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- Fachbereich Mathematik (14) (remove)

The various uses of fiber-reinforced composites, for example in the enclosures of planes, boats and cars, generates the demand for a detailed analysis of these materials. The final goal is to optimize fibrous materials by the means of “virtual material design”. New fibrous materials are virtually created as realizations of a stochastic model and evaluated with physical simulations. In that way, materials can be optimized for specific use cases, without constructing expensive prototypes or performing mechanical experiments. In order to design a practically fabricable material, the stochastic model is first adapted to an existing material and then slightly modified. The virtual reconstruction of the existing material requires a precise knowledge of the geometry of its microstructure. The first part of this thesis describes a fiber quantification method by the means of local measurements of the fiber radius and orientation. The combination of a sparse chord length transform and inertia moments leads to an efficient and precise new algorithm. It outperforms existing approaches with the possibility to treat different fiber radii within one sample, with high precision in continuous space and comparably fast computing time. This local quantification method can be directly applied on gray value images by adapting the directional distance transforms on gray values. In this work, several approaches of this kind are developed and evaluated. Further characterization of the fiber system requires a segmentation of each single fiber. Using basic morphological operators with specific structuring elements, it is possible to derive a probability for each pixel describing if the pixel belongs to a fiber core in a region without overlapping fibers. Tracking high probabilities leads to a partly reconstruction of the fiber cores in non crossing regions. These core parts are then reconnected over critical regions, if they fulfill certain conditions ensuring the affiliation to the same fiber. In the second part of this work, we develop a new stochastic model for dense systems of non overlapping fibers with a controllable level of bending. Existing approaches in the literature have at least one weakness in either achieving high volume fractions, producing non overlapping fibers, or controlling the bending or the orientation distribution. This gap can be bridged by our stochastic model, which operates in two steps. Firstly, a random walk with the multivariate von Mises-Fisher orientation distribution defines bent fibers. Secondly, a force-biased packing approach arranges them in a non overlapping configuration. Furthermore, we provide the estimation of all parameters needed for the fitting of this model to a real microstructure. Finally, we simulate the macroscopic behavior of different microstructures to derive their mechanical and thermal properties. This part is mostly supported by existing software and serves as a summary of physical simulation applied to random fiber systems. The application on a glass fiber reinforced polymer proves the quality of the reconstruction by our stochastic model, as the effective properties match for both the real microstructure and the realizations of the fitted model. This thesis includes all steps to successfully perform virtual material design on various data sets. With novel and efficient algorithms it contributes to the science of analysis and modeling of fiber reinforced materials.

A Multi-Phase Flow Model Incorporated with Population Balance Equation in a Meshfree Framework
(2011)

This study deals with the numerical solution of a meshfree coupled model of Computational Fluid Dynamics (CFD) and Population Balance Equation (PBE) for liquid-liquid extraction columns. In modeling the coupled hydrodynamics and mass transfer in liquid extraction columns one encounters multidimensional population balance equation that could not be fully resolved numerically within a reasonable time necessary for steady state or dynamic simulations. For this reason, there is an obvious need for a new liquid extraction model that captures all the essential physical phenomena and still tractable from computational point of view. This thesis discusses a new model which focuses on discretization of the external (spatial) and internal coordinates such that the computational time is drastically reduced. For the internal coordinates, the concept of the multi-primary particle method; as a special case of the Sectional Quadrature Method of Moments (SQMOM) is used to represent the droplet internal properties. This model is capable of conserving the most important integral properties of the distribution; namely: the total number, solute and volume concentrations and reduces the computational time when compared to the classical finite difference methods, which require many grid points to conserve the desired physical quantities. On the other hand, due to the discrete nature of the dispersed phase, a meshfree Lagrangian particle method is used to discretize the spatial domain (extraction column height) using the Finite Pointset Method (FPM). This method avoids the extremely difficult convective term discretization using the classical finite volume methods, which require a lot of grid points to capture the moving fronts propagating along column height.

The interest of the exploration of new hydrocarbon fields as well as deep geothermal reservoirs is permanently growing. The analysis of seismic data specific for such exploration projects is very complex and requires the deep knowledge in geology, geophysics, petrology, etc from interpreters, as well as the ability of advanced tools that are able to recover some particular properties. There again the existing wavelet techniques have a huge success in signal processing, data compression, noise reduction, etc. They enable to break complicate functions into many simple pieces at different scales and positions that makes detection and interpretation of local events significantly easier.
In this thesis mathematical methods and tools are presented which are applicable to the seismic data postprocessing in regions with non-smooth boundaries. We provide wavelet techniques that relate to the solutions of the Helmholtz equation. As application we are interested in seismic data analysis. A similar idea to construct wavelet functions from the limit and jump relations of the layer potentials was first suggested by Freeden and his Geomathematics Group.
The particular difficulty in such approaches is the formulation of limit and
jump relations for surfaces used in seismic data processing, i.e., non-smooth
surfaces in various topologies (for example, uniform and
quadratic). The essential idea is to replace the concept of parallel surfaces known for a smooth regular surface by certain appropriate substitutes for non-smooth surfaces.
By using the jump and limit relations formulated for regular surfaces, Helmholtz wavelets can be introduced that recursively approximate functions on surfaces with edges and corners. The exceptional point is that the construction of wavelets allows the efficient implementation in form of
a tree algorithm for the fast numerical computation of functions on the boundary.
In order to demonstrate the
applicability of the Helmholtz FWT, we study a seismic image obtained by the reverse time migration which is based on a finite-difference implementation. In fact, regarding the requirements of such migration algorithms in filtering and denoising the wavelet decomposition is successfully applied to this image for the attenuation of low-frequency
artifacts and noise. Essential feature is the space localization property of
Helmholtz wavelets which numerically enables to discuss the velocity field in
pointwise dependence. Moreover, the multiscale analysis leads us to reveal additional geological information from optical features.

