##
Pattern Formation with Nonlocal Diffusivity

Lisa Ott, 2016

In this thesis we are interested in spatial patterns formed by a system
of interacting particles where the mobility of any individual is determined
by the distribution of all other individuals. Fluctuations of the population
density influence the intensity of the random motion of every individual. More
precisely, the extent of the particular motion of every individual depends of
the average density in its surrounding. This effect leads to nonlocal individual
based and PDE models. To model these conflicting incentives, we take two different approaches.
On the one hand, we will extend the standard aggregation model, which is
based on the average density on one length scale, by using a non-monotonous
weighting function. This introduces movement for two different ranges of local
density. On the other
hand, we will extend the model to have two different local densities for two
different sampling radii. This way we can separate the two incentives, such
that one radius corresponds to the tendency to aggregate and the other radius
corresponds to the tendency to disperse.
[pdf-file]

##
Rekonstruktionsverfahren der
thermoakustischen Tomographie

Paul Striewski, 2014 (Supervision: Frank Wübbeling)

Thermoacoustic Tomography (TAT) is a promising non-invasive, hybrid
medical imaging modality which aims at combining the advantages of
microwave and ultrasound imaging. In this thesis, the mathematical
background of TAT was studied. The focus was laid on the inverse
problem of TAT, which is an inverse problem for the scalar wave
equation in two and three dimensions. The derivation of different
inversion techniques was discussed and their numerical realization
treated. The application of the derived methods to synthetic data and a
comparison of the achieved reconstruction results concluded the thesis.
[pdf-file]

##
A Multiscale Method for Interacting
Particle Systems

Jörg Sauter, 2013

Motivated by biological ion channels, we develop a new multiscale method for interacting
particle systems coupling a particle and a continuum model. The method aims at
the computation of particle interactions on the molecular scale and therefore incorporates
particle correlations induced by short and long range interactions. To construct
the multiscale method we use a stochastic particle model and derive a macroscopic
equation hierarchy using the Fokker-Planck equation. Instead of closing the hierarchy
by the molecular chaos assumption, we use the particle model to compute a closure to
the equation hierarchy. We perform numerical experiments comparing the multiscale
method to the stochastic particle model and a mean-field equation which can be interpreted
as the hierarchy closed by the molecular chaos assumption. We find that under
some restrictions the method is capable to handle long range and also additional short
range interactions.
[pdf-file]

##
Blind Unmixing von
Raman-Mikrospektroskopie-Daten

Andre Overesch, 2013

Raman-Spectroscopie has recently become a very powerful tool to analyse biological samples.
Unfortunately mixing of the spectra from different components to a mixed result is an often
reported problem. The underlying spectra are of course necessary for the further treatment and
interpretation.
This thesis deals with the unmixing of the received (mixed) result in order to reveal the spectra of
the hidden components. Therefor we discuss different Non-negative Matrix Factorization
approaches and provide the derivation of these methods. At the end we give their implementation
and apply our algorithms on synthetic and real raman-spectra datasets.
[pdf-file]

##
Linear Functionals in ECG and VCG

Joanna Tendera, 2013 (Collaboration with: Eric Schulze-Bahr, Sven Zumhagen UKM)

This thesis deals with the diagnostic method called vectorcardiogram, which is an
extension of the well known electrocardiogram. Based on the dipole theory, we express
the heart vector on the body surface as dot product of potential difference and lead
matrix. The linear functional strategy enables us to find a second representation of
the heart vector on the heart boundary. Thus, we analyze the heart vectors on the
body boundary and on the heart boundary for two different lead matrices and two
electrode configurations. Finally, we establish different vectorcardiogram parameters
and examine the relationship between the heart disease area and the corresponding
vectorcardiogram.
[pdf-file]

##
Mathematical Modelling of Lane Formation in Crowds

Andrea Riberi, 2013

The steadily increasing number of people living in urban areas has led to a growing
interest to understand pedestrian motion. This thesis is concerned with the modelling
of crowd motion. Based on a simple exclusion process, a microscopic model for two
different types of pedestrians moving on a finite number of coupled lanes is presented.
The transition to a macroscopic model leads to a system of continuous drift-diffusion
equations. In order to analyse the stationary states of these partial differential equations,
an initial value problem is derived and solved numerically. Different crowd densities and
lane transition rates are compared and discussed.
[pdf-file]

##
Numerische Analyse eines
Lucas-Tree-Modells mit zwei
heterogenen Investoren

Jan Philipp Bensch, 2013 (Collaboration with: Nicole Branger, WWU)

In this diploma thesis, differential equations for two heterogeneous Epstein-Zin investors are derived with the help of modern option-price theory.
These differential equations describe the wealth-consumption ratio of the two investors. They are restricted numerically under two different restrictions.
Under the stronger restriction, the equations are simplified to ordinary differential equations. Here, the derivatives of the equations are approximated with the Upwind-procedure.
Then, the ordinary differential equations are solved numerically with the Newton's method. Under the weaker restriction, the partial characteristic of the differential equations
is received. In this case, differential equations are dependent on the consumption share and the long-run risk variable. These partial differential equations are solved numerically
with the Forward-Explicit Euler method. The two different solution methods presented both were tested on their exactness and speed.
[pdf-file]

