## Diploma Thesis

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Satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG), respectively, are two measurement principles in modern satellite geodesy which yield knowledge of the first and second order radial derivative of the earth's gravitational potential at satellite altitude, respectively. A numerical method to compute the gravitational potential on the earth's surface from those observations should be capable of processing huge amounts of observational data. Moreover, it should yield a reconstruction of the gravitational potential at different levels of detail, and it should be possible to reconstruct the gravitational potential from only locally given data. SST and SGG are modeled as ill-posed linear pseudodifferential operator equations with an injective but non-surjective compact operator, which operates between Sobolev spaces of harmonic functions and such ones consisting of their first and second order radial derivatives, respectively. An immediate discretization of the operator equation is obtained by replacing the signal on its right-hand-side either by an interpolating or a smoothing spline which approximates the observational data. Here the noise level and the spatial distribution of the data determine whether spline-interpolation or spline-smoothing is appropriate. The large full linear equation system with positive definite matrix which occurs in the spline-interplation and spline-smoothing problem, respectively, is efficiently solved with the help of the Schwarz alternating algorithm, a domain decomposition method which allows it to split the large linear equation system into several smaller ones which are then solved alernatingly in an iterative procedure. Strongly space-localizing regularization scaling functions and wavelets are used to obtain a multiscale reconstruction of the gravitational potential on the earth's surface. In a numerical experiment the advocated method is successfully applied to reconstruct the earth's gravitational potential from simulated 'exact' and 'error-affected' SGG data on a spherical orbit, using Tikhonov regularization. The applicability of the numerical method is, however, not restricted to data given on a closed orbit but it can also cope with realistic satellite data.

A hub location problem consists of locating p hubs in a network in order to collect and consolidate flow between node pairs. This thesis deals with the uncapacitated single allocation p-hub center problem (USApHCP) as a special type of hub location problem with min max objective function. Using the so-called radius formulation of the problem, the dimension of the polyhedron of USApHCP is derived. The formulation constraints are investigated to find out which of these define facets. Then, three new classes of facet-defining inequalities are derived. Finally, efficient procedures to separate facets in a branch-and-cut algorithm are proposed. The polyhedral analysis of USApHCP is based on a tight relation to the uncapacitated facility location problem (UFL). Hence, many results stated in this thesis also hold for UFL.

This diploma thesis examines logistic problems occurring in a container terminal. The thesis focuses on the scheduling of cranes handling containers in a port. Two problems are discussed in detail: the yard crane scheduling of rubber-tired gantry cranes (RMGC) which move freely among the container blocks, and the scheduling of rail-mounted gantry cranes (RMGC) which can only move within a yard zone. The problems are formulated as integer programs. For each of the two problems discussed, two models are presented: In one model, the crane tasks are interpreted as jobs with release times and processing times while in the other model, it is assumed that the tasks can be modeled as generic workload measured in crane minutes. It is shown that the problems are NP-hard in the strong sense. Heuristic solution procedures are developed and evaluated by numerical results. Further ideas which could lead to other solution procedures are presented and some interesting special cases are discussed.

The scope of this diploma thesis is to examine the four generations of asset pricing models and the corresponding volatility dynamics which have been devepoled so far. We proceed as follows: In chapter 1 we give a short repetition of the Black-Scholes first generation model which assumes a constant volatility and we show that volatility should not be modeled as constant by examining statistical data and introducing the notion of implied volatility. In chapter 2, we examine the simplest models that are able to produce smiles or skews - local volatility models. These are called second generation models. Local volatility models model the volatility as a function of the stock price and time. We start with the work of Dupire, show how local volatility models can be calibrated and end with a detailed discussion of the constant elasticity of volatility model. Chapter 3 focuses on the Heston model which represents the class of the stochastic volatility models, which assume that the volatility itself is driven by a stochastic process. These are called third generation models. We introduce the model structure, derive a partial differential pricing equation, give a closed-form solution for European calls by solving this equation and explain how the model is calibrated. The last part of chapter 3 then deals with the limits and the mis-specifications of the Heston model, in particular for recent exotic options like reverse cliquets, Accumulators or Napoleons. In chapter 4 we then introduce the Bergomi forward variance model which is called fourth generation model as a consequence of the limits of the Heston model explained in chapter 3. The Bergomi model is a stochastic local volatility model - the spot price is modeled as a constant elasticity of volatility diffusion and its volatility parameters are functions of the so called forward variances which are specified as stochastic processes. We start with the model specification, derive a partial differential pricing equation, show how the model has to be calibrated and end with pricing examples and a concluding discussion.

