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.

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.

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.