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- A multiscale modeling approach to glioma invasion with therapy (2015)
- We consider the multiscale model for glioma growth introduced in a previous work and extend it to account for therapy effects. Thereby, three treatment strategies involving surgical resection, radio-, and chemotherapy are compared for their efficiency. The chemotherapy relies on inhibiting the binding of cell surface receptors to the surrounding tissue, which impairs both migration and proliferation.

- A stochastic model featuring acid induced gaps during tumor progression. (2015)
- In this paper we propose a phenomenological model for the formation of an interstitial gap between the tumor and the stroma. The gap is mainly filled with acid produced by the progressing edge of the tumor front. Our setting extends existing models for acid-induced tumor invasion models to incorporate several features of local invasion like formation of gaps, spikes, buds, islands, and cavities. These behaviors are obtained mainly due to the random dynamics at the intracellular level, the go-or-grow-or-recede dynamics on the population scale, together with the nonlinear coupling between the microscopic (intracellular) and macroscopic (population) levels. The wellposedness of the model is proved using the semigroup technique and 1D and 2D numerical simulations are performed to illustrate model predictions and draw conclusions based on the observed behavior.

- A nonlocal sample dependence SDE-PDE system modeling proton dynamics in a tumor (2015)
- A nonlocal stochastic model for intra- and extracellular proton dynamics in a tumor is proposed. The intracellular dynamics is governed by an SDE coupled to a reaction-diffusion equation for the extracellular proton concentration on the macroscale. In a more general context the existence and uniqueness of solutions for local and nonlocal SDE-PDE systems are established allowing, in particular, to analyze the proton dynamics model both, in its local version and the case with nonlocal path dependence. Numerical simulations are performed to illustrate the behavior of solutions, providing some insights into the effects of randomness on tumor acidity.

- On the History of Differential-Algebraic Equations (2015)
- To write about the history of a subject is a challenge that grows with the number of pages as the original goal of completeness is turning more and more into an impossibility. With this in mind, the present article takes a very narrow approach and uses personal side trips and memories on conferences, workshops, and summer schools as the stage for some of the most important protagonists and their contributions to the field of Differential-Algebraic Equations (DAEs).

- 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.

- 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.

- Hierarchical Edge Colorings and Rehabilitation Therapy Planning in Germany (2014)
- 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.

- Robust Geometric Programming is co-NP hard (2014)
- 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.

- Sink Location to Find Optimal Shelters in Evacuation Planning (2014)
- 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.

- Empirical Evaluation for the Conceptual Interoperability Analysis Approach: A Controlled Experiment Design (2014)
- Building interoperation among separately developed software units requires checking their conceptual assumptions and constraints. However, eliciting such assumptions and constraints is time consuming and is a challenging task as it requires analyzing each of the interoperating software units. To address this issue we proposed a new conceptual interoperability analysis approach which aims at decreasing the analysis cost and the conceptual mismatches between the interoperating software units. In this report we present the design of a planned controlled experiment for evaluating the effectiveness, efficiency, and acceptance of our proposed conceptual interoperability analysis approach. The design includes the study objectives, research questions, statistical hypotheses, and experimental design. It also provides the materials that will be used in the execution phase of the planned experiment.