## Fachbereich Mathematik

- Tropical Intersection Products and Families of Tropical Curves (2012)
- This thesis is devoted to furthering the tropical intersection theory as well as to applying the developed theory to gain new insights about tropical moduli spaces. We use piecewise polynomials to define tropical cocycles that generalise the notion of tropical Cartier divisors to higher codimensions, introduce an intersection product of cocycles with tropical cycles and use the connection to toric geometry to prove a Poincaré duality for certain cases. Our main application of this Poincaré duality is the construction of intersection-theoretic fibres under a large class of tropical morphisms. We construct an intersection product of cycles on matroid varieties which are a natural generalisation of tropicalisations of classical linear spaces and the local blocks of smooth tropical varieties. The key ingredient is the ability to express a matroid variety contained in another matroid variety by a piecewise polynomial that is given in terms of the rank functions of the corresponding matroids. In particular, this enables us to intersect cycles on the moduli spaces of n-marked abstract rational curves. We also construct a pull-back of cycles along morphisms of smooth varieties, relate pull-backs to tropical modifications and show that every cycle on a matroid variety is rationally equivalent to its recession cycle and can be cut out by a cocycle. Finally, we define families of smooth rational tropical curves over smooth varieties and construct a tropical fibre product in order to show that every morphism of a smooth variety to the moduli space of abstract rational tropical curves induces a family of curves over the domain of the morphism. This leads to an alternative, inductive way of constructing moduli spaces of rational curves.

- Quadrature for Path-dependent Functionals of Lévy-driven Stochastic Differential Equations (2012)
- The main topic of this thesis is to define and analyze a multilevel Monte Carlo algorithm for path-dependent functionals of the solution of a stochastic differential equation (SDE) which is driven by a square integrable, \(d_X\)-dimensional Lévy process \(X\). We work with standard Lipschitz assumptions and denote by \(Y=(Y_t)_{t\in[0,1]}\) the \(d_Y\)-dimensional strong solution of the SDE. We investigate the computation of expectations \(S(f) = \mathrm{E}[f(Y)]\) using randomized algorithms \(\widehat S\). Thereby, we are interested in the relation of the error and the computational cost of \(\widehat S\), where \(f:D[0,1] \to \mathbb{R}\) ranges in the class \(F\) of measurable functionals on the space of càdlàg functions on \([0,1]\), that are Lipschitz continuous with respect to the supremum norm. We consider as error \(e(\widehat S)\) the worst case of the root mean square error over the class of functionals \(F\). The computational cost of an algorithm \(\widehat S\), denoted \(\mathrm{cost}(\widehat S)\), should represent the runtime of the algorithm on a computer. We work in the real number model of computation and further suppose that evaluations of \(f\) are possible for piecewise constant functions in time units according to its number of breakpoints. We state strong error estimates for an approximate Euler scheme on a random time discretization. With this strong error estimates, the multilevel algorithm leads to upper bounds for the convergence order of the error with respect to the computational cost. The main results can be summarized in terms of the Blumenthal-Getoor index of the driving Lévy process, denoted by \(\beta\in[0,2]\). For \(\beta <1\) and no Brownian component present, we almost reach convergence order \(1/2\), which means, that there exists a sequence of multilevel algorithms \((\widehat S_n)_{n\in \mathbb{N}}\) with \(\mathrm{cost}(\widehat S_n) \leq n\) such that \( e(\widehat S_n) \precsim n^{-1/2}\). Here, by \( \precsim\), we denote a weak asymptotic upper bound, i.e. the inequality holds up to an unspecified positive constant. If \(X\) has a Brownian component, the order has an additional logarithmic term, in which case, we reach \( e(\widehat S_n) \precsim n^{-1/2} \, (\log(n))^{3/2}\). For the special subclass of $Y$ being the Lévy process itself, we also provide a lower bound, which, up to a logarithmic term, recovers the order \(1/2\), i.e., neglecting logarithmic terms, the multilevel algorithm is order optimal for \( \beta <1\). An empirical error analysis via numerical experiments matches the theoretical results and completes the analysis.

- On selected efficient numerical methods for multiscale problems with stochastic coefficients (2013)
- Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients. We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods. The work is divided into three parts. In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients. The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations.