##
Mathematische Methoden zur Segmentierung von Kardio-MR Zeitsequenzen

Dirk Mannweiler, 2012

The aim of this thesis is to automate and optimize
the segmentation of MRI time sequence images and to connect them
into a time sequence. In the first step of this dissertation the theory of segmentation
and processing of MRI images is presented.
In the second step this theory has been used to create a MATLAB user interface
with which can generate from a set of DICOM slices a volume and segment it into
a human heart. Based on several segmented images, these time sequences could
be assembled to an animation of a beating heart. [pdf-file]

##
Anisotropic Conduction in Electrocardiology

Stefanie Kälz, 2012

In this thesis we have applied the theory of periodic homogenization in
order to calculate the macroscopic electrical conductivity values within the
heart. Macroscopic electrical conductivity values are an important factor in
the modeling and simulation of the electrical activity of the heart. The
base for our calculations is a micro-CT image of a pig heart. We solve the
so-called cell problem of homogenization and calculate the homogenized
macroscopic conductivity values. Instead of periodic boundary conditions, we
have used homogeneous Dirichlet and homogeneous Neumann boundary conditions.
Subsequently, we have compared the results. [pdf-file]

##
Numerische Analyse von LRR-Modellen
mit zwei Lucas-B\"aumen und heterogenen Investoren

Michael Dörr, 2012 (Collaboration with: Nicole Branger, WWU)

This thesis was concerned with a numerical approximation of the wealth-consumption ratios of two heterogeneous EZ investors. The market is
modelled by two Lucas trees and the consumption share of the investors plays an important role in our model. We derive an initial value
problem from the partial differential equations to search for a stationary solution based on the market parameters. We simulate market
situations for different degrees of risk, elasticities of intertemporal substitution and time preferences of the investors. Varied parameters
of the two Lucas trees are also taken into account. We provide a thorough analysis of the results.
[pdf-file]

##
Numerische Analyse
von Long Run Risk Modellen
mit zwei Bäumen und Sprungrisiko

Johannes Härtel, 2012 (Collaboration with: Nicole Branger, WWU)

Nowadays considerable attention is given to general equilibrium models of the
pricing of capital assets. This thesis was concerned with a numerical analysis
of a long-run risk model with rare events. The partial dierential equation for
the wealth-consumption ratio arises from a model with two stochastic consumption
growth rates, a stochastic volatility and a stochastic jump intensity.
In particular, the partial dierential equation was analysed with respect to
existence and uniqueness as well as to convergence of a numerical solution.
In addition, the numerical approximation was implemented and tested in a
realistic setup.
[pdf-file]

##
TV-basierte Entrauschung von 3D Bildern mit Anwendung in der Fluoreszenzmikroskopie

Nina Schumacher, 2012

This thesis deals with the reconstruction of images corrupted by Poisson noise, which
occurs in several applied sciences e.g. in medical imaging, such as fluorescence
microscopy and positron emission tomography (PET), or in geophysics or astronomy. We
combine expectation-maximization (EM) and total-variation (TV) to present an algorithm
for computing images with homogeneous regions and sharp edges. Our approach is
based on the minimization of the Kullback-Leibler divergence and we further verify the
convergence of the algorithm. Finally, we present an implementation and apply it on
several synthetic and experimental datasets in 2D and 3D.
[pdf-file]

##
Sparse model-based reconstruction in dynamic positron emission tomography

Pia Heins, 2011

This thesis deals with the inverse problem of dynamic positron emission tomography,
which we attempt to solve with variational methods. Based on kinetic modeling we
express the unknown image as a linear combination of a large dictionary, which contains
the given basis functions. Since only a few of them are necessary to express the data
for each pixel, we motivate the usage of `1;1-regularization to promote sparsity of the
coecients. In a more general approach we analyze, which conditions are necessary
for exact recovery of the coecients. The main difficulty is that the basis functions
are extremely similar. Thus, most of the known conditions for exact recovery are
not applicable in this case. We provide three different splitting algorithms, which
demonstrate that the use of l-1-infinity-regularization is a reasonable approach. In order to
analyze these algorithms, we finally test them on artificial data.
[pdf-file]

##
Mathematische Modellierung Neuronaler Polarisierung

Natalie Emken, 2011 (Collaboration with Andreas Püschel, WWU Münster)

This thesis was concerned with the development of neuronal polarity. This
event is initiated by a symmetry-breaking event whereby one out of multiple
minor neurites undergoes rapid outgrowth and becomes the axon. Many signalling
pathways, involving the phosphatidylinositol 3-kinase (PI3K) and Rho GTPases,
regulates this mechanism. To understand how such signalling pathways can
generate polarity, basal models of cell polarity were presented. Following a
mathematical model of neuronal polarity based on the local positive feedback
between the GTPase Rac and PI3K were developed. The analysis has shown that
this feedback is able to initiate robust symmetry breaking and formation of a
single axon in the presence of extracellular signals or stochastic processes.
Including anterograde transport of molecules numerical simulations eventually
has shown the importance of proceeding transport, which enables neurons to
polarize in the absence of extrinsic spatial cues. Furthermore it is shown
that this transport guarantees the polarization of the longest neurite.
[pdf-file]

