## Fachbereich Mathematik

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#### Keywords

- Multiobjective optimization (3)
- Hypervolume (2)
- Subset selection (2)
- isogeometric analysis (2)
- k-link shortest path (2)
- pH-taxis (2)
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- Change analysis, nonparametric regression, nonlinear regression, autoregressive time series, sequential test, integer-valued time series (1)
- Debt Management (1)
- Eikonal equation (1)
- Erwarteter Nutzen (1)
- NURBS (1)
- Pedestrian FLow (1)
- Portfolio Selection (1)
- Timetabling; Scheduling; Graph coloring; Hierarchies; Rehabilitation Clinics (1)
- acid-mediated tumor invasion (1)
- adjoint approach (1)
- chemotherapy (1)
- coverage error (1)
- harmonic balance (1)
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- multiscale model, acid-mediated tumor invasion, reaction-diffusion equations, random differential equations (1)
- multiscale models (1)
- network flow (1)
- network location (1)
- radiotherapy (1)
- reaction-diffusion-taxis equations (1)
- reaction-diffusion-transport equations (1)
- shape optimization (1)
- sink location (1)
- time-delayed carrying capacities (1)
- tumor cell migration (1)
- weight optimization (1)

This thesis focuses on dealing with some new aspects of continuous time portfolio optimization by using the stochastic control method.
First, we extend the Busch-Korn-Seifried model for a large investor by using the Vasicek model for the short rate, and that problem is solved explicitly for two types of intensity functions.
Next, we justify the existence of the constant proportion portfolio insurance (CPPI) strategy in a framework containing a stochastic short rate and a Markov switching parameter. The effect of Vasicek short rate on the CPPI strategy has been studied by Horsky (2012). This part of the thesis extends his research by including a Markov switching parameter, and the generalization is based on the B\"{a}uerle-Rieder investment problem. The explicit solutions are obtained for the portfolio problem without the Money Market Account as well as the portfolio problem with the Money Market Account.
Finally, we apply the method used in Busch-Korn-Seifried investment problem to explicitly solve the portfolio optimization with a stochastic benchmark.

Starting from the two-scale model for pH-taxis of cancer cells introduced in [1], we consider here an extension accounting for tumor heterogeneity w.r.t. treatment sensitivity and a treatment approach including chemo- and radiotherapy. The effect of peritumoral region alkalinization on such therapeutic combination is investigated with the aid of numerical simulations.

In this paper we give an overview on the system of rehabilitation clinics in Germany in general and the literature on patient scheduling applied to rehabilitation facilities in particular.
We apply a class-teacher model developed to this environment and then generalize it to meet some of the specific constraints of inpatient rehabilitation clinics. To this end we introduce a restricted edge coloring on undirected bipartite graphs which is called group-wise balanced. The problem considered is called patient-therapist-timetable problem with group-wise balanced constraints (PTTPgb). In order to specify weekly schedules further such that they produce a reasonable allocation to morning/afternoon (second level decision) and to the single periods (third level decision) we introduce (hierarchical PTTPgb). For the corresponding model, the hierarchical edge coloring problem, we present some first feasibility results.

We develop a framework for shape optimization problems under state equation con-
straints where both state and control are discretized by B-splines or NURBS. In other
words, we use isogeometric analysis (IGA) for solving the partial differential equation and a nodal approach to change domains where control points take the place of nodes and where thus a quite general class of functions for representing optimal shapes and their boundaries becomes available. The minimization problem is solved by a gradient descent method where the shape gradient will be defined in isogeometric terms. This
gradient is obtained following two schemes, optimize first–discretize then and, reversely,
discretize first–optimize then. We show that for isogeometric analysis, the two schemes yield the same discrete system. Moreover, we also formulate shape optimization with respect to NURBS in the optimize first ansatz which amounts to finding optimal control points and weights simultaneously. Numerical tests illustrate the theory.

This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts.
The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm.
Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations.
The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research.
In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused.
Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.

