## Dissertation

### Filtern

#### Erscheinungsjahr

#### Dokumenttyp

- Dissertation (226) (entfernen)

#### Schlagworte

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (226) (entfernen)

Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems.
Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task.
The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain.
The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained.
The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.

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.

Numerical Godeaux surfaces are minimal surfaces of general type with the smallest possible numerical invariants. It is known that the torsion group of a numerical Godeaux surface is cyclic of order \(m\leq 5\). A full classification has been given for the cases \(m=3,4,5\) by the work of Reid and Miyaoka. In each case, the corresponding moduli space is 8-dimensional and irreducible.
There exist explicit examples of numerical Godeaux surfaces for the orders \(m=1,2\), but a complete classification for these surfaces is still missing.
In this thesis we present a construction method for numerical Godeaux surfaces which is based on homological algebra and computer algebra and which arises from an experimental approach by Schreyer. The main idea is to consider the canonical ring \(R(X)\) of a numerical Godeaux surface \(X\) as a module over some graded polynomial ring \(S\). The ring \(S\) is chosen so that \(R(X)\) is finitely generated as an \(S\)-module and a Gorenstein \(S\)-algebra of codimension 3. We prove that the canonical ring of any numerical Godeaux surface, considered as an \(S\)-module, admits a minimal free resolution whose middle map is alternating. Moreover, we show that a partial converse of this statement is true under some additional conditions.
Afterwards we use these results to construct (canonical rings of) numerical Godeaux surfaces. Hereby, we restrict our study to surfaces whose bicanonical system has no fixed component but 4 distinct base points, in the following referred to as marked numerical Godeaux surfaces.
The particular interest of this thesis lies on marked numerical Godeaux surfaces whose torsion group is trivial. For these surfaces we study the fibration of genus 4 over \(\mathbb{P}^1\) induced by the bicanonical system. Catanese and Pignatelli showed that the general fibre is non-hyperelliptic and that the number \(\tilde{h}\) of hyperelliptic fibres is bounded by 3. The two explicit constructions of numerical Godeaux surfaces with a trivial torsion group due to Barlow and Craighero-Gattazzo, respectively, satisfy \(\tilde{h} = 2\).
With the method from this thesis, we construct an 8-dimensional family of numerical Godeaux surfaces with a trivial torsion group and whose general element satisfy \(\tilde{h}=0\).
Furthermore, we establish a criterion for the existence of hyperelliptic fibres in terms of a minimal free resolution of \(R(X)\). Using this criterion, we verify experimentally the
existence of a numerical Godeaux surface with \(\tilde{h}=1\).

In modern algebraic geometry solutions of polynomial equations are studied from a qualitative point of view using highly sophisticated tools such as cohomology, \(D\)-modules and Hodge structures. The latter have been unified in Saito’s far-reaching theory of mixed Hodge modules, that has shown striking applications including vanishing theorems for cohomology. A mixed Hodge module can be seen as a special type of filtered \(D\)-module, which is an algebraic counterpart of a system of linear differential equations. We present the first algorithmic approach to Saito’s theory. To this end, we develop a Gröbner basis theory for a new class of algebras generalizing PBW-algebras.
The category of mixed Hodge modules satisfies Grothendieck’s six-functor formalism. In part these functors rely on an additional natural filtration, the so-called \(V\)-filtration. A key result of this thesis is an algorithm to compute the \(V\)-filtration in the filtered setting. We derive from this algorithm methods for the computation of (extraordinary) direct image functors under open embeddings of complements of pure codimension one subvarieties. As side results we show
how to compute vanishing and nearby cycle functors and a quasi-inverse of Kashiwara’s equivalence for mixed Hodge modules.
Describing these functors in terms of local coordinates and taking local sections, we reduce the corresponding computations to algorithms over certain bifiltered algebras. It leads us to introduce the class of so-called PBW-reduction-algebras, a generalization of the class of PBW-algebras. We establish a comprehensive Gröbner basis framework for this generalization representing the involved filtrations by weight vectors.

Certain brain tumours are very hard to treat with radiotherapy due to their irregular shape caused by the infiltrative nature of the tumour cells. To enhance the estimation of the tumour extent one may use a mathematical model. As the brain structure plays an important role for the cell migration, it has to be included in such a model. This is done via diffusion-MRI data. We set up a multiscale model class accounting among others for integrin-mediated movement of cancer cells in the brain tissue, and the integrin-mediated proliferation. Moreover, we model a novel chemotherapy in combination with standard radiotherapy.
Thereby, we start on the cellular scale in order to describe migration. Then we deduce mean-field equations on the mesoscopic (cell density) scale on which we also incorporate cell proliferation. To reduce the phase space of the mesoscopic equation, we use parabolic scaling and deduce an effective description in the form of a reaction-convection-diffusion equation on the macroscopic spatio-temporal scale. On this scale we perform three dimensional numerical simulations for the tumour cell density, thereby incorporating real diffusion tensor imaging data. To this aim, we present programmes for the data processing taking the raw medical data and processing it to the form to be included in the numerical simulation. Thanks to the reduction of the phase space, the numerical simulations are fast enough to enable application in clinical practice.

Composite materials are used in many modern tools and engineering applications and
consist of two or more materials that are intermixed. Features like inclusions in a matrix
material are often very small compared to the overall structure. Volume elements that
are characteristic for the microstructure can be simulated and their elastic properties are
then used as a homogeneous material on the macroscopic scale.
Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated
trigonometric polynomials on a tensor product grid as ansatz functions.
In this thesis, we generalize their approach to anisotropic lattices and extend it to
anisotropic translation invariant spaces. We discretize the partial differential equation
on these spaces and prove the convergence rate. The speed of convergence depends on
the smoothness of the coefficients and the regularity of the ansatz space. The spaces of
translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and
periodic Box splines, including the constant finite element discretization of Brisard and
Dormieux [1].
For finely resolved images, sampling on a coarser lattice reduces the computational
effort. We introduce mixing rules as the means to transfer fine-grid information to the
smaller lattice.
Finally, we show the effect of the anisotropic pattern, the space of translates, and the
convergence of the method, and mixing rules on two- and three-dimensional examples.
References
[1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites:
A general variational framework”. In: Computational Materials Science 49.3 (2010),
pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009.
[2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response
of nonlinear composites with complex microstructure”. In: Computer Methods in
Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.

