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The focus of this work is to provide and evaluate a novel method for multifield topology-based analysis and visualization. Through this concept, called Pareto sets, one is capable to identify critical regions in a multifield with arbitrary many individual fields. It uses ideas found in graph optimization to find common behavior and areas of divergence between multiple optimization objectives. The connections between the latter areas can be reduced into a graph structure allowing for an abstract visualization of the multifield to support data exploration and understanding.
The research question that is answered in this dissertation is about the general capability and expandability of the Pareto set concept in context of visualization and application. Furthermore, the study of its relations, drawbacks and advantages towards other topological-based approaches. This questions is answered in several steps, including consideration and comparison with related work, a thorough introduction of the Pareto set itself as well as a framework for efficient implementation and an attached discussion regarding limitations of the concept and their implications for run time, suitable data, and possible improvements.
Furthermore, this work considers possible simplification approaches like integrated single-field simplification methods but also using common structures identified through the Pareto set concept to smooth all individual fields at once. These considerations are especially important for real-world scenarios to visualize highly complex data by removing small local structures without destroying information about larger, global trends.
To further emphasize possible improvements and expandability of the Pareto set concept, the thesis studies a variety of different real world applications. For each scenario, this work shows how the definition and visualization of the Pareto set is used and improved for data exploration and analysis based on the scenarios.
In summary, this dissertation provides a complete and sound summary of the Pareto set concept as ground work for future application of multifield data analysis. The possible scenarios include those presented in the application section, but are found in a wide range of research and industrial areas relying on uncertainty analysis, time-varying data, and ensembles of data sets in general.

Novel image processing techniques have been in development for decades, but most
of these techniques are barely used in real world applications. This results in a gap
between image processing research and real-world applications; this thesis aims to
close this gap. In an initial study, the quantification, propagation, and communication
of uncertainty were determined to be key features in gaining acceptance for
new image processing techniques in applications.
This thesis presents a holistic approach based on a novel image processing pipeline,
capable of quantifying, propagating, and communicating image uncertainty. This
work provides an improved image data transformation paradigm, extending image
data using a flexible, high-dimensional uncertainty model. Based on this, a completely
redesigned image processing pipeline is presented. In this pipeline, each
step respects and preserves the underlying image uncertainty, allowing image uncertainty
quantification, image pre-processing, image segmentation, and geometry
extraction. This is communicated by utilizing meaningful visualization methodologies
throughout each computational step.
The presented methods are examined qualitatively by comparing to the Stateof-
the-Art, in addition to user evaluation in different domains. To show the applicability
of the presented approach to real world scenarios, this thesis demonstrates
domain-specific problems and the successful implementation of the presented techniques
in these domains.

Destructive diseases of the lung like lung cancer or fibrosis are still often lethal. Also in case of fibrosis in the liver, the only possible cure is transplantation.
In this thesis, we investigate 3D micro computed synchrotron radiation (SR\( \mu \)CT) images of capillary blood vessels in mouse lungs and livers. The specimen show so-called compensatory lung growth as well as different states of pulmonary and hepatic fibrosis.
During compensatory lung growth, after resecting part of the lung, the remaining part compensates for this loss by extending into the empty space. This process is accompanied by an active vessel growing.
In general, the human lung can not compensate for such a loss. Thus, understanding this process in mice is important to improve treatment options in case of diseases like lung cancer.
In case of fibrosis, the formation of scars within the organ's tissue forces the capillary vessels to grow to ensure blood supply.
Thus, the process of fibrosis as well as compensatory lung growth can be accessed by considering the capillary architecture.
As preparation of 2D microscopic images is faster, easier, and cheaper compared to SR\( \mu \)CT images, they currently form the basis of medical investigation. Yet, characteristics like direction and shape of objects can only properly be analyzed using 3D imaging techniques. Hence, analyzing SR\( \mu \)CT data provides valuable additional information.
For the fibrotic specimen, we apply image analysis methods well-known from material science. We measure the vessel diameter using the granulometry distribution function and describe the inter-vessel distance by the spherical contact distribution. Moreover, we estimate the directional distribution of the capillary structure. All features turn out to be useful to characterize fibrosis based on the deformation of capillary vessels.
It is already known that the most efficient mechanism of vessel growing forms small torus-shaped holes within the capillary structure, so-called intussusceptive pillars. Analyzing their location and number strongly contributes to the characterization of vessel growing. Hence, for all three applications, this is of great interest. This thesis provides the first algorithm to detect intussusceptive pillars in SR\( \mu \)CT images. After segmentation of raw image data, our algorithm works automatically and allows for a quantitative evaluation of a large amount of data.
The analysis of SR\( \mu \)CT data using our pillar algorithm as well as the granulometry, spherical contact distribution, and directional analysis extends the current state-of-the-art in medical studies. Although it is not possible to replace certain 3D features by 2D features without losing information, our results could be used to examine 2D features approximating the 3D findings reasonably well.