##
Numerical Methods for Mean-Field Games

Philipp Schmauck, 2011 (Collaboration with Marie-Therese Wolfram, Vienna)

This thesis was concerned with the numerical implementation of a Fokker-Planck equation coupled with a Hamilton-Jacobi-Bellman equation. This system is called Mean-Field Games and models the behavior of a large number of interacting agents.
We use the fact that under certain assumptions the equations form an optimality system of a convex optimization problem. The well-posedness of such a problem is carried out with optimal control theory. The discretization is done using Raviart-Thomas elements
and an implicit difference scheme for the time derivative. To solve the resulting system, we use Newton's method.
[pdf-file]

##
Medical Image Segmentation including Particle Methods

Hendrik Dirks, 2011

Segmentation plays an important role in modern image processing. In this diploma
thesis we present an edge tracking algorithm based on charged particles. We discuss
the reconstruction from a point-cloud to a set of boundaries, perform analysis on the
discrete model and establish a mean-eld limit which results in a PDE. Also topological
aspects of point-clouds are analyzed using persistent homology and applied to the reconstruction.
We give a short overview of current particle models and nally introduce
an algorithm that converts a set of boundaries into a level-set representation.
[pdf-file]

Hendrik Dirks is now a PhD-Student at the Institute for Computational and Applied Mathematics, WWU Münster.

##
An adjoint FEM approach for the EEG
forward problem

Sven Wagner, 2011 (Collaboration
with Carsten Wolters, WWU Münster)

This thesis is concerned with an investigation of the adjoint approach for the EEG forward problem. It is deduced from the adjoint method and compared to
the partial integration approach using transfer matrices for tetrahedral and hexahedral 4-layered spherical shell models. Furthermore a realistic
hexahedral head model with anisotropic gray and white matter compartments is used to investigate the $L_2$ sensitivity distribution for a given lead positioned at the surface of the head model.
This head model is used to investigate the effect on the $L_2$ sensitivity distribution and the orientation of the lead field if the CSF compartment within the volume conductor is neglected.
The adjoint approach and the partial integration approach attain exactly the same results concerning required arithmetic operations, relative difference error
and magnitude error and both approaches use a continuous source space. The adjoint approach can be further used to investigate the $L_2$ sensitivity for a given lead for dipoles in
the source space.
[pdf-file]

Sven Wagner is now a PhD-Student at the Institute for Biomagnetism and Biosignalanalysis, WWU Münster.

##
Comparison of Numerical Approaches to the
EEG Forward Problem

Johannes Vorwerk, 2011 (Collaboration
with Carsten Wolters, WWU Münster)

This thesis was devoted to the study and optimization of solution approaches to the forward problem of electroencephalography (EEG). We investigated different numerical approaches regarding time consumption and particularly accuracy, two aspects that are crucial for solving the inverse problem of EEG, i.e., reconstructing source activity from measurement data.
Several studies have shown that the accuracy of the solution highly depends on a realistic modeling of the human head. Besides others, two different numerical techniques are commonly used in combination with realistically shaped head models nowadays: Boundary element methods (BEM) and finite element methods (FEM). We evaluated a recently proposed BE approach, the symmetric BEM, and demonstrated that it leads to a considerably improved accuracy in multi-layer sphere models compared to former BE formulations. Furthermore, we enhanced accuracy and speed of different FE approaches by the application of leadfield interpolation, a technique originally proposed for the speed-up of BE calculations. Various investigations regarding the accuracy of the FE approaches in
multi-layer sphere models were carried out before leadfield interpolation was applied to further optimize the results.
[pdf-file]

Johannes Vorwerk is now a PhD-Student at the Institute for Biomagnetism and Biosignalanalysis, WWU Münster.

##
Optimal Consumption Using a Continuous
Income Model

Daniel Raß, 2011 (Collaboration
with Mark Trede, WWU Münster)

This thesis was concerned with the developement of a Hamilton-Jacobi-Bellman like equation arising in the optimal consumption problem
using a continuous income model with stochastic uncertainty. Later on a finite differencing scheme was developed to determine the
associated viscosity solution numerically. The same scheme was used to compute the lower border of an area with allowed income/wealth
combinations under the additional condition no debts are allowed at the end of the life. With this border we were able to compute the maximal
utilty gained by allowed income/wealth combinations.
[pdf-file]

Daniel Raß is now working for Saracus Consulting, Münster.

##
Numerical methods for the
determination of relative
inverter positions in large
PV-plants

Matthias Gröne, 2011 (Collaboration
with Jens Klein, SMA Technologies, Kassel)

We provide an algorithm that is capable of determining relative inverter and string positions
in PV-power plants. For this purpose, we harness clouds whose shadows cause
declines in the power and current data allowing us to estimate the relative distances
between the strings.
The estimation of the positions from the relative distances leads to a problem of matrix
factorization and is solved through an alternating least squares algorithm. This
algorithm is analyzed theoretically and then applied to synthetic data.
Among other procedures, we have to linearize cloud fronts to match the mathematical
model in order to obtain reliable time differences from real data. We will show results
from real data and also deal with problems that occur within them.