Multiphase materials combine properties of several materials, which makes them interesting for high-performing components. This thesis considers a certain set of multiphase materials, namely silicon-carbide (SiC) particle-reinforced aluminium (Al) metal matrix composites and their modelling based on stochastic geometry models.
Stochastic modelling can be used for the generation of virtual material samples: Once we have fitted a model to the material statistics, we can obtain independent three-dimensional “samples” of the material under investigation without the need of any actual imaging. Additionally, by changing the model parameters, we can easily simulate a new material composition.
The materials under investigation have a rather complicated microstructure, as the system of SiC particles has many degrees of freedom: Size, shape, orientation and spatial distribution. Based on FIB-SEM images, that yield three-dimensional image data, we extract the SiC particle structure using methods of image analysis. Then we model the SiC particles by anisotropically rescaled cells of a random Laguerre tessellation that was fitted to the shapes of isotropically rescaled particles. We fit a log-normal distribution for the volume distribution of the SiC particles. Additionally, we propose models for the Al grain structure and the Aluminium-Copper (\({Al}_2{Cu}\)) precipitations occurring on the grain boundaries and on SiC-Al phase boundaries.
Finally, we show how we can estimate the parameters of the volume-distribution based on two-dimensional SEM images. This estimation is applied to two samples with different mean SiC particle diameters and to a random section through the model. The stereological estimations are within acceptable agreement with the parameters estimated from three-dimensional image data
as well as with the parameters of the model.

Using valuation theory we associate to a one-dimensional equidimensional semilocal Cohen-Macaulay ring \(R\) its semigroup of values, and to a fractional ideal of \(R\) we associate its value semigroup ideal. For a class of curve singularities (here called admissible rings) including algebroid curves the semigroups of values, respectively the value semigroup ideals, satisfy combinatorial properties defining good semigroups, respectively good semigroup ideals. Notably, the class of good semigroups strictly contains the class of value semigroups of admissible rings. On good semigroups we establish combinatorial versions of algebraic concepts on admissible rings which are compatible with their prototypes under taking values. Primarily we examine duality and quasihomogeneity.
We give a definition for canonical semigroup ideals of good semigroups which characterizes canonical fractional ideals of an admissible ring in terms of their value semigroup ideals. Moreover, a canonical semigroup ideal induces a duality on the set of good semigroup ideals of a good semigroup. This duality is compatible with the Cohen-Macaulay duality on fractional ideals under taking values.
The properties of the semigroup of values of a quasihomogeneous curve singularity lead to a notion of quasihomogeneity on good semigroups which is compatible with its algebraic prototype. We give a combinatorial criterion which allows to construct from a quasihomogeneous semigroup \(S\) a quasihomogeneous curve singularity having \(S\) as semigroup of values.
As an application we use the semigroup of values to compute endomorphism rings of maximal ideals of algebroid curves. This yields an explicit description of the intermediate rings in an algorithmic normalization of plane central arrangements of smooth curves based on a criterion by Grauert and Remmert. Applying this result to hyperplane arrangements we determine the number of steps needed to compute the normalization of a the arrangement in terms of its Möbius function.

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.

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.

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.

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.

Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.

In this thesis we address two instances of duality in commutative algebra.
In the first part, we consider value semigroups of non irreducible singular algebraic curves
and their fractional ideals. These are submonoids of Z^n closed under minima, with a conductor and which fulfill special compatibility properties on their elements. Subsets of Z^n
fulfilling these three conditions are known in the literature as good semigroups and their ideals, and their class strictly contains the class of value semigroup ideals. We examine
good semigroups both independently and in relation with their algebraic counterpart. In the combinatoric setting, we define the concept of good system of generators, and we
show that minimal good systems of generators are unique. In relation with the algebra side, we give an intrinsic definition of canonical semigroup ideals, which yields a duality
on good semigroup ideals. We prove that this semigroup duality is compatible with the Cohen-Macaulay duality under taking values. Finally, using the duality on good semigroup ideals, we show a symmetry of the Poincaré series of good semigroups with special properties.
In the second part, we treat Macaulay’s inverse system, a one-to-one correspondence
which is a particular case of Matlis duality and an effective method to construct Artinian k-algebras with chosen socle type. Recently, Elias and Rossi gave the structure of the inverse system of positive dimensional Gorenstein k-algebras. We extend their result by establishing a one-to-one correspondence between positive dimensional level k-algebras and certain submodules of the divided power ring. We give several examples to illustrate
our result.

Nonwoven materials are used as filter media which are the key component of automotive filters such as air filters, oil filters, and fuel filters. Today, the advanced engine technologies require innovative filter media with higher performances. A virtual microstructure of the nonwoven filter medium, which has similar filter properties as the existing material, can be used to design new filter media from existing media. Nonwoven materials considered in this thesis prominently feature non-overlapping fibers, curved fibers, fibers with circular cross section, fibers of apparently infinite length, and fiber bundles. To this end, as part of this thesis, we extend the Altendorf-Jeulin individual fiber model to incorporate all the above mentioned features. The resulting novel stochastic 3D fiber model can generate geometries with good visual resemblance of real filter media. Furthermore, pressure drop, which is one of the important physical properties of the filter, simulated numerically on the computed tomography (CT) data of the real nonwoven material agrees well (with a relative error of 8%) with the pressure drop simulated in the generated microstructure realizations from our model.
Generally, filter properties for the CT data and generated microstructure realizations are computed using numerical simulations. Since numerical simulations require extensive system memory and computation time, it is important to find the representative domain size of the generated microstructure for a required filter property. As part of this thesis, simulation and a statistical approach are used to estimate the representative domain size of our microstructure model. Precisely, the representative domain size with respect to the packing density, the pore size distribution, and the pressure drop are considered. It turns out that the statistical approach can be used to estimate the representative domain size for the given property more precisely and using less generated microstructures than the purely simulation based approach.
Among the various properties of fibrous filter media, fiber thickness and orientation are important characteristics which should be considered in design and quality assurance of filter media. Automatic analysis of images from scanning electron microscopy (SEM) is a suitable tool in that context. Yet, the accuracy of such image analysis tools cannot be judged based on images of real filter media since their true fiber thickness and orientation can never be known accurately. A solution is to employ synthetically generated models for evaluation. By combining our 3D fiber system model with simulation of the SEM imaging process, quantitative evaluation of the fiber thickness and orientation measurements becomes feasible. We evaluate the state-of-the-art automatic thickness and orientation estimation method that way.

The thesis studies change points in absolute time for censored survival data with some contributions to the more common analysis of change points with respect to survival time. We first introduce the notions and estimates of survival analysis, in particular the hazard function and censoring mechanisms. Then, we discuss change point models for survival data. In the literature, usually change points with respect to survival time are studied. Typical examples are piecewise constant and piecewise linear hazard functions. For that kind of models, we propose a new algorithm for numerical calculation of maximum likelihood estimates based on a cross entropy approach which in our simulations outperforms the common Nelder-Mead algorithm.
Our original motivation was the study of censored survival data (e.g., after diagnosis of breast cancer) over several decades. We wanted to investigate if the hazard functions differ between various time periods due, e.g., to progress in cancer treatment. This is a change point problem in the spirit of classical change point analysis. Horváth (1998) proposed a suitable change point test based on estimates of the cumulative hazard function. As an alternative, we propose similar tests based on nonparametric estimates of the hazard function. For one class of tests related to kernel probability density estimates, we develop fully the asymptotic theory for the change point tests. For the other class of estimates, which are versions of the Watson-Leadbetter estimate with censoring taken into account and which are related to the Nelson-Aalen estimate, we discuss some steps towards developing the full asymptotic theory. We close by applying the change point tests to simulated and real data, in particular to the breast cancer survival data from the SEER study.

