Anhand des vom Gutachterausschuß der Stadt Kaiserlautern zur Verfügung gestellten Datenmaterials soll untersucht werden, welche Faktoren den Verkehrswert eines bebauten Grundstücks beeinflussen. Mit diesen Erkenntnissen soll eine möglichst einfache Formel ermittelt werden, die eine Schätzung für den Verkehrswert liefert, und die dabei die in der Vergangenheit erzielten Kaufpreise berücksichtigt. Für die Lösung dieser Aufgabe bietet sich das Verfahren der multiplen linearen Regression an. Auf die theoretischen Grundlagen soll hier nicht näher eingegangen werden, man findet sie in jedem Buch über mathematische Statistik, oder in [1]. Bei der Analyse der Daten wurde im großen und ganzen der Weg eingeschlagen, den Angelika Schwarz in [1] beschreibt. Ihre Ergebnisse lassen sich jedoch nicht direkt übertragen, da die dort betrachteten Grundstücke unbebaut waren. Da bei der statistischen Auswertung großer Datenmengen ein immenser Rechenaufwand anfällt, ist es unverzichtbar, professionelle statistische Software einzusetzen. Es stand das Programm S-Plus 2.0 (PC-Version für Windows) zur Verfügung. Sämtliche Berechnungen und alle Grafiken in diesem Bericht wurden in S-Plus erstellt.

We consider the problem to evacuate several regions due to river flooding, where sufficient time is given to plan ahead. To ensure a smooth evacuation procedure, our model includes the decision which regions to assign to which shelter, and when evacuation orders should be issued, such that roads do not become congested. Due to uncertainty in weather forecast, several possible scenarios are simultaneously considered in a robust optimization framework. To solve the resulting integer program, we apply a Tabu search algorithm based on decomposing the problem into better tractable subproblems. Computational experiments on random instances and an instance based on Kulmbach, Germany, data show considerable improvement compared to an MIP solver provided with a strong starting solution.

Die Arbeit ist zu verstehen als ein Teil im großen Projekt der Universität Kaiserslautern, das sich unter dem Namen Technomathematik um die dringend erforderliche Verständigung zwischen Technik und Mathematik bemüht.; Der große Leitfaden war das Buch von Natke: Einführung in Theorie und Praxis der Zeitreihen- und Modalanalyse, Schilderung der wesentlichen dort verwendeten Ideen der indirekten Systemidentifikation sowie des wahrscheinlichkeitstheoretischen und physikalisch-technischen Hintergrundes.

Gegenstand dieser Arbeit ist die Entwicklung eines Wärmetransportmodells für tiefe geothermische (hydrothermale) Reservoire. Existenz- und Eindeutigkeitsaussagen bezüglich einer schwachen Lösung des vorgestellten Modells werden getätigt. Weiterhin wird ein Verfahren zur Approximation dieser Lösung basierend auf einem linearen Galerkin-Schema dargelegt, wobei sowohl die Konvergenz nachgewiesen als auch eine Konvergenzrate erarbeitet werden.

In 2002, Korn and Wilmott introduced the worst-case scenario optimal portfolio approach. They extend a Black-Scholes type security market, to include the possibility of a crash. For the modeling of the possible stock price crash they use a Knightian uncertainty approach and thus make no probabilistic assumption on the crash size or the crash time distribution. Based on an indifference argument they determine the optimal portfolio process for an investor who wants to maximize the expected utility from final wealth. In this thesis, the worst-case scenario approach is extended in various directions to enable the consideration of stress scenarios, to include the possibility of asset defaults and to allow for parameter uncertainty. Insurance companies and banks regularly have to face stress tests performed by regulatory instances. In the first part we model their investment decision problem that includes stress scenarios. This leads to optimal portfolios that are already stress test prone by construction. The solution to this portfolio problem uses the newly introduced concept of minimum constant portfolio processes. In the second part we formulate an extended worst-case portfolio approach, where asset defaults can occur in addition to asset crashes. In our model, the strictly risk-averse investor does not know which asset is affected by the worst-case scenario. We solve this problem by introducing the so-called worst-case crash/default loss. In the third part we set up a continuous time portfolio optimization problem that includes the possibility of a crash scenario as well as parameter uncertainty. To do this, we combine the worst-case scenario approach with a model ambiguity approach that is also based on Knightian uncertainty. We solve this portfolio problem and consider two concrete examples with box uncertainty and ellipsoidal drift ambiguity.