Matthias Gröne is now working for SMA Technologies, Kassel.

##
Reconstruction of the
Epicardial Potential from
Body Surface Potential Maps

Patrick Verfürth, 2011 (Collaboration
with Alexander Samol, UKM and Christian Vahlhaus, Klinikum Leer)

The standard 12-lead electrocardiogram only yields information about the electrical potential on the body surface.
These potentials are generated by the electric currents along the heart surface and these currents determine the health status of the heart.
Applying a larger number (120) of electrodes on the torso surface we record a Body Surface Potential Map (BSPM) and formulate the inverse problem of ECG,
i.e. reconstruct the epicardial potential. This problem is severely ill-posed and regularization techniques are required. We applied different Tikhonov
regularizations penalizing the derivations of the solutions. With the reconstructed electrical potential along the heart surface a better diagnosis might be possible in future.
[pdf-file]

Patrick Verfürth is now working for Mentz DV, Münster.

##
Bildbasierte Lösung von Partiellen Differentialgleichungen mit Composite Finite Elements

Sebastian Westerheide, 2011 (Collaboration
with Mario Ohlberger, WWU)

This thesis was concerned with the approximate solution of partial
differential equations via discretization with composite finite
elements. It deals with the application of this approach to problems
where the computational domain is given through image data and thus
possibly is of complex shape. We summarize the method's mathematical
background, namely the Galerkin method and classical finite elements,
and describe in detail the theory of composite finite elements as
introduced by Hackbusch and Sauter. Based on simple elliptic model
problems we further present an efficient strategy for calculating
arbitrarily coarse approximations which nevertheless resolve the
problem's computational domain and preserve characteristic properties of
the exact solution. As part of the work, a program realizing the
described method for Poisson-like problems in up to three dimensions was
implemented in C++ using the DUNE framework. We use this program to
practically verify the theoretical convergence properties of composite
finite element discretizations and compare the potential of those
discretizations with the potential of the classical finite element
method.
[pdf-file]

Sebastian Westerheide is now a PhD Student at the Institute for Computational and Applied Mathematics, WWU Münster.

##
Hierarchical Bayesian Approaches to the
Inverse Problem of EEG/MEG Current
Density Reconstruction

Felix Lucka, 2011 (Collaboration
with Carsten Wolters, UKM)

This thesis was concerned with hierarchical Bayesian approaches to the inverse problem of EEG/MEG current density
reconstruction. The estimation of the activity-related ion currents by measuring the induced electromagnetic fields
outside the skull by means of current density reconstruction is both severely under-determined and ill-conditioned.
A special class of statistical models, called hierarchical Bayesian models, were introduced. Different inference strategies
based on these models were proposed and algorithmical aspects of the implementation and their practical use and
properties within simulation studies were examined. The corresponding forward modeling was done with a realistic
high resolution finite element (FE) model of a human head.
[pdf-file]

Felix Lucka is now a PhD Student at the Institute for Computational and Applied Mathematics and the Institute for Biomagnetism and Biosignal-Analysis, WWU Münster.

##
Variationsmethoden in der Datenassimilation zur Wetter- und Preisvorhersage

Fabian Lenz, 2011 (Collaboration with Peter Markowich, Cambridge)

This thesis was concerned with an application of the 4DVAR model to weather prediction and price formation. In both situations we have an initial value problem where the initial condition has to be estimated using appropriate observations. The 4DVAR method reduces this task to a minimisation problem. This can be solved with the help of Gauss-Newton or regularisation methods. We have compared the numerical solutions obtained with this methods.
[pdf-file]

##
Fokker-Planck Equations for Heterogeneous Trader Models

Kathrin Schulte, 2010

This thesis was concerned with the derivation, analysis, and numerical
simulation of multi-dimensional Fokker-Planck equations arising from agent-based models of financial markets
with heterogenous agents. The derivation of steady states and their existence and uniqueness is shown. The entropy
approach yields convergence of the solution to the steady state. The positivity of the solution is shown for
appropriate conditions. The derived systems in two and three dimensions are implemented with the Scharfetter-Gummel
scheme and an adjusted Upwind scheme.
[pdf-file]

##
Adjoint Methods for Hamilton-Jacobi Equations

Jan-Michael Schulte, 2010

This thesis was concerned with a novel approach for
Hamilton-Jacobi-Bellman
equations.
These partial differential equations arise as a central aspect in optimal
control
theory. Furthermore, they play an important part in the recently introduced
concept of mean field games, an approach for modelling the behavior of large numbers
of interacting agents. We take advantage of this by deducing stable
discretizations of Hamilton-Jacobi-Bellman equations and by applying the derived schemes to carry
out mean field games simulations. As a last result, we show a new achievement in
the theory of viscosity solutions and how these ideas may be generalized.
[pdf-file]

Jan Michael Schulte is now working for Oliver Wyman.