The main theme of this thesis is the interplay between algebraic and tropical intersection
theory, especially in the context of enumerative geometry. We begin by exploiting
well-known results about tropicalizations of subvarieties of algebraic tori to give a
simple proof of Nishinou and Siebert’s correspondence theorem for rational curves
through given points in toric varieties. Afterwards, we extend this correspondence
by additionally allowing intersections with psi-classes. We do this by constructing
a tropicalization map for cycle classes on toroidal embeddings. It maps algebraic
cycle classes to elements of the Chow group of the cone complex of the toroidal
embedding, that is to weighted polyhedral complexes, which are balanced with respect
to an appropriate map to a vector space, modulo a naturally defined equivalence relation.
We then show that tropicalization respects basic intersection-theoretic operations like
intersections with boundary divisors and apply this to the appropriate moduli spaces
to obtain our correspondence theorem.
Trying to apply similar methods in higher genera inevitably confronts us with moduli
spaces which are not toroidal. This motivates the last part of this thesis, where we
construct tropicalizations of cycles on fine logarithmic schemes. The logarithmic point of
view also motivates our interpretation of tropical intersection theory as the dualization
of the intersection theory of Kato fans. This duality gives a new perspective on the
tropicalization map; namely, as the dualization of a pull-back via the characteristic
morphism of a logarithmic scheme.

In this thesis we explicitly solve several portfolio optimization problems in a very realistic setting. The fundamental assumptions on the market setting are motivated by practical experience and the resulting optimal strategies are challenged in numerical simulations.
We consider an investor who wants to maximize expected utility of terminal wealth by trading in a high-dimensional financial market with one riskless asset and several stocks.
The stock returns are driven by a Brownian motion and their drift is modelled by a Gaussian random variable. We consider a partial information setting, where the drift is unknown to the investor and has to be estimated from the observable stock prices in addition to some analyst’s opinion as proposed in [CLMZ06]. The best estimate given these observations is the well known Kalman-Bucy-Filter. We then consider an innovations process to transform the partial information setting into a market with complete information and an observable Gaussian drift process.
The investor is restricted to portfolio strategies satisfying several convex constraints.
These constraints can be due to legal restrictions, due to fund design or due to client's specifications. We cover in particular no-short-selling and no-borrowing constraints.
One popular approach to constrained portfolio optimization is the convex duality approach of Cvitanic and Karatzas. In [CK92] they introduce auxiliary stock markets with shifted market parameters and obtain a dual problem to the original portfolio optimization problem that can be better solvable than the primal problem.
Hence we consider this duality approach and using stochastic control methods we first solve the dual problems in the cases of logarithmic and power utility.
Here we apply a reverse separation approach in order to obtain areas where the corresponding Hamilton-Jacobi-Bellman differential equation can be solved. It turns out that these areas have a straightforward interpretation in terms of the resulting portfolio strategy. The areas differ between active and passive stocks, where active stocks are invested in, while passive stocks are not.
Afterwards we solve the auxiliary market given the optimal dual processes in a more general setting, allowing for various market settings and various dual processes.
We obtain explicit analytical formulas for the optimal portfolio policies and provide an algorithm that determines the correct formula for the optimal strategy in any case.
We also show optimality of our resulting portfolio strategies in different verification theorems.
Subsequently we challenge our theoretical results in a historical and an artificial simulation that are even closer to the real world market than the setting we used to derive our theoretical results. However, we still obtain compelling results indicating that our optimal strategies can outperform any benchmark in a real market in general.

In this dissertation convergence of binomial trees for option pricing is investigated. The focus is on American and European put and call options. For that purpose variations of the binomial tree model are reviewed.
In the first part of the thesis we investigated the convergence behavior of the already known trees from the literature (CRR, RB, Tian and CP) for the European options. The CRR and the RB tree suffer from irregular convergence, so our first aim is to find a way to get the smooth convergence. We first show what causes these oscillations. That will also help us to improve the rate of convergence. As a result we introduce the Tian and the CP tree and we proved that the order of convergence for these trees is \(O \left(\frac{1}{n} \right)\).
Afterwards we introduce the Split tree and explain its properties. We prove the convergence of it and we found an explicit first order error formula. In our setting, the splitting time \(t_{k} = k\Delta t\) is not fixed, i.e. it can be any time between 0 and the maturity time \(T\). This is the main difference compared to the model from the literature. Namely, we show that the good properties of the CRR tree when \(S_{0} = K\) can be preserved even without this condition (which is mainly the case). We achieved the convergence of \(O \left(n^{-\frac{3}{2}} \right)\) and we typically get better results if we split our tree later.

Non–woven materials consist of many thousands of fibres laid down on a conveyor belt
under the influence of a turbulent air stream. To improve industrial processes for the
production of non–woven materials, we develop and explore novel mathematical fibre and
material models.
In Part I of this thesis we improve existing mathematical models describing the fibres on the
belt in the meltspinning process. In contrast to existing models, we include the fibre–fibre
interaction caused by the fibres’ thickness which prevents the intersection of the fibres and,
hence, results in a more accurate mathematical description. We start from a microscopic
characterisation, where each fibre is described by a stochastic functional differential
equation and include the interaction along the whole fibre path, which is described by a
delay term. As many fibres are required for the production of a non–woven material, we
consider the corresponding mean–field equation, which describes the evolution of the fibre
distribution with respect to fibre position and orientation. To analyse the particular case of
large turbulences in the air stream, we develop the diffusion approximation which yields a
distribution describing the fibre position. Considering the convergence to equilibrium on
an analytical level, as well as performing numerical experiments, gives an insight into the
influence of the novel interaction term in the equations.
In Part II of this thesis we model the industrial airlay process, which is a production method
whereby many short fibres build a three–dimensional non–woven material. We focus on
the development of a material model based on original fibre properties, machine data and
micro computer tomography. A possible linking of these models to other simulation tools,
for example virtual tensile tests, is discussed.
The models and methods presented in this thesis promise to further the field in mathematical
modelling and computational simulation of non–woven materials.