In the first part of the thesis we develop the theory of standard bases in free modules over (localized) polynomial rings. Given that linear equations are solvable in the coefficients of the polynomials, we introduce an algorithm to compute standard bases with respect to arbitrary (module) monomial orderings. Moreover, we take special care to principal ideal rings, allowing zero divisors. For these rings we design modified algorithms which are new and much faster than the general ones. These algorithms were motivated by current limitations in formal verification of microelectronic System-on-Chip designs. We show that our novel approach using computational algebra is able to overcome these limitations in important classes of applications coming from industrial challenges.
The second part is based on research in collaboration with Jason Morton, Bernd Sturmfels and Anne Shiu. We devise a general method to describe and compute a certain class of rank tests motivated by statistics. The class of rank tests may loosely be described as being based on computing the number of linear extensions to given partial orders. In order to apply these tests to actual data we developed two algorithms and used our implementations to apply the methodology to gene expression data created at the Stowers Institute for Medical Research. The dataset is concerned with the development of the vertebra. Our rankings proved valuable to the biologists.

The goal of this thesis is to find ways to improve the analysis of hyperspectral Terahertz images. Although it would be desirable to have methods that can be applied on all spectral areas, this is impossible. Depending on the spectroscopic technique, the way the data is acquired differs as well as the characteristics that are to be detected. For these reasons, methods have to be developed or adapted to be especially suitable for the THz range and its applications. Among those are particularly the security sector and the pharmaceutical industry.
Due to the fact that in many applications the volume of spectra to be organized is high, manual data processing is difficult. Especially in hyperspectral imaging, the literature is concerned with various forms of data organization such as feature reduction and classification. In all these methods, the amount of necessary influence of the user should be minimized on the one hand and on the other hand the adaption to the specific application should be maximized.
Therefore, this work aims at automatically segmenting or clustering THz-TDS data. To achieve this, we propose a course of action that makes the methods adaptable to different kinds of measurements and applications. State of the art methods will be analyzed and supplemented where necessary, improvements and new methods will be proposed. This course of action includes preprocessing methods to make the data comparable. Furthermore, feature reduction that represents chemical content in about 20 channels instead of the initial hundreds will be presented. Finally the data will be segmented by efficient hierarchical clustering schemes. Various application examples will be shown.
Further work should include a final classification of the detected segments. It is not discussed here as it strongly depends on specific applications.

This thesis is devoted to constructive module theory of polynomial
graded commutative algebras over a field.
It treats the theory of Groebner bases (GB), standard bases (SB) and syzygies as well as algorithms
and their implementations.
Graded commutative algebras naturally unify exterior and commutative polynomial algebras.
They are graded non-commutative, associative unital algebras over fields and may contain zero-divisors.
In this thesis
we try to make the most use out of _a priori_ knowledge about
their characteristic (super-commutative) structure
in developing direct symbolic methods, algorithms and implementations,
which are intrinsic to graded commutative algebras and practically efficient.
For our symbolic treatment we represent them as polynomial algebras
and redefine the product rule in order to allow super-commutative structures
and, in particular, to allow zero-divisors.
Using this representation we give a nice characterization
of a GB and an algorithm for its computation.
We can also tackle central localizations of graded commutative algebras by allowing commutative variables to be _local_,
generalizing Mora algorithm (in a similar fashion as G.M.Greuel and G.Pfister by allowing local or mixed monomial orderings)
and working with SBs.
In this general setting we prove a generalized Buchberger's criterion,
which shows that syzygies of leading terms play the utmost important role
in SB and syzygy module computations.
Furthermore, we develop a variation of the La Scala-Stillman free resolution algorithm,
which we can formulate particularly close to our implementation.
On the implementation side
we have further developed the Singular non-commutative subsystem Plural
in order to allow polynomial arithmetic
and more involved non-commutative basic Computer Algebra computations (e.g. S-polynomial, GB)
to be easily implementable for specific algebras.
At the moment graded commutative algebra-related algorithms
are implemented in this framework.
Benchmarks show that our new algorithms and implementation are practically efficient.
The developed framework has a lot of applications in various
branches of mathematics and theoretical physics.
They include computation of sheaf cohomology, coordinate-free verification of affine geometry
theorems and computation of cohomology rings of p-groups, which are partially described in this thesis.

Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces
(2011)

We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In particular, we show that every tropical cycle can be degenerated into a sum of torus-invariant cycles. This allows us to tropicalize algebraic cycles of toric varieties over an algebraically closed field with non-Archimedean valuation. We see that the tropicalization map is a homomorphism on cycles and an isomorphism on cycle classes. Furthermore, we can use projective toric varieties to compactify known tropical varieties and study their combinatorics. We do this for the tropical Grassmannian in the Plücker embedding and compactify the tropical parameter space of rational degree d curves in tropical projective space using Chow quotients of the tropical Grassmannian.

Mrázek et al. [14] proposed a unified approach to curve estimation which combines
localization and regularization. In this thesis we will use their approach to study
some asymptotic properties of local smoothers with regularization. In Particular, we
shall discuss the regularized local least squares (RLLS) estimate with correlated errors
(more precisely with stationary time series errors), and then based on this approach
we will discuss the case when the kernel function is dirac function and compare our
smoother with the spline smoother. Finally, we will do some simulation study.

Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
(2011)

Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially Gröbner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.