##
Mean-Field and Kinetic Market Models

Veronika Penner, 2010

This thesis was concerned with the illustration and development of kinetic
models, related to financial markets. Three models with particular attention
to their long time behaviour were presented. The derivation of a novel model
with an explicit analysis based on results from particle and statistical
physics was the main purpose. In favour of a realistic illustration we
considered homogeneous as well as heterogeneous conditions. In conclusion we
yielded mean-field partial differential equations for the corresponding
models.
[pdf-file]

Veronika Penner is now a PhD-Student at University of Kiel.

##
Numerical Simulation of Water Transport into Porous Membranes

Thomas Grosser, 2010

In biophysics, one is often interested in the osmotic water flow through
membranes, i.e., in investigating a membrane's osmotic permeability
coefficient. As this coefficient cannot always be determined experimentally, a
numerical simulation of this process is of interest. In this thesis, the
process is modelled by partial differential equations which are then solved
numerically. The solvers are implemented in C++ for the DOLFIN interface of
FEniCS.
.
[pdf-file]

##
Rekonstruktionsbasierte Partialvolumenkorrektur
in der Positronen-Emissions-Tomographie

Andre Gripshöfer, 2010 (Collaboration with: Klaus Schäfers, EIMI)

This thesis was concerned with the correction of the partial volume effect in positron emission tomography.
For this purpose a new method was presented which corrects the partial volume effect directly in the reconstruction
process. This method was tested on synthetically generated pet-images and the results were compared to those of
standard correction methods which are applied to pet-images after reconstruction.
[pdf-file]

##
NuBAkO
- Entwicklung und Anwendung einer
Toolbox zur Bewertung von Derivaten

Christoph Heuer, 2010 (Collaboration with: Nicole Branger, WWU)

This thesis was concerned with valuing European and American calls and puts.
In order to calculate the prices of these options the Black-Scholes model,
the Merton model, the Stochastic Volatility model, the SVJ model and the
Duffie-Pan-Singleton model were derived. After the derivation of these models
the resulting partial differential equations were discussed numerically and
implemented in Matlab. The implementation also includes the imlied volatility with respect
to the Black-Scholes model and the price with respect to moneyness.
[pdf-file]

Christof Heuer is now a PhD-Student at the Department of Mathematics, University of Sussex.

##
Numerische Berechnung von Preis-Konsum
Quotienten in verallgemeinerten
Gleichgewichtsmodellen

Jann-Philipp Zocher, 2010 (Collaboration with: Nicole Branger, WWU)

Asset pricing models are an important part of today's world of finance. This
thesis was concerned with the numerical approximation of the price-
consumption ratio for a general equilibrium. The paper "Explaining pre- and
post-1987 crash asset prices Within a unified general equilibrium framework"
from Benzoni, Collin-Dufresne and Goldstein yields the basic model for this
thesis. This financial market model contains the price-consumption ratio as
a fundamental part of it. In particular the price-consumption ratio was
analyzed with respect to the existence and uniqueness of an analytic
solution as well as to the convergence of a numerical solution. The accuracy
of the price-consumption ratio computation is improved by this thesis.
Thereby the simulation results of the asset pricing model are more
sophisticated.
[pdf-file]

Jann-Philipp Zocher is now working for Accenture.

##
Fokker-Planck Equations
for Agent-based Models of Financial Markets

Jörg Hagenberg, 2010

This thesis deals with an alternative method to analyze a financial market
model. Instead of exploiting a time series, the Fokker-Planck equation is
derived from the agent based model of Lux, Marchesi and Sheng. Some of its
properties as the existence of a stationary solution, the large time behavior
and the possibility of "fat tails'' are proven. Then this equation is compared
with the Fokker-Planck equation obtained from the GARCH(1,1) model and a
second Fokker-Planck equation that describes the dynamics of the price.
[pdf-file]

##
Mathematische Modelle der Meinungsbildung

Lisa Richter, 2010

This thesis was concerned with the presentation and comparison of several models of opinion formation and some of their extensions. In particular the behaviour and characteristics of the popular models of Sznajd-Weron, Hegselmann-Krause and Deffuant-Weisbuch were considered. For some models the processes of opinion formation were described by partial differential
equations. Especially for the Deffuant-Weisbuch-Model the dynamics for the time variations of opinions' density were derived and analyzed by computing stationary solutions. Finally the Deffuant-Weisbuch-Model was simulated numerically.
[pdf-file]

Lisa Richter is now working for Alte Leipziger Lebensversicherung.

##
Mathematische Methoden zur QRS-Detektion in EKG-BSPM Signalen

Jan Bäumker, 2010 (Collaboration with: Frank Wübbeling, WWU, Alex Samol, UKM, Christian Vahlhaus, Klinikum Leer)

This thesis describes a variety of algorithms for automatical QRS-detection in ECG/BSPM-Signals.
In it we give a overview of the heart and its structure, and how signalanalysis is used to work with ECG-signals.
We explain what types of algorithms exist, how they work, and after that we give an in-depth explanation of
two chosen algorithms. One of the algorithms, the Pan/Tompkins-algorithm, we code in a new version in MATLAB.
[pdf-file]

##
Mass-preserving Registration of
Medical Images

Lars Ruthotto, 2010 (Collaboration with: Carsten Wolters, WWU)

Image Registration is one of today's most challenging tasks in medical imaging. This thesis develops and investigates
tailored registration approaches for two important problems in real life medicine demanding mass-preserving
transformations. A variational approach to susceptibility correction of echo planar images (EPI) is proposed,
validated on phantom data and applied to functional magnetic resonance images (fMRI) and diffusion tensor images (DTI).
Secondly, gated positron emission tomography (PET) images are corrected for respiratory motion using a novel non linear
mass preserving registration algorithm.
[pdf-file]

Lars Ruthotto is now a PhD-Student at MIC, University of Lübeck and the Institute for Computational and Applied Mathematics, WWU Münster.