We introduce and investigate a product pricing model in social networks where the value a possible buyer assigns to a product is influenced by the previous buyers. The selling proceeds in discrete, synchronous rounds for some set price and the individual values are additively altered. Whereas computing the revenue for a given price can be done in polynomial time, we show that the basic problem PPAI, i.e., is there a price generating a requested revenue, is weakly NP-complete. With algorithm Frag we provide a pseudo-polynomial time algorithm checking the range of prices in intervals of common buying behavior we call fragments. In some special cases, e.g., solely positive influences, graphs with bounded in-degree, or graphs with bounded path length, the amount of fragments is polynomial. Since the run-time of Frag is polynomial in the amount of fragments, the algorithm itself is polynomial for these special cases. For graphs with positive influence we show that every buyer does also buy for lower prices, a property that is not inherent for arbitrary graphs. Algorithm FixHighest improves the run-time on these graphs by using the above property.
Furthermore, we introduce variations on this basic model. The version of delaying the propagation of influences and the awareness of the product can be implemented in our basic model by substituting nodes and arcs with simple gadgets. In the chapter on Dynamic Product Pricing we allow price changes, thereby raising the complexity even for graphs with solely positive or negative influences. Concerning Perishable Product Pricing, i.e., the selling of products that are usable for some time and can be rebought afterward, the principal problem is computing the revenue that a given price can generate in some time horizon. In general, the problem is #P-hard and algorithm Break runs in pseudo-polynomial time. For polynomially computable revenue, we investigate once more the complexity to find the best price.
We conclude the thesis with short results in topics of Cooperative Pricing, Initial Value as Parameter, Two Product Pricing, and Bounded Additive Influence.

This thesis brings together convex analysis and hyperspectral image processing.
Convex analysis is the study of convex functions and their properties.
Convex functions are important because they admit minimization by efficient algorithms
and the solution of many optimization problems can be formulated as
minimization of a convex objective function, extending much beyond
the classical image restoration problems of denoising, deblurring and inpainting.
\(\hspace{1mm}\)
At the heart of convex analysis is the duality mapping induced within the
class of convex functions by the Fenchel transform.
In the last decades efficient optimization algorithms have been developed based
on the Fenchel transform and the concept of infimal convolution.
\(\hspace{1mm}\)
The infimal convolution is of similar importance in convex analysis as the
convolution in classical analysis. In particular, the infimal convolution with
scaled parabolas gives rise to the one parameter family of Moreau-Yosida envelopes,
which approximate a given function from below while preserving its minimum
value and minimizers.
The closely related proximal mapping replaces the gradient step
in a recently developed class of efficient first-order iterative minimization algorithms
for non-differentiable functions. For a finite convex function,
the proximal mapping coincides with a gradient step of its Moreau-Yosida envelope.
Efficient algorithms are needed in hyperspectral image processing,
where several hundred intensity values measured in each spatial point
give rise to large data volumes.
\(\hspace{1mm}\)
In the \(\textbf{first part}\) of this thesis, we are concerned with
models and algorithms for hyperspectral unmixing.
As part of this thesis a hyperspectral imaging system was taken into operation
at the Fraunhofer ITWM Kaiserslautern to evaluate the developed algorithms on real data.
Motivated by missing-pixel defects common in current hyperspectral imaging systems,
we propose a
total variation regularized unmixing model for incomplete and noisy data
for the case when pure spectra are given.
We minimize the proposed model by a primal-dual algorithm based on the
proximum mapping and the Fenchel transform.
To solve the unmixing problem when only a library of pure spectra is provided,
we study a modification which includes a sparsity regularizer into model.
\(\hspace{1mm}\)
We end the first part with the convergence analysis for a multiplicative
algorithm derived by optimization transfer.
The proposed algorithm extends well-known multiplicative update rules
for minimizing the Kullback-Leibler divergence,
to solve a hyperspectral unmixing model in the case
when no prior knowledge of pure spectra is given.
\(\hspace{1mm}\)
In the \(\textbf{second part}\) of this thesis, we study the properties of Moreau-Yosida envelopes,
first for functions defined on Hadamard manifolds, which are (possibly) infinite-dimensional
Riemannian manifolds with negative curvature,
and then for functions defined on Hadamard spaces.
\(\hspace{1mm}\)
In particular we extend to infinite-dimensional Riemannian manifolds an expression
for the gradient of the Moreau-Yosida envelope in terms of the proximal mapping.
With the help of this expression we show that a sequence of functions
converges to a given limit function in the sense of Mosco
if the corresponding Moreau-Yosida envelopes converge pointwise at all scales.
\(\hspace{1mm}\)
Finally we extend this result to the more general setting of Hadamard spaces.
As the reverse implication is already known, this unites two definitions of Mosco convergence
on Hadamard spaces, which have both been used in the literature,
and whose equivalence has not yet been known.

In this thesis, we consider a problem from modular representation theory of finite groups. Lluís Puig asked the question whether the order of the defect groups of a block \( B \) of the group algebra of a given finite group \( G \) can always be bounded in terms of the order of the vertices of an arbitrary simple module lying in \( B \).
In characteristic \( 2 \), there are examples showing that this is not possible in general, whereas in odd characteristic, no such examples are known. For instance, it is known that the answer to Puig's question is positive in case that \( G \) is a symmetric group, by work of Danz, Külshammer, and Puig.
Motivated by this, we study the cases where \( G \) is a finite classical group in non-defining characteristic or one of the finite groups \( G_2(q) \) or \( ³D_4(q) \) of Lie type, again in non-defining characteristic. Here, we generalize Puig's original question by replacing the vertices occurring in his question by arbitrary self-centralizing subgroups of the defect groups. We derive positive and negative answers to this generalized question.
\[\]
In addition to that, we determine the vertices of the unipotent simple \( GL_2(q) \)-module labeled by the partition \( (1,1) \) in characteristic \( 2 \). This is done using a method known as Brauer construction.

Abstract
The main theme of this thesis is about Graph Coloring Applications and Defining Sets in Graph Theory.
As in the case of block designs, finding defining sets seems to be difficult problem, and there is not a general conclusion. Hence we confine us here to some special types of graphs like bipartite graphs, complete graphs, etc.
In this work, four new concepts of defining sets are introduced:
• Defining sets for perfect (maximum) matchings
• Defining sets for independent sets
• Defining sets for edge colorings
• Defining set for maximal (maximum) clique
Furthermore, some algorithms to find and construct the defining sets are introduced. A review on some known kinds of defining sets in graph theory is also incorporated, in chapter 2 the basic definitions and some relevant notations used in this work are introduced.
chapter 3 discusses the maximum and perfect matchings and a new concept for a defining set for perfect matching.
Different kinds of graph colorings and their applications are the subject of chapter 4.
Chapter 5 deals with defining sets in graph coloring. New results are discussed along with already existing research results, an algorithm is introduced, which enables to determine a defining set of a graph coloring.
In chapter 6, cliques are discussed. An algorithm for the determination of cliques using their defining sets. Several examples are included.

