## Fachbereich Mathematik

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#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (200)
- Fraunhofer (ITWM) (2)

#### Erscheinungsjahr

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- Dissertation (200) (entfernen)

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- Englisch (200) (entfernen)

#### Schlagworte

- Local stationarity for spatial data (2017)
- 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.

- Two instances of duality in commutative algebra (2017)
- 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.

- Stochastic geometry models for interacting fibers (2017)
- 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.

- Nonparametric Tests for Change Points in Hazard Functions under Random Censorship in Survival Analysis (2017)
- 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.

- An Intersection-Theoretic Approach to Correspondence Problems in Tropical Geometry (2016)
- 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.

- Continuous-Time Portfolio Optimization under Partial Information and Convex Constraints: Deriving Explicit Results (2017)
- 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.

- The Split tree for option pricing (2017)
- 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.

- Mathematical modelling of interacting fibre structures and non-woven materials (2017)
- 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.

- Product Pricing with Additive Influences - Algorithms and Complexity Results for Pricing in Social Networks (2017)
- 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.

- Convex Analysis for Processing Hyperspectral Images and Data from Hadamard Spaces (2017)
- 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.