##
Opinion Dynamcis with Heterogeneous Agents

Ina Josek, 2009

This thesis investigated the modeling of dynamics of opinion formation between
heterogeneous agents. The classical Deffuant-Weisbuch-Model was modified and
extended to consider both the homogeneous and heterogeneous case. The
development of opinions in time was characterized each with partial differential
equations. In particular the heterogeneous model was analyzed with respect to the
existence and uniqueness of solutions, the occurrence of stationary states and their
stability. Furthermore numerical simulations of the model were carried out.
[pdf-file]

Ina Josek is now working for R+V Versicherung.

##
SEGMEDIX: Development and Application of a Medical Image Segmentation Framework

Daniel Wirtz, 2009 (Collaboration with: Xiaoyi Jiang, WWU)

One recent aim of the Imaging Workgroup is the processing of medical MR/CT
images in combination with ECG data to support
noninvasive medical analysis and thus treatment of patients. As a first step
this thesis is concerned with the development of
a segmentation framework containing a collection of both standard and modern
segmentation algorithms. Moreover, suitable pre- and
post processing algorithms complete the MATLAB based program to a fully
self-contained segmentation framework. In the second part
the framework is applied to the MR patient data obtained through the current
cooperation with several clinics and successful results
are presented along with some experiments.
[pdf-file]

Daniel Wirtz is now a PhD-Student at University of Stuttgart.

##
Spatial Structures in Geographical Economics: Mathematical Modelling, Simulation, and Inverse Problems

Ralf Engbers, 2009 (Collaboration with: Vincenzo Capasso, Milano)

This thesis was concerned with an extension of the Solow model which
is used to describe economic growth. A spatial dimension was added to
the model and a convex-concave production function was used instead of
a strictly concave production function. The resulting direct problem
was solved numerically and the parameter identification problem was
solved via the adjoint method with exact and noisy data.
[pdf-file]

Ralf Engbers is now a PhD-Student at the Institute for Computational and Applied Mathematics, WWU Münster.

##
Asset Allocation with Multiple State Variables

Tobias Neugebauer, 2009 (Collaboration with: Nicole Branger, WWU)

This thesis was concerned with the analysis and numerical solution of asset pricing problems with multiple underlyings, which are
formulated as Hamilton-Jacobi-Bellman equations.
The results are applied to an asset allocation problem with stochastic interest rate and income.
[pdf-file]

## Kollisionswahrscheinlichkeiten aus monokularen Bildfolgen

Jan Jatzkowski, 2009 (Collaboration with Bosch GmbH)

This thesis was concerned with the computation of a collision probability from motion scenes acquired with single cameras,
for the potential use in a driving assistance system. The method presented combines the optical flow and the result of
a segmentation to give the user an qualitative estimation of the risk, that an object will collide with the ego
vehicle (i.e. the vehicle the user is driving). The method was tested by sequences of both contrived and real traffic
scenes.

## An Adaptive Hybrid Method for 2D Crack Growth Simulation

Jan Hegemann, 2009 (Collaboration with: Joseph Teran, UCLA)

This thesis was concerned with the simulation of two dimensional quasi-static crack growth. The method presented combines the extended finite element method (XFEM) with a general cutting algorithm that provides the degrees of freedom for the crack to open as well as determines material connectivity. Also, a general and easy quadrature scheme was introduced, which uses an adaptively refined integration mesh embedded into a coarse simulation mesh.
The approach is applied to several test settings, which highlight varying considerations, including accuracy, crack propagation, and the ability of the method to handle complicated crack geometries.
[pdf-file]

Jan Hegemann is now a PhD-Student at the Department of Mathemtics, UCLA,and the Institute for Computational and Applied Mathematics, WWU Münster.

## A Variational Approach for Sharpening High-Dimensional Images

Michael Möller, 2009 (Collaboration with: Andrea Bertozzi, Todd Whitman, UCLA)

This thesis was concerned with the pan-sharpening of high-dimensional and hyperspectral images by variational techniques. A novel variational model including wavelet based energies as well as an edge alignemnt term based on total variation is analyzed and solved numerically. The approach is applied to satellite images. [pdf-file]

Michael Möller is now a PhD-Student at the Institute for Computational and Applied Mathematics, WWU Münster and UCLA.