We discuss the portfolio selection problem of an investor/portfolio manager in an arbitrage-free financial market where a money market account, coupon bonds and a stock are traded continuously. We allow for stochastic interest rates and in particular consider one and two-factor Vasicek models for the instantaneous
short rates. In both cases we consider a complete and an incomplete market setting by adding a suitable number of bonds.
The goal of an investor is to find a portfolio which maximizes expected utility
from terminal wealth under budget and present expected short-fall (PESF) risk
constraints. We analyze this portfolio optimization problem in both complete and
incomplete financial markets in three different cases: (a) when the PESF risk is
minimum, (b) when the PESF risk is between minimum and maximum and (c) without risk constraints. (a) corresponds to the portfolio insurer problem, in (b) the risk constraint is binding, i.e., it is satisfied with equality, and (c) corresponds
to the unconstrained Merton investment.
In all cases we find the optimal terminal wealth and portfolio process using the
martingale method and Malliavin calculus respectively. In particular we solve in the incomplete market settings the dual problem explicitly. We compare the
optimal terminal wealth in the cases mentioned using numerical examples. Without
risk constraints, we further compare the investment strategies for complete
and incomplete market numerically.

In change-point analysis the point of interest is to decide if the observations follow one model
or if there is at least one time-point, where the model has changed. This results in two sub-
fields, the testing of a change and the estimation of the time of change. This thesis considers
both parts but with the restriction of testing and estimating for at most one change-point.
A well known example is based on independent observations having one change in the mean.
Based on the likelihood ratio test a test statistic with an asymptotic Gumbel distribution was
derived for this model. As it is a well-known fact that the corresponding convergence rate is
very slow, modifications of the test using a weight function were considered. Those tests have
a better performance. We focus on this class of test statistics.
The first part gives a detailed introduction to the techniques for analysing test statistics and
estimators. Therefore we consider the multivariate mean change model and focus on the effects
of the weight function. In the case of change-point estimators we can distinguish between
the assumption of a fixed size of change (fixed alternative) and the assumption that the size
of the change is converging to 0 (local alternative). Especially, the fixed case in rarely analysed
in the literature. We show how to come from the proof for the fixed alternative to the
proof of the local alternative. Finally, we give a simulation study for heavy tailed multivariate
observations.
The main part of this thesis focuses on two points. First, analysing test statistics and, secondly,
analysing the corresponding change-point estimators. In both cases, we first consider a
change in the mean for independent observations but relaxing the moment condition. Based on
a robust estimator for the mean, we derive a new type of change-point test having a randomized
weight function. Secondly, we analyse non-linear autoregressive models with unknown
regression function. Based on neural networks, test statistics and estimators are derived for
correctly specified as well as for misspecified situations. This part extends the literature as
we analyse test statistics and estimators not only based on the sample residuals. In both
sections, the section on tests and the one on the change-point estimator, we end with giving
regularity conditions on the model as well as the parameter estimator.
Finally, a simulation study for the case of the neural network based test and estimator is
given. We discuss the behaviour under correct and mis-specification and apply the neural
network based test and estimator on two data sets.

A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process.
Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment.
This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models.
Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies.
The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field.
We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently.
The model is also applied to simulated and real world data to show the strengths of our approach.

By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1.

Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field.
In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\).
To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the
modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\).
On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation.
At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.

This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.

Functional data analysis is a branch of statistics that deals with observations \(X_1,..., X_n\) which are curves. We are interested in particular in time series of dependent curves and, specifically, consider the functional autoregressive process of order one (FAR(1)), which is defined as \(X_{n+1}=\Psi(X_{n})+\epsilon_{n+1}\) with independent innovations \(\epsilon_t\). Estimates \(\hat{\Psi}\) for the autoregressive operator \(\Psi\) have been investigated a lot during the last two decades, and their asymptotic properties are well understood. Particularly difficult and different from scalar- or vector-valued autoregressions are the weak convergence properties which also form the basis of the bootstrap theory.
Although the asymptotics for \(\hat{\Psi}{(X_{n})}\) are still tractable, they are only useful for large enough samples. In applications, however, frequently only small samples of data are available such that an alternative method for approximating the distribution of \(\hat{\Psi}{(X_{n})}\) is welcome. As a motivation, we discuss a real-data example where we investigate a changepoint detection problem for a stimulus response dataset obtained from the animal physiology group at the Technical University of Kaiserslautern.
To get an alternative for asymptotic approximations, we employ the naive or residual-based bootstrap procedure. In this thesis, we prove theoretically and show via simulations that the bootstrap provides asymptotically valid and practically useful approximations of the distributions of certain functions of the data. Such results may be used to calculate approximate confidence bands or critical bounds for tests.

Since the early days of representation theory of finite groups in the 19th century, it was known that complex linear representations of finite groups live over number fields, that is, over finite extensions of the field of rational numbers.
While the related question of integrality of representations was answered negatively by the work of Cliff, Ritter and Weiss as well as by Serre and Feit, it was not known how to decide integrality of a given representation.
In this thesis we show that there exists an algorithm that given a representation of a finite group over a number field decides whether this representation can be made integral.
Moreover, we provide theoretical and numerical evidence for a conjecture, which predicts the existence of splitting fields of irreducible characters with integrality properties.
In the first part, we describe two algorithms for the pseudo-Hermite normal form, which is crucial when handling modules over ring of integers.
Using a newly developed computational model for ideal and element arithmetic in number fields, we show that our pseudo-Hermite normal form algorithms have polynomial running time.
Furthermore, we address a range of algorithmic questions related to orders and lattices over Dedekind domains, including computation of genera, testing local isomorphism, computation of various homomorphism rings and computation of Solomon zeta functions.
In the second part we turn to the integrality of representations of finite groups and show that an important ingredient is a thorough understanding of the reduction of lattices at almost all prime ideals.
By employing class field theory and tools from representation theory we solve this problem and eventually describe an algorithm for testing integrality.
After running the algorithm on a large set of examples we are led to a conjecture on the existence of integral and nonintegral splitting fields of characters.
By extending techniques of Serre we prove the conjecture for characters with rational character field and Schur index two.

The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\]
In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.

