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.

In this thesis we consider the directional analysis of stationary point processes. We focus on three non-parametric methods based on second order analysis which we have defined as Integral method, Ellipsoid method, and Projection method. We present the methods in a general setting and then focus on their application in the 2D and 3D case of a particular type of anisotropy mechanism called geometric anisotropy. We mainly consider regular point patterns motivated by our application to real 3D data coming from glaciology. Note that directional analysis of 3D data is not so prominent in the literature. We compare the performance of the methods, which depends on the relative parameters, in a simulation study both in 2D and 3D. Based on the results we give recommendations on how to choose the methods´ parameters in practice. We apply the directional analysis to the 3D data coming from glaciology, which consist in the locations of air-bubbles in polar ice cores. The aim of this study is to provide information about the deformation rate in the ice and the corresponding thinning of ice layers at different depths. This information is substantial for the glaciologists in order to build ice dating models and consequently to give a correct interpretation of the climate information which can be found by analyzing ice cores. In this thesis we consider data coming from three different ice cores: the Talos Dome core, the EDML core and the Renland core. Motivated by the ice application, we study how isotropic and stationary noise influences the directional analysis. In fact, due to the relaxation of the ice after drilling, noise bubbles can form within the ice samples. In this context we take two classification algorithms into consideration, which aim to classify points in a superposition of a regular isotropic and stationary point process with Poisson noise. We introduce two methods to visualize anisotropy, which are particularly useful in 3D and apply them to the ice data. Finally, we consider the problem of testing anisotropy and the limiting behavior of the geometric anisotropy transform.

Various physical phenomenons with sudden transients that results into structrual changes can be modeled via switched nonlinear differential algebraic equations (DAEs) of the type \[ E_{\sigma}\dot{x}=A_{\sigma}x+f_{\sigma}+g_{\sigma}(x). \tag{DAE} \] where \(E_p,A_p \in \mathbb{R}^{n\times n}, x\mapsto g_p(x),\) is a mapping, \(p \in \{1,\cdots,P\}, P\in \mathbb{N} f \in \mathbb{R} \rightarrow \mathbb{R}^n , \sigma: \mathbb{R} \rightarrow \{1,\cdots, P\}\). Two related common tasks are: Task 1: Investigate if above (DAE) has a solution and if it is unique. Task 2: Find a connection among a solution of above (DAE) and solutions of related partial differential equations. In the linear case \(g(x) \equiv 0\) the task 1 has been tackeled already in a distributional solution framework. A main goal of the dissertation is to give contribution to task 1 for the nonlinear case \(g(x) \not \equiv 0\) ; also contributions to the task 2 are given for switched nonlinear DAEs arising while modeling sudden transients in water distribution networks. In addition, this thesis contains the following further contributions: The notion of structured switched nonlinear DAEs has been introduced, allowing also non regular distributions as solutions. This extend a previous framework that allowed only piecewise smooth functions as solutions. Further six mild conditions were given to ensure existence and uniqueness of the solution within the space of piecewise smooth distribution. The main condition, namely the regularity of the matrix pair \((E,A)\), is interpreted geometrically for those switched nonlinear DAEs arising from water network graphs. Another contribution is the introduction of these switched nonlinear DAEs as a simplication of the PDE model used classically for modeling water networks. Finally, with the support of numerical simulations of the PDE model it has been illustrated that this switched nonlinear DAE model is a good approximation for the PDE model in case of a small compressibility coefficient.

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.

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.

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

In this thesis, we 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.

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.

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.