## Influence of Volume Conduction on Beamformer Source Analysis in the Human Brain

Stephanie Sillekens, 2008 (Collaboration with: Carsten Wolters, Olaf Steinsträter, Markus Junghöfer, UKM)

This thesis was concerned with the investigation of beamforming in EEG/MEG source analysis. Beamforming is a variance-based technique for source localization, particular emphasis was laid on synthetic aperture magnetometry (SAM) here. The beamforming was applied together with different forward models at three different resolutions (sphere model, realistic geometry with isotropic conductivities, realistic geometry with anisotropic conductivities). In a series of tests the quality of beamforming reconstructions with the different models is tested in order to evaluate the potential need for realistic head models in certain source reconstruction tasks. [pdf-file]

Stefanie Sillekens is now working for ALDI-Sued.

## Long-Time Behaviour of Nonlinear Fokker-Planck Equations

Jan-Frederik Pietschmann, 2008 (Collaboration with: Peter Markowich, DAMTP, Cambridge University)

This thesis deals with the convergence of solution of a system of non-linear
Fokker-Planck equations to a equilibrium or stationary state. To
explore the behaviour of these equations, two techniques are used.
The first one is the so-called entropy dissipation method. Here, an entropy
functional is definied and then used to control the distance between a solution
at time t and the steady state. The second aprroach is to define a
manifold such that the Fokker-Planck equation can be
understood as a gradient flow on this manifold. If it is furthermore possible
to show that the Energy functional is convex, a contraction principle follows.
The proper sense of convexity here is the so-called displacement convexity
which reveals to connection to the Monge-Kantorovich problem of optimal
transport. [pdf-file]

Jan-Frederik Pietschmann is now a PhD-student at DAMTP, Cambridge University, and member of Trinity College.

## Parallel Total Variation Minimization

Jahn Müller, 2008 (Collaboration with: Sergej Gorlatch, Maraike Schellmann, Institut für Informatik, WWU Münster)

This thesis was developing a parallel algorithm for the solution of total variation denoising methods, to be used for high-resolution images. For this sake domain decomposition was applied to a primal-dual discretization of the variational problem. Optimal convergence behaviour was obtained by using a single primal-dual Newton step in each iteration of the domain decomposition. The scheme has been implemented using MPI and extensively tested on an HPC cluster. [pdf-file]

Jahn Müller is now a PhD-student at the Institute for Computational and Applied Mathematics and the European Institute for Molecular Imaging, WWU Münster.

## Valuation and Hedging of Carbon Derivatives

Matthias Tillmann, 2008 (Collaboration with: Magnus Wobben, , Institut für VWL, WWU Münster)

This thesis was concerned with options on carbon certificates, which receive growing interest due to climate preservation policies. The pricing is discussed based on jump-diffusion models with stochastic volatility. For several products explicit formulas can be derived, while American options are solved numerically using finite difference methods and appropriate iterative schemes. Calibration and application to real data is discussed, as well as dynamic hedging including transaction costs. [pdf-file]

Matthias Tillmann is now working for Deutsche Bank AG.

## Early-Exercise Kontrakte in der Stromwirtschaft - Alternative Vermarktung von Erzeugungspotentialen

Arvind Sarin, 2008 (Collaboration with: Magnus Wobben, Institut für VWL, WWU Münster)

This thesis was concerned with the pricing of American options on power markets, hence extending the results of the thesis of Oleg Reichmann. The free boundary problem for the pricing PIDE was analyzed using approximation techniques and semigroup theory. After discretization by finite element methods, an iterative scheme based on a semismooth Newton method was implemented and analyzed also with respect to global convergence. In addition calibration and application to real data is discussed. [pdf-file]

Arvind Sarin is now working for KPMG, Frankfurt.

## Fokker-Planck Equations for Agent-based Models of Financial Markets

Katharina Daniel, 2008

This thesis was concerned with the derivation, analysis, and numerical simulation of multi-dimensional Fokker-Planck equations arising from certain agent-based models of financial markets. The equations are derived by suitable asymptotics from the master equation and lead to a challenging structure with degenerate diffusivity and unbounded potential. Due to these issues standard results cannot be applied directly, and adaptations are carried out here in order to show existence, uniqueness and in order to analyze large-time behavior a relative entropy approach is carried out, and novel Hardy-Sobolev inequalities are derived. Moreover, a novel adjoint approach for computing autocorrelation functions in time is introduced, which allows to realize this task by solving a single Fokker-Planck equation with special initial value. For the robust numerical solution an implicit scheme based on operator splitting and Scharfetter-Gummel discretizations in time is introduced. [pdf-file]

## Bewertung von Stromoptionen in Modellen mit Sprüngen (Valuation of Power Options in Models including Jumps)

Oleg Reichmann, 2008 (Collaboration with: Magnus Wobben, , Institut für VWL, WWU Münster)

This thesis was concerned with the modelling, numerical simulation, and calibration of European options on energy markets, using special characteristics of the latter. Option pricing is investigated using an underlying model based on Ohrnstein-Uhlenbeck and jump processes. The resulting PIDE is analyzed and solved numerically. Calibration to marked data is performed by solving a PIDE-constraint optimization problem. As an alternative option pricing based on FPNIG-processes is discussed, and the results of both approaches are compared.
[pdf-file]

Oleg Reichmann is now a PhD student at ETH Zürich.