Inflation modeling is a very important tool for conducting an efficient monetary policy. This doctoral thesis reviewed inflation models, in particular the Phillips curve models of inflation dynamics. We focused on a well known and widely used model, the so-called three equation new Keynesian model which is a system of equations consisting of a new Keynesian Phillips curve (NKPC), an investment and saving (IS) curve and an interest rate rule.
We gave a detailed derivation of these equations. The interest rate rule used in this model is normally determined by using a Lagrangian method to solve an optimal control problem constrained by a standard discrete time NKPC which describes the inflation dynamics and an IS curve that represents the output gaps dynamics. In contrast to the real world, this method assumes that the policy makers intervene continuously. This means that the costs resulting from the change in the interest rates are ignored. We showed also that there are approximation errors made, when one log-linearizes non linear equations, by doing the derivation of the standard discrete time NKPC.
We agreed with other researchers as mentioned in this thesis, that errors which result from ignoring such log-linear approximation errors and the costs of altering interest rates by determining interest rate rule, can lead to a suboptimal interest rate rule and hence to non-optimal paths of output gaps and inflation rate.
To overcome such a problem, we proposed a stochastic optimal impulse control method. We formulated the problem as a stochastic optimal impulse control problem by considering the costs of change in interest rates and the approximation error terms. In order to formulate this problem, we first transform the standard discrete time NKPC and the IS curve into their high-frequency versions and hence into their continuous time versions where error terms are described by a zero mean Gaussian white noise with a finite and constant variance. After formulating this problem, we use the quasi-variational inequality approach to solve analytically a special case of the central bank problem, where an inflation rate is supposed to be on target and a central bank has to optimally control output gap dynamics. This method gives an optimal control band in which output gap process has to be maintained and an optimal control strategy, which includes the optimal size of intervention and optimal intervention time, that can be used to keep the process into the optimal control band.
Finally, using a numerical example, we examined the impact of some model parameters on optimal control strategy. The results show that an increase in the output gap volatility as well as in the fixed and proportional costs of the change in interest rate lead to an increase in the width of the optimal control band. In this case, the optimal intervention requires the central bank to wait longer before undertaking another control action.

In this thesis, mathematical research questions related to recursive utility and stochastic differential utility (SDU) are explored.
First, a class of backward equations under nonlinear expectations is investigated: Existence and uniqueness of solutions are established, and the issues of stability and discrete-time approximation are addressed. It is then shown that backward equations of this class naturally appear as a continuous-time limit in the context of recursive utility with nonlinear expectations.
Then, the Epstein-Zin parametrization of SDU is studied. The focus is on specifications with both relative risk aversion and elasitcity of intertemporal substitution greater that one. A concave utility functional is constructed and a utility gradient inequality is established.
Finally, consumption-portfolio problems with recursive preferences and unspanned risk are investigated. The investor's optimal strategies are characterized by a specific semilinear partial differential equation. The solution of this equation is constructed by a fixed point argument, and a corresponding efficient and accurate method to calculate optimal strategies numerically is given.

This thesis deals with risk measures based on utility functions and time consistency of dynamic risk measures. It is therefore aimed at readers interested in both, the theory of static and dynamic financial risk measures in the sense of Artzner, Delbaen, Eber and Heath [7], [8] and the theory of preferences in the tradition of von Neumann and Morgenstern [134].
A main contribution of this thesis is the introduction of optimal expected utility (OEU) risk measures as a new class of utility-based risk measures. We introduce OEU, investigate its main properties, and its applicability to risk measurement and put it in perspective to alternative risk measures and notions of certainty equivalents. To the best of our knowledge, OEU is the only existing utility-based risk measure that is (non-trivial and) coherent if the utility function u has constant relative risk aversion. We present several different risk measures that can be derived with special choices of u and illustrate that OEU reacts in a more sensitive way to slight changes of the probability of a financial loss than value at risk (V@R) and average value at risk.
Further, we propose implied risk aversion as a coherent rating methodology for retail structured products (RSPs). Implied risk aversion is based on optimal expected utility risk measures and, in contrast to standard V@R-based ratings, takes into account both the upside potential and the downside risks of such products. In addition, implied risk aversion is easily interpreted in terms of an individual investor's risk aversion: A product is attractive (unattractive) for an investor if its implied risk aversion is higher (lower) than his individual risk aversion. We illustrate this approach in a case study with more than 15,000 warrants on DAX ® and find that implied risk aversion is able to identify favorable products; in particular, implied risk aversion is not necessarily increasing with respect to the strikes of call warrants.
Another main focus of this thesis is on consistency of dynamic risk measures. To this end, we study risk measures on the space of distributions, discuss concavity on the level of distributions and slightly generalize Weber's [137] findings on the relation of time consistent dynamic risk measures to static risk measures to the case of dynamic risk measures with time-dependent parameters. Finally, this thesis investigates how recursively composed dynamic risk measures in discrete time, which are time consistent by construction, can be related to corresponding dynamic risk measures in continuous time. We present different approaches to establish this link and outline the theoretical basis and the practical benefits of this relation. The thesis concludes with a numerical implementation of this theory.

We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.

Advantage of Filtering for Portfolio Optimization in Financial Markets with Partial Information
(2016)

In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
for boundedness.
All results are implemented and illustrated with the corresponding numerical
findings.

In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use
isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite
general class of functions for representing optimal shapes is available. For the
gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.

The central topic of this thesis is Alperin's weight conjecture, a problem concerning the representation theory of finite groups.
This conjecture, which was first proposed by J. L. Alperin in 1986, asserts that for any finite group the number of its irreducible Brauer characters coincides with the number of conjugacy classes of its weights. The blockwise version of Alperin's conjecture partitions this problem into a question concerning the number of irreducible Brauer characters and weights belonging to the blocks of finite groups.
A proof for this conjecture has not (yet) been found. However, the problem has been reduced to a question on non-abelian finite (quasi-) simple groups in the sense that there is a set of conditions, the so-called inductive blockwise Alperin weight condition, whose verification for all non-abelian finite simple groups implies the blockwise Alperin weight conjecture. Now the objective is to prove this condition for all non-abelian finite simple groups, all of which are known via the classification of finite simple groups.
In this thesis we establish the inductive blockwise Alperin weight condition for three infinite series of finite groups of Lie type: the special linear groups \(SL_3(q)\) in the case \(q>2\) and \(q \not\equiv 1 \bmod 3\), the Chevalley groups \(G_2(q)\) for \(q \geqslant 5\), and Steinberg's triality groups \(^3D_4(q)\).

In this thesis, we investigate several upcoming issues occurring in the context of conceiving and building a decision support system. We elaborate new algorithms for computing representative systems with special quality guarantees, provide concepts for supporting the decision makers after a representative system was computed, and consider a methodology of combining two optimization problems.
We review the original Box-Algorithm for two objectives by Hamacher et al. (2007) and discuss several extensions regarding coverage, uniformity, the enumeration of the whole nondominated set, and necessary modifications if the underlying scalarization problem cannot be solved to optimality. In a next step, the original Box-Algorithm is extended to the case of three objective functions to compute a representative system with desired coverage error. Besides the investigation of several theoretical properties, we prove the correctness of the algorithm, derive a bound on the number of iterations needed by the algorithm to meet the desired coverage error, and propose some ideas for possible extensions.
Furthermore, we investigate the problem of selecting a subset with desired cardinality from the computed representative system, the Hypervolume Subset Selection Problem (HSSP). We provide two new formulations for the bicriteria HSSP, a linear programming formulation and a \(k\)-link shortest path formulation. For the latter formulation, we propose an algorithm for which we obtain the currently best known complexity bound for solving the bicriteria HSSP. For the tricriteria HSSP, we propose an integer programming formulation with a corresponding branch-and-bound scheme.
Moreover, we address the issue of how to present the whole set of computed representative points to the decision makers. Based on common illustration methods, we elaborate an algorithm guiding the decision makers in choosing their preferred solution.
Finally, we step back and look from a meta-level on the issue of how to combine two given optimization problems and how the resulting combinations can be related to each other. We come up with several different combined formulations and give some ideas for the practical approach.

