## Fachbereich Mathematik

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- Testrig optimization by block loads: Remodelling of damage as Gaussian functions and their clustering method (2014)
- In automotive testrigs we apply load time series to components such that the outcome is as close as possible to some reference data. The testing procedure should in general be less expensive and at the same time take less time for testing. In my thesis, I propose a testrig damage optimization problem (WSDP). This approach improves upon the testrig stress optimization problem (TSOP) used as a state of the art by industry experts. In both (TSOP) and (WSDP), we optimize the load time series for a given testrig configuration. As the name suggests, in (TSOP) the reference data is the stress time series. The detailed behaviour of the stresses as functions of time are sometimes not the most important topic. Instead the damage potential of the stress signals are considered. Since damage is not part of the objectives in the (TSOP) the total damage computed from the optimized load time series is not optimal with respect to the reference damage. Additionally, the load time series obtained is as long as the reference stress time series and the total damage computation needs cycle counting algorithms and Goodmann corrections. The use of cycle counting algorithms makes the computation of damage from load time series non-differentiable. To overcome the issues discussed in the previous paragraph this thesis uses block loads for the load time series. Using of block loads makes the damage differentiable with respect to the load time series. Additionally, in some special cases it is shown that damage is convex when block loads are used and no cycle counting algorithms are required. Using load time series with block loads enables us to use damage in the objective function of the (WSDP). During every iteration of the (WSDP), we have to find the maximum total damage over all plane angles. The first attempt at solving the (WSDP) uses discretization of the interval for plane angle to find the maximum total damage at each iteration. This is shown to give unreliable results and makes maximum total damage function non-differentiable with respect to the plane angle. To overcome this, damage function for a given surface stress tensor due to a block load is remodelled by Gaussian functions. The parameters for the new model are derived. When we model the damage by Gaussian function, the total damage is computed as a sum of Gaussian functions. The plane with the maximum damage is similar to the modes of the Gaussian Mixture Models (GMM), the difference being that the Gaussian functions used in GMM are probability density functions which is not the case in the damage approximation presented in this work. We derive conditions for a single maximum for Gaussian functions, similar to the ones given for the unimodality of GMM by Aprausheva et al. in [1]. By using the conditions for a single maximum we give a clustering algorithm that merges the Gaussian functions in the sum as clusters. Each cluster obtained through clustering is such that it has a single maximum in the absence of other Gaussian functions of the sum. The approximate point of the maximum of each cluster is used as the starting point for a fixed point equation on the original damage function to get the actual maximum total damage at each iteration. We implement the method for the (TSOP) and the two methods (with discretization and with clustering) for (WSDP) on two example problems. The results obtained from the (WSDP) using discretization is shown to be better than the results obtained from the (TSOP). Furthermore we show that, (WSDP) using clustering approach to finding the maximum total damage, takes less number of iterations and is more reliable than using discretization.

- Mathematik für Physiker ... und Mathematiker (2015)
- Eine Vorlesung für Studenten der Physik oder Mathematik im ersten Studienjahr: lineare Algebra und Analysis in einer und mehreren Veränderlichen.

- Bicriteria approach to the optimal location of surveillance cameras (2014)
- We consider the problem of finding efficient locations of surveillance cameras, where we distinguish between two different problems. In the first, the whole area must be monitored and the number of cameras should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range of vision are taken into account. We present solution approaches for these problems and report on results of their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives: maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.

- Freeness of hyperplane arrangements with multiplicities (2015)
- This bachelor thesis is concerned with arrangements of hyperplanes, that is, finite collections of hyperplanes in a finite-dimensional vector space. Such arrangements can be studied using methods from combinatorics, topology or algebraic geometry. Our focus lies on an algebraic object associated to an arrangement \(\mathcal{A}\), the module \(\mathcal{D(A)}\) of logarithmic derivations along \(\mathcal{A}\). It was introduced by K. Saito in the context of singularity theory, and intensively studied by Terao and others. If \(\mathcal{D(A)}\) admits a basis, the arrangement \(\mathcal{A}\) is called free. Ziegler generalized the concept of freeness to so-called multiarrangements, where each hyperplane carries a multiplicity. Terao conjectured that freeness of arrangements can be decided based on the combinatorics. We pursue the analogous question for multiarrangements in special cases. Firstly, we give a new proof of a result of Ziegler stating that generic multiarrangements are totally non-free, that is, non-free for any multiplicity. Our proof relies on the new concept of unbalanced multiplicities. Secondly, we consider freeness asymptotically for increasing multiplicity of a fixed hyperplane. We give an explicit bound for the multiplicity where the freeness property has stabilized.

- Bicriteria approach to the optimal location of surveillance cameras (2014)
- We consider the problem of finding efficient locations of surveillance cameras, where we distinguish between two different problems. In the first, the whole area must be monitored and the number of cameras should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range of vision are taken into account. We present solution approaches for these problems and report on results of their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives: maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.

- Combinations of Boolean Groebner Bases and SAT Solvers (2014)
- 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.

- Robust Flows with Losses and Improvability in Evacuation Planning (2014)
- 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.

- Transit Dependent Evacuation Planning for Kathmandu Valley: A Case Study (2014)
- Due to the increasing number of natural or man-made disasters, the application of operations research methods in evacuation planning has seen a rising interest in the research community. From the beginning, evacuation planning has been highly focused on car-based evacuation. Recently, also the evacuation of transit depended evacuees with the help of buses has been considered. In this case study, we apply two such models and solution algorithms to evacuate a core part of the metropolitan capital city Kathmandu of Nepal as a hypothetical endangered region, where a large part of population is transit dependent. We discuss the computational results for evacuation time under a broad range of possible scenarios, and derive planning suggestions for practitioners.

- Multilevel Constructions (2014)
- The thesis consists of the two chapters. The first chapter is addressed to make a deep investigation of the MLMC method. In particular we take an optimisation view at the estimate. Rather than fixing the number of discretisation points \(n_i\) to be a geometric sequence, we are trying to find an optimal set up for \(n_i\) such that for a fixed error the estimate can be computed within a minimal time. In the second chapter we propose to enhance the MLMC estimate with the weak extrapolation technique. This technique helps to improve order of a weak convergence of a scheme and as a result reduce CC of an estimate. In particular we study high order weak extrapolation approach, which is know not be inefficient in the standard settings. However, a combination of the MLMC and the weak extrapolation yields an improvement of the MLMC.

- A Bicriteria Approach to Robust Optimization (2014)
- 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.