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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.

We consider the problem of evacuating an urban area caused by a natural or man-made disaster. There are several planning aspects that need to be considered in such a scenario, which are usually considered separately, due to their computational complexity. These aspects include: Which shelters are used to accommodate evacuees? How to schedule public transport for transit-dependent evacuees? And how do public and individual traffic interact? Furthermore, besides evacuation time, also the risk of the evacuation needs to be considered.
We propose a macroscopic multi-criteria optimization model that includes all of these questions simultaneously. As a mixed-integer programming formulation cannot handle instances of real-world size, we develop a genetic algorithm of NSGA-II type that is able to generate feasible solutions of good quality in reasonable computation times.
We extend the applicability of these methods by also considering how to aggregate instance data, and how to generate solutions for the original instance starting from a reduced solution.
In computational experiments using real-world data modelling the cities of Nice in France and Kaiserslautern in Germany, we demonstrate the effectiveness of our approach and compare the trade-off between different levels of data aggregation.

A new algorithm for optimization problems with three objective functions is presented which computes a representation for the set of nondominated points. This representation is guaranteed to have a desired coverage error and a bound on the number of iterations needed by the algorithm to meet this coverage error is derived. Since the representation does not necessarily contain nondominated points only, ideas to calculate bounds for the representation error are given. Moreover, the incorporation of domination during the algorithm and other quality measures are discussed.

A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.

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.

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.

We propose a model for acid-mediated tumor invasion involving two different scales: the microscopic one, for the dynamics of intracellular protons and their exchange with their extracellular counterparts, and the macroscopic scale of interactions between tumor cell and normal cell populations, along with the evolution of extracellular protons. We also account for the tactic behavior of cancer cells, the latter being assumed to biase their motion according to a gradient of extracellular protons (following [2,31] we call this pH taxis). A time dependent (and also time delayed) carrying capacity for the tumor cells in response to the effects of acidity is considered as well. The global well posedness of the resulting multiscale model is proved with a regularization and fixed point argument. Numerical simulations are performed in order to illustrate the behavior of the model.

Minmax regret optimization aims at finding robust solutions that perform best in the worst-case, compared to the respective optimum objective value in each scenario. Even for simple uncertainty sets like boxes, most polynomially solvable optimization problems have strongly NP-hard minmax regret counterparts. Thus, heuristics with performance guarantees can potentially be of great value, but only few such guarantees exist.
A very easy but effective approximation technique is to compute the midpoint solution of the original optimization problem, which aims at optimizing the average regret, and also the average nominal objective. It is a well-known result that the regret of the midpoint solution is at most 2 times the optimal regret. Besides some academic instances showing that this bound is tight, most instances reveal a way better approximation ratio.
We introduce a new lower bound for the optimal value of the minmax regret problem. Using this lower bound we state an algorithm that gives an instance dependent performance guarantee of the midpoint solution for combinatorial problems that is at most 2. The computational complexity of the algorithm depends on the minmax regret problem under consideration; we show that the sharpened guarantee can be computed in strongly polynomial time for several classes of combinatorial optimization problems.
To illustrate the quality of the proposed bound, we use it within a branch and bound framework for the robust shortest path problem. In an experimental study comparing this approach with a bound from the literature, we find a considerable improvement in computation times.

Cancer research is not only a fast growing field involving many branches of science, but also an intricate and diversified field rife with anomalies. One such anomaly is the
consistent reliance of cancer cells on glucose metabolism for energy production even in a normoxic environment. Glycolysis is an inefficient pathway for energy production and normally is used during hypoxic conditions. Since cancer cells have a high demand for energy
(e.g. for proliferation) it is somehow paradoxical for them to rely on such a mechanism. An emerging conjecture aiming to explain this behavior is that cancer cells
preserve this aerobic glycolytic phenotype for its use in invasion and metastasis. We follow this hypothesis and propose a new model
for cancer invasion, depending on the dynamics of extra- and intracellular protons, by building upon the existing ones. We incorporate random perturbations in the intracellular proton dynamics to account
for uncertainties affecting the cellular machinery. Finally, we address the well-posedness of our setting and use numerical simulations to illustrate the model predictions.

In this paper we construct a numerical solver for the Saint Venant equations. Special attention
is given to the balancing of the source terms, including the bottom slope and variable cross-
sectional profiles. Therefore a special discretization of the pressure law is used, in order to
transfer analytical properties to the numerical method. Based on this approximation a well-
balanced solver is developed, assuring the C-property and depth positivity. The performance
of this method is studied in several test cases focusing on accurate capturing of steady states.