis as well as molecular adhesion at the solid-liquid-air interface, for a faithful representation of the droplets velocity and cover different aspects properly. Due to the non-trivial dependencies on the moving interface in the sharp interface context, the proof of existence of an optimal control remains impossible without further restrictive assumptions or constraints, e.g., on the geometry, and the complexity of the phase field model poses severe challenges for a fast (real-time) numerical solution as needed for EWOD devices. For these reasons, instead of computing time-discrete or optimal controls the project work pursues an idea from model predictive control (MPC).
  
  http://www2.mathematik.hu-berlin.de/~hajianso/ot6/

• OT7

  Model-based geometry reconstruction of quantum dots from TEM

  Dr. Thomas Koprucki / Dr. Karsten Tabelow

  Project heads: Dr. Thomas Koprucki / Dr. Karsten Tabelow
  Project members: Anieza Maltsi
  Duration: 01.06.2017 - 31.12.2018
  Status: completed
  Located at: Weierstraß-Institut
  

  Description

  Semiconductor quantum dots are nanostructures that form a technological path to innovative optoelectronic and photonic devices. Among them single quantum dots are promising candidates for single and entangled photon sources which are of importance for future quantum technologies such as quantum information processing, quantum cryptography, and quantum metrology. The growth of QDs with desired electronic properties would highly benefit from the assessment of QD geometry, distribution, and strain profile in a feedback loop between growth and analysis of their properties. In this project, we will therefore develop a novel 3D model-based geometry reconstruction (MBGR) of QDs. This will include an appropriate model for the QD configuration in real space, a characterization of corresponding simulated TEM images as well as a statistical procedure for the estimation of QD properties and classification of QD types based on acquired TEM image data. The MBGR approach will enable a high-throughput characterization of QD samples by TEM via QD geometry, distribution and strain field. Furthermore, it will provide a guiding example for mathematically enhanced microscopy for the reconstruction of other nanoscale objects in different applications.
  
  https://www.wias-berlin.de/projects/ECMath-OT7/

• OT8

  Modeling, analysis, and optimization of optoelectronic semiconductor devices driven by experimental data

  Dr. Marita Thomas

  Project heads: Dr. Marita Thomas
  Project members: Dr. Dirk Peschka
  Duration: 01.06.2017 - 31.12.2018
  Status: completed
  Located at: Weierstraß-Institut
  

  Description

  The goal of the Matheon project D-OT8: Modeling, analysis, and optimization of optoelectronic semiconductor devices driven by experimental data is to optimize the design of a strained germanium microbridge with respect to the light emission. In the funding period June 2016-December 2018 we will develop tools for the parameter identification and optimal design of experiment for optoelectronic applications. The project also involves the close collaboration with the Department for Materials Research at IHP (Leibniz-Institute for Innovative High Performance Microelectronics, Frankfurt Oder).
  
  http://www.wias-berlin.de/projects/ECMath-OT8/

• OT9

  From single photon sources to tailored multi-photon states

  Dr. Sven Burger / Prof. Dr. Frank Schmidt

  Project heads: Dr. Sven Burger / Prof. Dr. Frank Schmidt
  Project members: Felix Binkowski
  Duration: 01.06.2017 - 30.09.2019
  Status: completed
  Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin
  

  Description

  This project investigates methods to model and simulate nanoscale light emitters in complex environments. Semiconductor quantum dots can be used as light sources in quantum information processing. Typical applications like secure communication or quantum-computing require integration of quantum dots into optical nanostructures. For the analysis and design of such structures and their interaction with radiation emitted by the quantum dots, numerical modeling and simulations are essential. In this project we concentrate on the specific problems arising when pointlike sources like quantum dots are interacting with nanostructures which support optical resonances. We plan to develop, extend and analyze methods for efficiently simulating coupling to optical resonators with material dispersion and for methods for handling coupled resonators.
  
  https://www.zib.de/projects/single-photon-sources-tailored-multi-photon-states

• 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-TU26

  Asymptotic analysis of the wave-propagation in realistic photonic crystal wave-guides

  Dr. Kersten Schmidt

  Project heads: Dr. Kersten Schmidt
  Project members: Dirk Klindworth / Dr. Adrien Semin
  Duration: 01.06.2014 - 31.12.2016
  Status: completed
  Located at: Technische Universität Berlin
  

  Description

  Photonic crystal wave-guides are devices that allow for exceptional tailoring of the properties of light propagation. Currently, the prediction of the properties relies mainly on models for infinite, perfect photonic crystal wave-guides. For photonic crystal circuits scattering matrix approaches have been proposed. In this project we study imperfect photonic crystal wave-guides and circuits of finite lengths with techniques of asymptotic expansion.
  
  http://www.math.tu-berlin.de/?Matheon-d26

• OT-AP2

  Direct and inverse interaction problems with unbounded interfaces between acoustic, electromagnetic and elastic waves

  Dr. Guanghui Hu

  Project heads: Dr. Guanghui Hu
  Project members: -
  Duration: 01.08.2012 - 31.07.2015
  Status: completed
  Located at: Weierstraß-Institut
  

  Description

  Direct and inverse interaction problems between acoustic, electromagnetic and elastic waves occur in many applications in natural sciences and engineering. The project is devoted to the investigation of scattering of time harmonic acoustic and electromagnetic waves by an unbounded elastic body in the case of periodic structures (diffraction gratings) as well as in the non-periodic case (rough surfaces). This leads to direct and inverse transmission problems between the Helmholtz (or Maxwell) equations and the Navier equation in unbounded domains, the analytical and numerical treatment of which is challenging. One objective of the project is to develop a new solvability theory (existence and uniqueness of solutions, Fredholm property) for the direct scattering problems using variational methods. In the more general and difficult case of rough interfaces, this requires the derivation of novel a priori estimates in weighted Sobolev spaces. The second goal of the project is the development and theoretical justification of efficient numerical methods for the solution of the direct and inverse interaction problems. The approximate solution of the direct problems will be based on finite element and boundary element methods, whereas for the solution of the inverse problem of reconstructing the interface from near and far field measurements of the scattered acoustic or electromagnetic field, optimization and factorization methods will be used. For both tasks, inspiration should be taken from recent results on electromagnetic and elastic diffraction gratings and rough surfaces and on interaction problems with bounded elastic obstacles.
  
  http://www.wias-berlin.de/projects/interaction/

• 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/