The overall goal of the work is to simulate rarefied flows inside geometries with moving boundaries. The behavior of a rarefied flow is characterized through the Knudsen number \(Kn\), which can be very small (\(Kn < 0.01\) continuum flow) or larger (\(Kn > 1\) molecular flow). The transition region (\(0.01 < Kn < 1\)) is referred to as the transition flow regime.
Continuum flows are mainly simulated by using commercial CFD methods, which are used to solve the Euler equations. In the case of molecular flows one uses statistical methods, such as the Direct Simulation Monte Carlo (DSMC) method. In the transition region Euler equations are not adequate to model gas flows. Because of the rapid increase of particle collisions the DSMC method tends to fail, as well
Therefore, we develop a deterministic method, which is suitable to simulate problems of rarefied gases for any Knudsen number and is appropriate to simulate flows inside geometries with moving boundaries. Thus, the method we use is the Finite Pointset Method (FPM), which is a mesh-free numerical method developed at the ITWM Kaiserslautern and is mainly used to solve fluid dynamical problems.
More precisely, we develop a method in the FPM framework to solve the BGK model equation, which is a simplification of the Boltzmann equation. This equation is mainly used to describe rarefied flows.
The FPM based method is implemented for one and two dimensional physical and velocity space and different ranges of the Knudsen number. Numerical examples are shown for problems with moving boundaries. It is seen, that our method is superior to regular grid methods with respect to the implementation of boundary conditions. Furthermore, our results are comparable to reference solutions gained through CFD- and DSMC methods, respectevly.

In this dissertation, we discuss how to price American-style options. Our aim is to study and improve the regression-based Monte Carlo methods. In order to have good benchmarks to compare with them, we also study the tree methods.
In the second chapter, we investigate the tree methods specifically. We do research firstly within the Black-Scholes model and then within the Heston model. In the Black-Scholes model, based on Müller's work, we illustrate how to price one dimensional and multidimensional American options, American Asian options, American lookback options, American barrier options and so on. In the Heston model, based on Sayer's research, we implement his algorithm to price one dimensional American options. In this way, we have good benchmarks of various American-style options and put them all in the appendix.
In the third chapter, we focus on the regression-based Monte Carlo methods theoretically and numerically. Firstly, we introduce two variations, the so called "Tsitsiklis-Roy method" and the "Longstaff-Schwartz method". Secondly, we illustrate the approximation of American option by its Bermudan counterpart. Thirdly we explain the source of low bias and high bias. Fourthly we compare these two methods using in-the-money paths and all paths. Fifthly, we examine the effect using different number and form of basis functions. Finally, we study the Andersen-Broadie method and present the lower and upper bounds.
In the fourth chapter, we study two machine learning techniques to improve the regression part of the Monte Carlo methods: Gaussian kernel method and kernel-based support vector machine. In order to choose a proper smooth parameter, we compare fixed bandwidth, global optimum and suboptimum from a finite set. We also point out that scaling the training data to [0,1] can avoid numerical difficulty. When out-of-sample paths of stock prices are simulated, the kernel method is robust and even performs better in several cases than the Tsitsiklis-Roy method and the Longstaff-Schwartz method. The support vector machine can keep on improving the kernel method and needs less representations of old stock prices during prediction of option continuation value for a new stock price.
In the fifth chapter, we switch to the hardware (FGPA) implementation of the Longstaff-Schwartz method and propose novel reversion formulas for the stock price and volatility within the Black-Scholes and Heston models. The test for this formula within the Black-Scholes model shows that the storage of data is reduced and also the corresponding energy consumption.

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie.
Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der \(p\)-adischen Zahlen \(\mathbb{Q}_p\), dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.\[\phantom{newline}\]
In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist.
Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.\[\phantom{newline}\]
Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit \(p\)-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake.

In some processes for spinning synthetic fibers the filaments are exposed to highly turbulent air flows to achieve a high degree of stretching (elongation). The quality of the resulting filaments, namely thickness and uniformity, is thus determined essentially by the aerodynamic force coming from the turbulent flow. Up to now, there is a gap between the elongation measured in experiments and the elongation obtained by numerical simulations available in the literature.
The main focus of this thesis is the development of an efficient and sufficiently accurate simulation algorithm for the velocity of a turbulent air flow and the application in turbulent spinning processes.
In stochastic turbulence models the velocity is described by an \(\mathbb{R}^3\)-valued random field. Based on an appropriate description of the random field by Marheineke, we have developed an algorithm that fulfills our requirements of efficiency and accuracy. Applying a resulting stochastic aerodynamic drag force on the fibers then allows the simulation of the fiber dynamics modeled by a random partial differential algebraic equation system as well as a quantization of the elongation in a simplified random ordinary differential equation model for turbulent spinning. The numerical results are very promising: whereas the numerical results available in the literature can only predict elongations up to order \(10^4\) we get an order of \(10^5\), which is closer to the elongations of order \(10^6\) measured in experiments.

Motivated by the results of infinite dimensional Gaussian analysis and especially white noise analysis, we construct a Mittag-Leffler analysis. This is an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. Our results indicate that the Wick ordered polynomials, which play a key role in Gaussian analysis, cannot be generalized to this non-Gaussian case. We provide evidence that a system of biorthogonal polynomials, called generalized Appell system, is applicable to the Mittag-Leffler measures, instead of using Wick ordered polynomials. With the help of an Appell system, we introduce a test function and a distribution space. Furthermore we give characterizations of the distribution space and we characterize the weak integrable functions and the convergent sequences within the distribution space. We construct Donsker's delta in a non-Gaussian setting as an application.
In the second part, we develop a grey noise analysis. This is a special application of the Mittag-Leffler analysis. In this framework, we introduce generalized grey Brownian motion and prove differentiability in a distributional sense and the existence of generalized grey Brownian motion local times. Grey noise analysis is then applied to the time-fractional heat equation and the time-fractional Schrödinger equation. We prove a generalization of the fractional Feynman-Kac formula for distributional initial values. In this way, we find a Green's function for the time-fractional heat equation which coincides with the solutions given in the literature.