Die Akustik liefert einen interessanten Hintergrund, interdisziplinären und fächerverbindenen Unterricht zwischen Mathematik, Physik und Musik durchzuführen. SchülerInnen können hierbei beispielsweise experimentell tätig sein, indem sie Audioaufnahmen selbst erzeugen und sich mit Computersoftware Frequenzspektren erzeugen lassen. Genauso können die Schüler auch Frequenzspektren vorgeben und daraus Klänge erzeugen. Dies kann beispielsweise dazu dienen, den Begriff der Obertöne im Musikunterricht physikalisch oder mathematisch greifbar zu machen oder in der Harmonielehre Frequenzverhältnisse von Intervallen und Dreiklängen näher zu untersuchen. Der Computer ist hier ein sehr nützliches Hilfsmittel, da der mathematische Hintergrund dieser Aufgabe -- das Wechseln zwischen Audioaufnahme und ihrem Frequenzbild -- sich in der Fourier-Analysis findet, die für SchülerInnen äußerst anspruchsvoll ist. Indem man jedoch die Fouriertransformation als numerisches Hilfsmittel einführt, das nicht im Detail verstanden werden muss, lässt sich an anderer Stelle interessante Mathematik betreiben und die Zusammenhänge zwischen Akustik und Musik können spielerisch erfahren werden. Im folgenden Beitrag wird eine Herangehensweise geschildert, wie wir sie bereits bei der Felix-Klein-Modellierungswoche umgesetzt haben: Die SchülerInnen haben den Auftrag erhalten, einen Synthesizer zu entwickeln, mit dem verschiedene Musikinstrumente nachgeahmt werden können. Als Hilfsmittel haben sie eine kurze Einführung in die Eigenschaften der Fouriertransformation erhalten, sowie Audioaufnahmen verschiedener Instrumente.

In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.

We introduce a class of models for time series of counts which include INGARCH-type models as well as log linear models for conditionally Poisson distributed data. For those processes, we formulate simple conditions for stationarity and weak dependence with a geometric rate. The coupling argument used in the proof serves as a role model for a similar treatment of integer-valued time series models based on other types of thinning operations.

A wavelet technique, the wavelet-Mie-representation, is introduced for the analysis and modelling of the Earth's magnetic field and corresponding electric current distributions from geomagnetic data obtained within the ionosphere. The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz-decomposition of spherical vector fields and (ii) the Mie-representation of solenoidal vector fields in terms of poloidal and toroidal parts. The wavelet-Mie-representation is shown to provide an adequate tool for geomagnetic modelling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features. An important example are ionospheric currents flowing radially onto or away from the Earth. To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.

With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.

We consider wavelet estimation of the time-dependent (evolutionary) power spectrum of a locally stationary time series. Allowing for departures from stationary proves useful for modelling, e.g., transient phenomena, quasi-oscillating behaviour or spectrum modulation. In our work wavelets are used to provide an adaptive local smoothing of a short-time periodogram in the time-freqeuncy plane. For this, in contrast to classical nonparametric (linear) approaches we use nonlinear thresholding of the empirical wavelet coefficients of the evolutionary spectrum. We show how these techniques allow for both adaptively reconstructing the local structure in the time-frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which is motivated by minimax properties w.r.t. the integrated mean squared error. Our approach is based on a 2-d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time-varying spectrum motivated from mobile radio propagation.

This work is dedicated to the wavelet modelling of regional and temporal variations of the Earth's gravitational potential observed by GRACE. In the first part, all required mathematical tools and methods involving spherical wavelets are introduced. Then we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified, which is restricted to areas, where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time series of regionally obtained wavelet coefficients of the gravitational signal originated from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement to previous studies and illustrate that wavelets are an appropriate tool to investigate regional time-variable effects in the gravitational field.

In this paper we introduce a multiscale technique for the analysis of deformation phenomena of the Earth. Classically, the basis functions under use are globally defined and show polynomial character. In consequence, only a global analysis of deformations is possible such that, for example, the water load of an artificial reservoir is hardly to model in that way. Up till now, the alternative to realize a local analysis can only be established by assuming the investigated region to be flat. In what follows we propose a local analysis based on tools (Navier scaling functions and wavelets) taking the (spherical) surface of the Earth into account. Our approach, in particular, enables us to perform a zooming-in procedure. In fact, the concept of Navier wavelets is formulated in such a way that subregions with larger or smaller data density can accordingly be modelled with a higher or lower resolution of the model, respectively.

A multiscale method is introduced using spherical (vector) wavelets for the computation of the earth's magnetic field within source regions of ionospheric and magnetospheric currents. The considerations are essentially based on two geomathematical keystones, namely (i) the Mie representation of solenoidal vector fields in terms of toroidal and poloidal parts and (ii) the Helmholtz decomposition of spherical (tangential) vector fields. Vector wavelets are shown to provide adequate tools for multiscale geomagnetic modelling in form of a multiresolution analysis, thereby completely circumventing the numerical obstacles caused by vector spherical harmonics. The applicability and efficiency of the multiresolution technique is tested with real satellite data.