In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.

In this thesis we integrate discrete dividends into the stock model, estimate
future outstanding dividend payments and solve different portfolio optimization
problems. Therefore, we discuss three well-known stock models, including
discrete dividend payments and evolve a model, which also takes early
announcement into account.
In order to estimate the future outstanding dividend payments, we develop a
general estimation framework. First, we investigate a model-free, no-arbitrage
methodology, which is based on the put-call parity for European options. Our
approach integrates all available option market data and simultaneously calculates
the market-implied discount curve. We illustrate our method using stocks
of European blue-chip companies and show within a statistical assessment that
the estimate performs well in practice.
As American options are more common, we additionally develop a methodology,
which is based on market prices of American at-the-money options.
This method relies on a linear combination of no-arbitrage bounds of the dividends,
where the corresponding optimal weight is determined via a historical
least squares estimation using realized dividends. We demonstrate our method
using all Dow Jones Industrial Average constituents and provide a robustness
check with respect to the used discount factor. Furthermore, we backtest our
results against the method using European options and against a so called
simple estimate.
In the last part of the thesis we solve the terminal wealth portfolio optimization
problem for a dividend paying stock. In the case of the logarithmic utility
function, we show that the optimal strategy is not a constant anymore but
connected to the Merton strategy. Additionally, we solve a special optimal
consumption problem, where the investor is only allowed to consume dividends.
We show that this problem can be reduced to the before solved terminal wealth
problem.

Embedded reactive systems underpin various safety-critical applications wherein they interact with other systems and the environment with limited or even no human supervision. Therefore, design errors that violate essential system specifications can lead to severe unacceptable damages. For this reason, formal verification of such systems in their physical environment is of high interest. Synchronous programs are typically used to represent embedded reactive systems while hybrid systems serve to model discrete reactive system in a continuous environment. As such, both synchronous programs and hybrid systems play important roles in the model-based design of embedded reactive systems. This thesis develops induction-based techniques for safety property verification of synchronous and hybrid programs. The imperative synchronous language Quartz and its hybrid systems’ extensions are used to sustain the findings.
Deductive techniques for software verification typically use Hoare calculus. In this context, Verification Condition Generation (VCG) is used to apply Hoare calculus rules to a program whose statements are annotated with pre- and postconditions so that the validity of an obtained Verification Condition (VC) implies correctness of a given proof goal. Due to the abstraction of macro steps, Hoare calculus cannot directly generate VCs of synchronous programs unless it handles additional label variables or goto statements. As a first contribution, Floyd’s induction-based approach is employed to generate VCs for synchronous and hybrid programs. Five VCG methods are introduced that use inductive assertions to decompose the overall proof goal. Given the right assertions, the procedure can automatically generate a set of VCs that can then be checked by SMT solvers or automated theorem provers. The methods are proved sound and relatively complete, provided that the underlying assertion language is expressive enough. They can be applied to any program with a state-based semantics.
Property Directed Reachability (PDR) is an efficient method for synchronous hardware circuit verification based on induction rather than fixpoint computation. Crucial steps of the PDR method consist of deciding about the reachability of Counterexamples to Induction (CTIs) and generalizing them to clauses that cover as many unreachable states as possible. The thesis demonstrates that PDR becomes more efficient for imperative synchronous programs when using the distinction between the control- and dataflow. Before calling the PDR method, it is possible to derive additional program control-flow information that can be added to the transition relation such that less CTIs will be generated. Two methods to compute additional control-flow information are presented that differ in how precisely they approximate the reachable control-flow states and, consequently, in their required runtime. After calling the PDR method, the CTI identification work is reduced to its control-flow part and to checking whether the obtained control-flow states are unreachable in the corresponding extended finite state machine of the program. If so, all states of the transition system that refer to the same program locations can be excluded, which significantly increases the performance of PDR.