- Efficient time integration and nonlinear model reduction for incompressible hyperelastic materials (2013)
- This thesis deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements. We analyze the structure of the equations of motion and show that a differential-algebraic system of index 1 with a singular perturbation term needs to be solved. In the limit case the index may jump to index 3 and thus renders the time integration into a difficult problem. For the time integration we apply Rosenbrock methods and study their convergence behavior for a test problem, which highlights the importance of the well-known Scholz conditions for this problem class. Numerical tests demonstrate that such linear-implicit methods are an attractive alternative to established time integration methods in structural dynamics. In the second part we combine the simulation of nonlinear materials with a model reduction step. We use the method of proper orthogonal decomposition and apply it to the discretized system of second order. For a nonlinear model reduction to be efficient we approximate the nonlinearity by following the lookup approach. In a practical example we show that large CPU time savings can achieved. This work is in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications.

- Fibre Processes and their Applications (2013)
- The main purpose of the study was to improve the physical properties of the modelling of compressed materials, especially fibrous materials. Fibrous materials are finding increasing application in the industries. And most of the materials are compressed for different applications. For such situation, we are interested in how the fibre arranged, e.g. with which distribution. For given materials it is possible to obtain a three-dimensional image via micro computed tomography. Since some physical parameters, e.g. the fibre lengths or the directions for points in the fibre, can be checked under some other methods from image, it is beneficial to improve the physical properties by changing the parameters in the image. In this thesis, we present a new maximum-likelihood approach for the estimation of parameters of a parametric distribution on the unit sphere, which is various as some well known distributions, e.g. the von-Mises Fisher distribution or the Watson distribution, and for some models better fit. The consistency and asymptotic normality of the maximum-likelihood estimator are proven. As the second main part of this thesis, a general model of mixtures of these distributions on a hypersphere is discussed. We derive numerical approximations of the parameters in an Expectation Maximization setting. Furthermore we introduce a non-parametric estimation of the EM algorithm for the mixture model. Finally, we present some applications to the statistical analysis of fibre composites.

- Moduli spaces of rational tropical stable maps into smooth tropical varieties (2013)
- This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.

- Factorization of multivariate polynomials (2013)
- Factorization of multivariate polynomials is a cornerstone of many applications in computer algebra. To compute it, one uses an algorithm by Zassenhaus who used it in 1969 to factorize univariate polynomials over \(\mathbb{Z}\). Later Musser generalized it to the multivariate case. Subsequently, the algorithm was refined and improved. In this work every step of the algorithm is described as well as the problems that arise in these steps. In doing so, we restrict to the coefficient domains \(\mathbb{F}_{q}\), \(\mathbb{Z}\), and \(\mathbb{Q}(\alpha)\) while focussing on a fast implementation. The author has implemented almost all algorithms mentioned in this work in the C++ library factory which is part of the computer algebra system Singular. Besides, a new bound on the coefficients of a factor of a multivariate polynomial over \(\mathbb{Q}(\alpha)\) is proven which does not require \(\alpha\) to be an algebraic integer. This bound is used to compute Hensel lifting and recombination of factors in a modular fashion. Furthermore, several sub-steps are improved. Finally, an overview on the capability of the implementation is given which includes benchmark examples as well as random generated input which is supposed to give an impression of the average performance.

- Overlapping Domain Decomposition Preconditioners for Multi-Phase Elastic Composites (2013)
- The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established. The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids. We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions. Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.

- Multivariate Polynomial Interpolation and the Lifting Scheme with an Application to Scattered Data Approximation (2013)
- This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme. Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.

- Trading to stops: The investigation of state-based stopping rules (2013)
- The use of trading stops is a common practice in financial markets for a variety of reasons: it provides a simple way to control losses on a given trade, while also ensuring that profit-taking is not deferred indefinitely; and it allows opportunities to consider reallocating resources to other investments. In this thesis, it is explained why the use of stops may be desirable in certain cases. This is done by proposing a simple objective to be optimized. Some simple and commonly-used rules for the placing and use of stops are investigated; consisting of fixed or moving barriers, with fixed transaction costs. It is shown how to identify optimal levels at which to set stops, and the performances of different rules and strategies are compared. Thereby, uncertainty and altering of the drift parameter of the investment are incorporated.