We consider a network flow problem, where the outgoing flow is reduced by a certain percentage in each node. Given a maximum amount of flow that can leave the source node, the aim is to find a solution that maximizes the amount of flow which arrives at the sink.
Starting from this basic model, we include two new, additional aspects: On the one hand, we are able to reduce the loss at some of the nodes; on the other hand, the exact loss values are not known, but may come from a discrete uncertainty set of exponential size.
Applications for problems of this type can be found in evacuation planning, where one would like to improve the safety of nodes such that the number of evacuees reaching safety is maximized.
We formulate the resulting robust flow problem with losses and improvability as a mixed-integer program for finitely many scenarios, and present an iterative scenario-generation procedure that avoids the inclusion of all scenarios from the beginning. In a computational study using both randomly generated instance and realistic data based on the city of Nice, France, we compare our solution algorithms.

In this thesis, we combine Groebner basis with SAT Solver in different manners.
Both SAT solvers and Groebner basis techniques have their own strength and weakness.
Combining them could fix their weakness.
The first combination is using Groebner techniques to learn additional binary clauses for SAT solver from a selection of clauses. This combination is first proposed by Zengler and Kuechlin.
However, in our experiments, about 80 percent Groebner basis computations give no new binary clauses.
By selecting smaller and more compact input for Groebner basis computations, we can significantly
reduce the number of inefficient Groebner basis computations, learn much more binary clauses. In addition,
the new strategy can reduce the solving time of a SAT Solver in general, especially for large and hard problems.
The second combination is using all-solution SAT solver and interpolation to compute Boolean Groebner bases of Boolean elimination ideals of a given ideal. Computing Boolean Groebner basis of the given ideal is an inefficient method in case we want to eliminate most of the variables from a big system of Boolean polynomials.
Therefore, we propose a more efficient approach to handle such cases.
In this approach, the given ideal is translated to the CNF formula. Then an all-solution SAT Solver is used to find the projection of all solutions of the given ideal. Finally, an algorithm, e.g. Buchberger-Moeller Algorithm, is used to associate the reduced Groebner basis to the projection.
We also optimize the Buchberger-Moeller Algorithm for lexicographical ordering and compare it with Brickenstein's interpolation algorithm.
Finally, we combine Groebner basis and abstraction techniques to the verification of some digital designs that contain complicated data paths.
For a given design, we construct an abstract model.
Then, we reformulate it as a system of polynomials in the ring \({\mathbb Z}_{2^k}[x_1,\dots,x_n]\).
The variables are ordered in a way such that the system has already been a Groebner basis w.r.t lexicographical monomial ordering.
Finally, the normal form is employed to prove the desired properties.
To evaluate our approach, we verify the global property of a multiplier and a FIR filter using the computer algebra system Singular. The result shows that our approach is much faster than the commercial verification tool from Onespin on these benchmarks.

The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset \(W\) has to be selected such that given supplies can be transported to the nodes in \(W\). In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters.
We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status.
There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.

Geometric Programming is a useful tool with a wide range of applications in engineering. As in real-world problems input data is likely to be affected by uncertainty, Hsiung, Kim, and Boyd introduced robust geometric programming to include the uncertainty in the optimization process. They also developed a tractable approximation method to tackle this problem. Further, they pose the question whether there exists a tractable reformulation of their robust geometric programming model instead of only an approximation method. We give a negative answer to this question by showing that robust geometric programming is co-NP hard in its natural posynomial form.

The classic approach in robust optimization is to optimize the solution with respect to the worst case scenario. This pessimistic approach yields solutions that perform best if the worst scenario happens, but also usually perform bad on average. A solution that optimizes the average performance on the other hand lacks in worst-case performance guarantee.
In practice it is important to find a good compromise between these two solutions. We propose to deal with this problem by considering it from a bicriteria perspective. The Pareto curve of the bicriteria problem visualizes exactly how costly it is to ensure robustness and helps to choose the solution with the best balance between expected and guaranteed performance.
Building upon a theoretical observation on the structure of Pareto solutions for problems with polyhedral feasible sets, we present a column generation approach that requires no direct solution of the computationally expensive worst-case problem. In computational experiments we demonstrate the effectivity of both the proposed algorithm, and the bicriteria perspective in general.