Diese Doktorarbeit befasst sich mit Volatilitätsarbitrage bei europäischen Kaufoptionen und mit der Modellierung von Collateralized Debt Obligations (CDOs). Zuerst wird anhand einer Idee von Carr gezeigt, dass es stochastische Arbitrage in einem Black-Scholes-ähnlichen Modell geben kann. Danach optimieren wir den Arbitrage- Gewinn mithilfe des Erwartungswert-Varianz-Ansatzes von Markowitz und der Martingaltheorie. Stochastische Arbitrage im stochastischen Volatilitätsmodell von Heston wird auch untersucht. Ferner stellen wir ein Markoff-Modell für CDOs vor. Wir zeigen dann, dass man relativ schnell an die Grenzen dieses Modells stößt: Nach dem Ausfall einer Firma steigen die Ausfallintensitäten der überlebenden Firmen an, und kehren nie wieder zu ihrem Ausgangsniveau zurück. Dieses Verhalten stimmt aber nicht mit Beobachtungen am Markt überein: Nach Turbulenzen auf dem Markt stabilisiert sich der Markt wieder und daher würde man erwarten, dass die Ausfallintensitäten der überlebenden Firmen ebenfalls wieder abflachen. Wir ersetzen daher das Markoff-Modell durch ein Semi-Markoff-Modell, das den Markt viel besser nachbildet.

The present work deals with the (global and local) modeling of the windfield on the real topography of Rheinland-Pfalz. Thereby the focus is on the construction of a vectorial windfield from low, irregularly distributed data given on a topographical surface. The developed spline procedure works by means of vectorial (homogeneous, harmonic) polynomials (outer harmonics) which control the oscillation behaviour of the spline interpoland. In the process the characteristic of the spline curvature which defines the energy norm is assumed to be on a sphere inside the Earth interior and not on the Earth’s surface. The numerical advantage of this method arises from the maximum-minimum principle for harmonic functions.

In this thesis we classify simple coherent sheaves on Kodaira fibers of types II, III and IV (cuspidal and tacnode cubic curves and a plane configuration of three concurrent lines). Indecomposable vector bundles on smooth elliptic curves were classified in 1957 by Atiyah. In works of Burban, Drozd and Greuel it was shown that the categories of vector bundles and coherent sheaves on cycles of projective lines are tame. It turns out, that all other degenerations of elliptic curves are vector-bundle-wild. Nevertheless, we prove that the category of coherent sheaves of an arbitrary reduced plane cubic curve, (including the mentioned Kodaira fibers) is brick-tame. The main technical tool of our approach is the representation theory of bocses. Although, this technique was mainly used for purely theoretical purposes, we illustrate its computational potential for investigating tame behavior in wild categories. In particular, it allows to prove that a simple vector bundle on a reduced cubic curve is determined by its rank, multidegree and determinant, generalizing Atiyah's classification. Our approach leads to an interesting class of bocses, which can be wild but are brick-tame.

The shortest path problem in which the \((s,t)\)-paths \(P\) of a given digraph \(G =(V,E)\) are compared with respect to the sum of their edge costs is one of the best known problems in combinatorial optimization. The paper is concerned with a number of variations of this problem having different objective functions like bottleneck, balanced, minimum deviation, algebraic sum, \(k\)-sum and \(k\)-max objectives, \((k_1, k_2)-max, (k_1, k_2)\)-balanced and several types of trimmed-mean objectives. We give a survey on existing algorithms and propose a general model for those problems not yet treated in literature. The latter is based on the solution of resource constrained shortest path problems with equality constraints which can be solved in pseudo-polynomial time if the given graph is acyclic and the number of resources is fixed. In our setting, however, these problems can be solved in strongly polynomial time. Combining this with known results on \(k\)-sum and \(k\)-max optimization for general combinatorial problems, we obtain strongly polynomial algorithms for a variety of path problems on acyclic and general digraphs.

We present some new general results on the existence and form of value preserving portfolio strategies in a general semimartingale setting. The concept of value preservation will be derived via a mean-variance argument. It will also be embedded into a framework for local approaches to the problem of portfolio optimisation.

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.

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.

Handwritten digitized script to Prof. Schweigert's lecture "Universal Algebra"

An asymptotic preserving numerical scheme (with respect to diffusion scalings) for a linear transport equation is investigated. The scheme is adopted from a class of recently developped schemes. Stability is proven uniformly in the mean free path under a CFL type condition turning into a parabolic CFL condition in the diffusion limit.