Grape powdery mildew, Erysiphe necator, is one of the most significant plant pathogens, which affects grape growing regions world-wide. Because of its short generation time and the production of large amounts of conidia throughout the season, E. necator is classified as a moderate to high risk pathogen with respect to the development of fungicide resistance. The number of fungicidal mode of actions available to control powdery mildew is limited and for some of them resistances are already known. Aryl-phenyl-ketones (APKs), represented by metrafenone and pyriofenone, and succinate-dehydrogenase inhibitors (SDHIs), composed of numerous active ingredients, are two important fungicide classes used for the control of E. necator. Over the period 2014 to 2016, the emergence and development of metrafenone and SDHI resistant E. necator isolates in Europe was followed and evaluated. The distribution of resistant isolates was thereby strongly dependent on the European region. Whereas the north-western part is still predominantly sensitive, samples from east European countries showed higher resistance frequencies.
Classical sensitivity tests with obligate biotrophs can be challenging regarding sampling, transport and especially the maintenance of the living strains. Whenever possible, molecular genetic methods are preferred for a more efficient monitoring. Such methods require the knowledge of the resistance mechanisms. The exact molecular target and the resistance mechanism of metrafenone is still unknown. Whole genome sequencing of metrafenone sensitive and resistant wheat powdery mildew isolates, as well as adapted laboratory mutants of Aspergillus nidulans, where performed with the aim to identify proteins potentially linked to the mode of action or which contribute to metrafenone resistance. Based on comparative SNP analysis, four proteins potentially associated with metrafenone resistance were identified, but validation studies could not confirm their role in metrafenone resistance. In contrast to APKs, the mode of action of SDHIs is well understood. Sequencing of the sdh-genes of less sensitive E. necator isolates identified four different target-site mutations, the B-H242R, B-I244V, C-G169D and C-G169S, in sdhB and sdhC, respectively. Based on this information it was possible to develop molecular genetic monitoring methods for the mutations B-H242R and C-G169D. In 2016, the B-H242R was thereby identified as by far the most frequent mutation. Depending on the analysed SDH compound and the sdh-genotype, different sensitivities were observed and revealed a complex cross-resistance pattern.
Growth competition assays without selection pressure, with mixtures of sensitive and resistant E. necator isolates, were performed to determine potential fitness costs associated with fungicide resistance. With the experimental setups used, a clear fitness disadvantage associated with metrafenone resistance was not identified, although a strong variability of fitness was observed among the tested resistant E. necator isolates. For isolates with a reduced sensitivity towards SDHIs, associated fitness costs were dependent on the sdh-genotype analysed. Competition tests with the B-H242R genotypes gave evidence that there are no fitness costs associated with this mutation. In contrast, the C-G169D genotypes were less competitive, indicating a restricted fitness compared to the tested sensitive partners. Competition assays of field isolates, which exhibited several resistances towards different fungicide classes, indicated that there are no fitness costs associated with a multiple resistant phenotype in E. necator. Overall, these results clearly indicate the importance to analyse a representative number of isolates with sensitive and resistant phenotypes.

The screening of metagenomic datasets led to the identification of new phage-derived members of the heme oxygenase and the ferredoxin-dependent bilin reductase enzyme families.
The novel bilin biosynthesis genes were shown to form mini-cassettes on metagenomic scaffolds and further form distinct clusters in phylogenetic analyses (Ledermann et al., 2016). In this project, it was demonstrated that the discovered sequences actually encode for active enzymes. The biochemical characterization of a member of the heme oxygenases (ΦHemO) revealed that it possesses a regiospecificity for the α-methine bridge in the cleavage of the heme macrocycle. The reaction product biliverdin IXα was shown to function as the substrate for the novel ferredoxin-dependent bilin reductases (PcyX reductases), which catalyze its reduction to PEB via the intermediate 15,16-DHBV. While it was demonstrated that ΦPcyX, a phage-derived member of the PcyX reductases, is an active enzyme, it also became clear that the rate of the reaction is highly dependent on the employed redox partner. It turned out that the ferredoxin from the cyanophage P-SSM2 is to date the most suitable redox partner for the reductases of the PcyX group. Furthermore, the solution of the ΦPcyX crystal structure revealed that it adopts an α/β/α-sandwich fold, typical for the FDBR-family. Activity assays and subsequent HPLC analyses with different variants of the ΦPcyX protein demonstrated that, despite their similarity, PcyX and PcyA reductases must act via different reaction mechanisms.
Another part of this project focused on the biochemical characterization of the FDBR KflaHY2 from the streptophyte alga Klebsormidium flaccidum. Experiments with recombinant KflaHY2 showed that it is an active FDBR which produces 3(Z)-PCB as the main reaction product, like it can be found in reductases of the PcyA group. Moreover, it was shown that under the employed assay conditions the reaction of BV to PCB proceeds in two different ways: Both 3(Z)-PΦB and 18¹,18²-DHBV occur as intermediates. Activity assays with the purified intermediates yielded PCB. Hence, both compounds are suitable substrates for KflaHY2.
The results of this work highlight the importance of the biochemical experiments, as catalytic activity cannot solely be predicted by sequence analysis.

