## Fachbereich Mathematik

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- Preprint (549)
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- Englisch (921) (entfernen)

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- Wavelet (13)
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- Boltzmann Equation (8)
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- integer programming (7)

#### Fachbereich / Organisatorische Einheit

- Fachbereich Mathematik (921)
- Fraunhofer (ITWM) (2)

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.

In this thesis we extend the worst-case modeling approach as first introduced by Hua and Wilmott (1997) (option pricing in discrete time) and Korn and Wilmott (2002) (portfolio optimization in continuous time) in various directions.
In the continuous-time worst-case portfolio optimization model (as first introduced by Korn and Wilmott (2002)), the financial market is assumed to be under the threat of a crash in the sense that the stock price may crash by an unknown fraction at an unknown time. It is assumed that only an upper bound on the size of the crash is known and that the investor prepares for the worst-possible crash scenario. That is, the investor aims to find the strategy maximizing her objective function in the worst-case crash scenario.
In the first part of this thesis, we consider the model of Korn and Wilmott (2002) in the presence of proportional transaction costs. First, we treat the problem without crashes and show that the value function is the unique viscosity solution of a dynamic programming equation (DPE) and then construct the optimal strategies. We then consider the problem in the presence of crash threats, derive the corresponding DPE and characterize the value function as the unique viscosity solution of this DPE.
In the last part, we consider the worst-case problem with a random number of crashes by proposing a regime switching model in which each state corresponds to a different crash regime. We interpret each of the crash-threatened regimes of the market as states in which a financial bubble has formed which may lead to a crash. In this model, we prove that the value function is a classical solution of a system of DPEs and derive the optimal strategies.

Using particle methods to solve the Boltzmann equation for rarefied gases numerically, in realistic streaming problems, huge differences in the total number of particles per cell arise. In order to overcome the resulting numerical difficulties the application of a weighted particle concept is well-suited. The underlying idea is to use different particle masses in different cells depending on the macroscopic density of the gas. Discrepance estimates and numerical results are given.

Weighted k-cardinality trees
(1992)

We consider the k -CARD TREE problem, i.e., the problem of finding in a given undirected graph G a subtree with k edges, having minimum weight. Applications of this problem arise in oil-field leasing and facility layout. While the general problem is shown to be strongly NP hard, it can be solved in polynomial time if G is itself a tree. We give an integer programming formulation of k-CARD TREE, and an efficient exact separation routine for a set of generalized subtour elimination constraints. The polyhedral structure of the convex huLl of the integer solutions is studied.

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.

Given a finite set of points in the plane and a forbidden region R, we want to find a point X not an element of int(R), such that the weighted sum to all given points is minimized. This location problem is a variant of the well-known Weber Problem, where we measure the distance by polyhedral gauges and allow each of the weights to be positive or negative. The unit ball of a polyhedral gauge may be any convex polyhedron containing the origin. This large class of distance functions allows very general (practical) settings - such as asymmetry - to be modeled. Each given point is allowed to have its own gauge and the forbidden region R enables us to include negative information in the model. Additionally the use of negative and positive weights allows to include the level of attraction or dislikeness of a new facility. Polynomial algorithms and structural properties for this global optimization problem (d.c. objective function and a non-convex feasible set) based on combinatorial and geometrical methods are presented.

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.

By means of the limit and jump relations of classical potential theory the framework of a wavelet approach on a regular surface is established. The properties of a multiresolution analysis are verified, and a tree algorithm for fast computation is developed based on numerical integration. As applications of the wavelet approach some numerical examples are presented, including the zoom-in property as well as the detection of high frequency perturbations. At the end we discuss a fast multiscale representation of the solution of (exterior) Dirichlet's or Neumann's boundary-value problem corresponding to regular surfaces.

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.