Aggregation of Large-Scale Network Flow Problems with Application to Evacuation Planning at SAP
(2005)

Our initial situation is as follows: The blueprint of the ground floor of SAP’s main building the EVZ is given and the open question on how mathematic can support the evacuation’s planning process ? To model evacuation processes in advance as well as for existing buildings two models can be considered: macro- and microscopic models. Microscopic models emphasize the individual movement of evacuees. These models consider individual parameters such as walking speed, reaction time or physical abilities as well as the interaction of evacuees during the evacuation process. Because of the fact that the microscopic model requires lots of data, simulations are taken for implementation. Most of the current approaches concerning simulation are based on cellular automats. In contrast to microscopic models, macroscopic models do not consider individual parameters such as the physical abilities of the evacuees. This means that the evacuees are treated as a homogenous group for which only common characteristics are considered; an average human being is assumed. We do not have that much data as in the case of the microscopic models. Therefore, the macroscopic models are mainly based on optimization approaches. In most cases, a building or any other evacuation object is represented through a static network. A time horizon T is added, in order to be able to describe the evolution of the evacuation process over time. Connecting these two components we finally get a dynamic network. Based on this network, dynamic network flow problems are formulated, which can map evacuation processes. We focused on the macroscopic model in our thesis. Our main focus concerning the transfer from the real world problem (e.g. supporting the evacuation planning) will be the modeling of the blueprint as a dynamic network. After modeling the blueprint as a dynamic network, it will be no problem to give a formulation of a dynamic network flow problem, the so-called evacuation problem, which seeks for an optimal evacuation time. However, we have to solve a static large-scale network flow problem to derive a solution for this formulation. In order to reduce the network size, we will examine the possibility of applying aggregation to the evacuation problem. Aggregation (lat. aggregare = piling, affiliate; lat. aggregatio = accumulation, union; the act of gathering something together) was basically used to reduce the size of general large-scale linear or integer programs. The results gained for the general problem definitions were then applied to the transportation problem and the minimum cost network flow problem. We review this theory in detail and look on how results derived there can be used for the evacuation problem, too.

While there exist closed-form solutions for vanilla options in the presence of stochastic volatility for nearly a decade, practitioners still depend on numerical methods - in particular the Finite Difference and Monte Carlo methods - in the case of double barrier options. It was only recently that Lipton proposed (semi-)analytical solutions for this special class of path-dependent options. Although he presents two different approaches to derive these solutions, he restricts himself in both cases to a less general model, namely one where the correlation and the interest rate differential are assumed to be zero. Naturally the question arises, if these methods are still applicable for the general stochastic volatility model without these restrictions. In this paper we show that such a generalization fails for both methods. We will explain why this is the case and discuss the consequences of our results.

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.

Tropical geometry is a very new mathematical domain. The appearance of
tropical geometry was motivated by its deep relations to other mathematical
branches. These include algebraic geometry, symplectic geometry, complex
analysis, combinatorics and mathematical biology.
In this work we see some more relations between algebraic geometry and
tropical geometry. Our aim is to prove a one-to-one correspondence between
the divisor classes on the moduli space of n-pointed rational stable curves
and the divisors of the moduli space of n-pointed abstract tropical curves.
Thus we state some results of the algebraic case first. In algebraic geometry
these moduli spaces are well understood. In particular, the group of divisor
classes is calculated by S. Keel. We recall the needed results in chapter one.
For the proof of the correspondence we use some results of toric geometry.
Further we want to show an equality of the Chow groups of a special toric
variety and the algebraic moduli space. Thus we state some results of the
toric geometry as well.
This thesis tries to discover some connection between algebraic and tropical
geometry. Thus we also need the corresponding tropical objects to the
algebraic objects. Therefore we give some necessary definitions such as fan,
tropical fan, morphisms between tropical fans, divisors or the topical moduli
space of all n-marked tropical curves. Since we need it, we show that the
tropical moduli space can be embedded as a tropical fan.
After this preparatory work we prove that the group of divisor classes in
v
classical algebraic geometry has it equivalence in tropical geometry. For this
it is useful to give a map from the group of divisor classes of the algebraic
moduli space to the group of divisors of the tropical moduli space. Our aim is
to prove the bijectivity of this map in chapter three. On the way we discover
a deep connection between the algebraic moduli space and the toric variety
given by the tropical fan of the tropical moduli space.

The flow of a liquid into an empty channel is simulated. The simulation is based on a recently published model for general fluid/liquid/solid systems which eliminates the shear stress singularity at the moving contact line between the liquid/fluid interface and the solid. This model is carefully analyzed for low Reynolds and Capillary numbers, adapted to the channel inflow problem, and implemented. Very convincing numerical results are presented.