In this thesis, we focus on the application of the Heath-Platen (HP) estimator in option
pricing. In particular, we extend the approach of the HP estimator for pricing path dependent
options under the Heston model. The theoretical background of the estimator
was first introduced by Heath and Platen [32]. The HP estimator was originally interpreted
as a control variate technique and an application for European vanilla options was
presented in [32]. For European vanilla options, the HP estimator provided a considerable
amount of variance reduction. Thus, applying the technique for path dependent options
under the Heston model is the main contribution of this thesis.
The first part of the thesis deals with the implementation of the HP estimator for pricing
one-sided knockout barrier options. The main difficulty for the implementation of the HP
estimator is located in the determination of the first hitting time of the barrier. To test the
efficiency of the HP estimator we conduct numerical tests with regard to various aspects.
We provide a comparison among the crude Monte Carlo estimation, the crude control
variate technique and the HP estimator for all types of barrier options. Furthermore, we
present the numerical results for at the money, in the money and out of the money barrier
options. As numerical results imply, the HP estimator performs superior among others
for pricing one-sided knockout barrier options under the Heston model.
Another contribution of this thesis is the application of the HP estimator in pricing bond
options under the Cox-Ingersoll-Ross (CIR) model and the Fong-Vasicek (FV) model. As
suggested in the original paper of Heath and Platen [32], the HP estimator has a wide
range of applicability for derivative pricing. Therefore, transferring the structure of the
HP estimator for pricing bond options is a promising contribution. As the approximating
Vasicek process does not seem to be as good as the deterministic volatility process in the
Heston setting, the performance of the HP estimator in the CIR model is only relatively
good. However, for the FV model the variance reduction provided by the HP estimator is
again considerable.
Finally, the numerical result concerning the weak convergence rate of the HP estimator
for pricing European vanilla options in the Heston model is presented. As supported by
numerical analysis, the HP estimator has weak convergence of order almost 1.

A popular model for the locations of fibres or grains in composite materials
is the inhomogeneous Poisson process in dimension 3. Its local intensity function
may be estimated non-parametrically by local smoothing, e.g. by kernel
estimates. They crucially depend on the choice of bandwidths as tuning parameters
controlling the smoothness of the resulting function estimate. In this
thesis, we propose a fast algorithm for learning suitable global and local bandwidths
from the data. It is well-known, that intensity estimation is closely
related to probability density estimation. As a by-product of our study, we
show that the difference is asymptotically negligible regarding the choice of
good bandwidths, and, hence, we focus on density estimation.
There are quite a number of data-driven bandwidth selection methods for
kernel density estimates. cross-validation is a popular one and frequently proposed
to estimate the optimal bandwidth. However, if the sample size is very
large, it becomes computational expensive. In material science, in particular,
it is very common to have several thousand up to several million points.
Another type of bandwidth selection is a solve-the-equation plug-in approach
which involves replacing the unknown quantities in the asymptotically optimal
bandwidth formula by their estimates.
In this thesis, we develop such an iterative fast plug-in algorithm for estimating
the optimal global and local bandwidth for density and intensity estimation with a focus on 2- and 3-dimensional data. It is based on a detailed
asymptotics of the estimators of the intensity function and of its second
derivatives and integrals of second derivatives which appear in the formulae
for asymptotically optimal bandwidths. These asymptotics are utilised to determine
the exact number of iteration steps and some tuning parameters. For
both global and local case, fewer than 10 iterations suffice. Simulation studies
show that the estimated intensity by local bandwidth can better indicate
the variation of local intensity than that by global bandwidth. Finally, the
algorithm is applied to two real data sets from test bodies of fibre-reinforced
high-performance concrete, clearly showing some inhomogeneity of the fibre
intensity.