The Wilkie model is a stochastic asset model, developed by A.D. Wilkie in 1984 with a purpose to explore the behaviour of investment factors of insurers within the United Kingdom. Even so, there is still no analysis that studies the Wilkie model in a portfolio optimization framework thus far. Originally, the Wilkie model is considering a discrete-time horizon and we apply the concept of Wilkie model to develop a suitable ARIMA model for Malaysian data by using Box-Jenkins methodology. We obtained the estimated parameters for each sub model within the Wilkie model that suits the case of Malaysia, and permits us to analyse the result based on statistics and economics view. We then tend to review the continuous time case which was initially introduced by Terence Chan in 1998. The continuous-time Wilkie model inspired is then being employed to develop the wealth equation of a portfolio that consists of a bond and a stock. We are interested in building portfolios based on three well-known trading strategies, a self-financing strategy, a constant growth optimal strategy as well as a buy-and-hold strategy. In dealing with the portfolio optimization problems, we use the stochastic control technique consisting of the maximization problem itself, the Hamilton-Jacobi-equation, the solution to the Hamilton-Jacobi-equation and finally the verification theorem. In finding the optimal portfolio, we obtained the specific solution of the Hamilton-Jacobi-equation and proved the solution via the verification theorem. For a simple buy-and-hold strategy, we use the mean-variance analysis to solve the portfolio optimization problem.

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term \(\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and then solve one of the regularized minimization problems \begin{align*}
&{}(P_{1,\tau}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ \Phi(x) \;{\rm s.t.}\; \Psi(x) \leq \tau \big\} \\ &{}(P_{2,\lambda}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ \Phi(x) + \lambda \Psi(x) \}, \; \lambda > 0 \end{align*} with some function \(\Psi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets \({\rm SOL}(P_{1,\tau})\) and
\({\rm SOL}(P_{2,\lambda})\) of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals \((0,c)\) and \((0,d)\) such that the setvalued curves \begin{align*}
\tau \mapsto {}& {\rm SOL}(P_{1,\tau}), \; \tau \in (0,c) \\ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), \; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change \(g: (0,c) \rightarrow (0,d)\). Moreover we show that the solver sets are changing all the time while \(\tau\) runs from \(0\) to \(c\) and \(\lambda\) runs from \(d\) to \(0\).
In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity.
Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.

Lithium-ion batteries are broadly used nowadays in all kinds of portable electronics, such as laptops, cell phones, tablets, e-book readers, digital cameras, etc. They are preferred to other types of rechargeable batteries due to their superior characteristics, such as light weight and high energy density, no memory effect, and a big number of charge/discharge cycles. The high demand and applicability of Li-ion batteries naturally give rise to the unceasing necessity of developing better batteries in terms of performance and lifetime. The aim of the mathematical modelling of Li-ion batteries is to help engineers test different battery configurations and electrode materials faster and cheaper. Lithium-ion batteries are multiscale systems. A typical Li-ion battery consists of multiple connected electrochemical battery cells. Each cell has two electrodes - anode and cathode, as well as a separator between them that prevents a short circuit.
Both electrodes have porous structure composed of two phases - solid and electrolyte. We call macroscale the lengthscale of the whole electrode and microscale - the lengthscale at which we can distinguish the complex porous structure of the electrodes. We start from a Li-ion battery model derived on the microscale. The model is based on nonlinear diffusion type of equations for the transport of Lithium ions and charges in the electrolyte and in the active material. Electrochemical reactions on the solid-electrolyte interface couple the two phases. The interface kinetics is modelled by the highly nonlinear Butler-Volmer interface conditions. Direct numerical simulations with standard methods, such as the Finite Element Method or Finite Volume Method, lead to ill-conditioned problems with a huge number of degrees of freedom which are difficult to solve. Therefore, the aim of this work is to derive upscaled models on the lengthscale of the whole electrode so that we do not have to resolve all the small-scale features of the porous microstructure thus reducing the computational time and cost. We do this by applying two different upscaling techniques - the Asymptotic Homogenization Method and the Multiscale Finite Element Method (MsFEM). We consider the electrolyte and the solid as two self-complementary perforated domains and we exploit this idea with both upscaling methods. The first method is restricted only to periodic media and periodically oscillating solutions while the second method can be applied to randomly oscillating solutions and is based on the Finite Element Method framework. We apply the Asymptotic Homogenization Method to derive a coupled macro-micro upscaled model under the assumption of periodic electrode microstructure. A crucial step in the homogenization procedure is the upscaling of the Butler-Volmer interface conditions. We rigorously determine the asymptotic order of the interface exchange current densities and we perform a comprehensive numerical study in order to validate the derived homogenized Li-ion battery model. In order to upscale the microscale battery problem in the case of random electrode microstructure we apply the MsFEM, extended to problems in perforated domains with Neumann boundary conditions on the holes. We conduct a detailed numerical investigation of the proposed algorithm and we show numerical convergence of the method that we design. We also apply the developed technique to a simplified two-dimensional Li-ion battery problem and we show numerical convergence of the solution obtained with the MsFEM to the reference microscale one.

Lithium-ion batteries are increasingly becoming an ubiquitous part of our everyday life - they are present in mobile phones, laptops, tools, cars, etc. However, there are still many concerns about their longevity and their safety. In this work we focus on the simulation of several degradation mechanisms on the microscopic scale, where one can resolve the active materials inside the electrodes of the lithium-ion batteries as porous structures. We mainly study two aspects - heat generation and mechanical stress. For the former we consider an electrochemical non-isothermal model on the spatially resolved porous scale to observe the temperature increase inside a battery cell, as well as to observe the individual heat sources to assess their contributions to the total heat generation. As a result from our experiments, we determined that the temperature has very small spatial variance for our test cases and thus allows for an ODE formulation of the heat equation.
The second aspect that we consider is the generation of mechanical stress as a result of the insertion of lithium ions in the electrode materials. We study two approaches - using small strain models and finite strain models. For the small strain models, the initial geometry and the current geometry coincide. The model considers a diffusion equation for the lithium ions and equilibrium equation for the mechanical stress. First, we test a single perforated cylindrical particle using different boundary conditions for the displacement and with Neumann boundary conditions for the diffusion equation. We also test for cylindrical particles, but with boundary conditions for the diffusion equation in the electrodes coming from an isothermal electrochemical model for the whole battery cell. For the finite strain models we take in consideration the deformation of the initial geometry as a result of the intercalation and the mechanical stress. We compare two elastic models to study the sensitivity of the predicted elastic behavior on the specific model used. We also consider a softening of the active material dependent on the concentration of the lithium ions and using data for silicon electrodes. We recover the general behavior of the stress from known physical experiments.
Some models, like the mechanical models we use, depend on the local values of the concentration to predict the mechanical stress. In that sense we perform a short comparative study between the Finite Element Method with tetrahedral elements and the Finite Volume Method with voxel volumes for an isothermal electrochemical model.
The spatial discretizations of the PDEs are done using the Finite Element Method. For some models we have discontinuous quantities where we adapt the FEM accordingly. The time derivatives are discretized using the implicit Backward Euler method. The nonlinear systems are linearized using the Newton method. All of the discretized models are implemented in a C++ framework developed during the thesis.