## Identification of Cardiac Infarctions from ECG Measurements

Melanie Schröter 2008 (Collaboration with Bjorn Nielsen, SIMULA Research Labs, Oslo)

This thesis was concerned with the identification of infarcted region in the human heart from high-level ECG data. As a model for the electrical activity of the heart the bidomain equations have been used, where infarctions or ischemia can be modeled via unknown distributed parameters or subregions. The identification of these unknowns from ECG data was treated as a nonlinear inverse problem, which was analyzed in detail. An adjoint-based method for Tikhonov functionals was developed, which includes the first sensitivity analysis of the cardiac bidomain model. The Tikhonov regularízation has been adapted to a-priori knowledge about infarcted regions via a nonconvex term. Numerical implementations have been discussed as well as the asymptotic determination of small infarctions via topological derivatives. [pdf-file]

Melanie Schröter is now working for LBS, Münster,

## A Nonlinear Variational Method for Improved Quantification of Myocardial Blood Flow using dynamic H_{2}^{1}^{5}O PET

Martin Benning, 2008 (Collaboration with Klaus Schäfers, UKM & EIMI)

This thesis was concerned with the quantitative imaging of physiological parameters in the heart such as perfusion from PET data collected with radioactive water. The parameters are linked to the image intensity via nonlinear differential equations, hence the reconstruction from PET data becomes a nonlinear inverse problem, which is treated by nonlinear variational methods. The numerical solution is performed by a two-step method alternating EM iterations with the (approximate) solution of nonlinear parameter identification problems. [pdf-file]

Martin Benning is now a PhD-student at the Institute for Computational and Applied Mathematics, WWU Münster.

## Numerical Methods for Equilibrium Problems in General Markets

Anna Weisweiler, 2008 (Collaboration with Nicole Branger, BWL)

This thesis was concerned with equilibria in general markets with heterogeneous agents or firms. The heterogeneity leads to coupled systems of partial differential equations of reaction-diffussion-convection type, usually with diffusion coefficients degenerating at the boundary. Such problems have been analyzed using fixed-point arguments and maximum principles. For the numerical solution robust iterative schemes have been derived and implemented. The approach is applied favourably to two particular market cases. [pdf-file]

Anna Weisweiler is now working for GAD eg, Münster.

## Mathematical Models for Pedrestian Motion

Bärbel Schlake, 2008

This thesis was concerned with an investigation of models for the movement of pedestrians, in particular in evacuation situations, which have been proposed in the last years. A literature review has been given, writing the models into a unified mathematical form. Stability analyses for the most important models have been carried out as well as numerical simulations and some modifications. Moreover, continuum models have been derived in crowded situations, via an approach based on a BBGKY-hierarchy. Due to the crowding molecular chaos does not hold and suitable correlations between pedestrian positions needed to be introduced. [pdf-file]

Bärbel Schlake is now a PhD-student at the Institute for Computational and Applied Mathematics, WWU Münster.

## Numerical simulation of oscillatory zoning

Tanja Mues, 2008 (Collaboration with Andreas Heuer, Institute for Physical Chemistry)

This thesis considered a model for oscillatory zoning, a particular case of crystal growth, based on partial diffferential equations and its extension to multiple spatial dimensions. Numerical algorithms for the simulation of the model have been developed and linear stability analyses have been carried out in various cases. Diffusion regularizations curing potential instabilities appearing in the original formulation of the model have been incorporated and investigated. [pdf-file]

Tanja Mues is now a PhD-Student in the group of Andreas Heuer, Institute for Physical Chemistry, WWU Münster..

## Iterative Total Variation Methods for Nonlinear Inverse Problems

Markus Bachmayr, 2007 (JKU Linz)

This thesis considered an extension of iterative total variation methods previously developed for imaging applications to nonlinear inverse problems. A construction analogous to the linear case led to three different procedures, which can in a similar way be regarded as generalizations of nonlinear iterated Tikhonov regularization, the Levenberg-Marquardt method, and the Landweber iteration, respectively. Like standard total variation regularization, the methods are especially suitable for problems with discontinuous solutions. The convergence of the methods for exact and noisy data was analyzed, the required assumptions verified in the case of a parameter identification problem for an elliptic partial differential equation. Moreover the numerical realization has been discussed. The thesis was awarded the Erwin Wenzl Prize 2007. [pdf-file]

After finishing his civil service, Markus Bachmayr is now a PhD-Student at RWTH Aachen.

## Semiconductor Inverse Dopant Profiling from Transient Measurements

Marie-Therese Wolfram, 2005 (JKU Linz)

This thesis was concerned with the identification of semiconductor doping profiles based on transient current and capacitance measurements. The underlying model considered here were the transient Poisson-drift-diffusion equations. Numerical schemes based on descent algorithms for the Tikhonov functional have been developed and implemented (based on adjoint techniques). Moroever, asymptotic expansions have been used for the solution of the inverse problems in the case of highly-doped devices. [pdf-file]

After finishing her PhD in 2008, Marie-Therese Wolfram spent two years as a Post-Doc researcher at Cambridge University, and is now with the University of Vienna, Austria.