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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 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.

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).

We study an online flow shop scheduling problem where each job consists of several tasks that have to be completed in t different stages and the goal is to maximize the total weight of accepted jobs.
The set of tasks of a job contains one task for each stage and each stage has a dedicated set of identical parallel machines corresponding to it that can only process tasks of this stage. In order to gain the weight (profit) associated with a job j, each of its tasks has to be executed between a task-specific release date and deadline subject to the constraint that all tasks of job j from stages 1, …, i-1 have to be completed before the task of the ith stage can be started. In the online version, jobs arrive over time and all information about the tasks of a job becomes available at the release date of its first task. This model can be used to describe production processes in supply chains when customer orders arrive online.
We show that even the basic version of the offline problem with a single machine in each stage, unit weights, unit processing times, and fixed execution times for all tasks (i.e., deadline minus release date equals processing time) is APX-hard. Moreover, we show that the approximation ratio of any polynomial-time approximation algorithm for this basic version of the problem must depend on the number t of stages.
For the online version of the basic problem, we provide a (2t-1)-competitive deterministic online algorithm and a matching lower bound. Moreover, we provide several (sometimes tight) upper and lower bounds on the competitive ratio of online algorithms for several generalizations of the basic problem involving different weights, arbitrary release dates and deadlines, different processing times of tasks, and several identical machines per stage.

A new solution approach for solving the 2-facility location problem in the plane with block norms
(2015)

Motivated by the time-dependent location problem over T time-periods introduced in
Maier and Hamacher (2015) we consider the special case of two time-steps, which was shown
to be equivalent to the static 2-facility location problem in the plane. Geometric optimality
conditions are stated for the median objective. When using block norms, these conditions
are used to derive a polygon grid inducing a subdivision of the plane based on normal cones,
yielding a new approach to solve the 2-facility location problem in polynomial time. Combinatorial algorithms for the 2-facility location problem based on geometric properties are
deduced and their complexities are analyzed. These methods differ from others as they are
completely working on geometric objects to derive the optimal solution set.

Scheduling-Location (ScheLoc) Problems integrate the separate fields of
scheduling and location problems. In ScheLoc Problems the objective is to
find locations for the machines and a schedule for each machine subject to
some production and location constraints such that some scheduling object-
ive is minimized. In this paper we consider the Discrete Parallel Machine
Makespan (DPMM) ScheLoc Problem where the set of possible machine loc-
ations is discrete and a set of n jobs has to be taken to the machines and
processed such that the makespan is minimized. Since the separate location
and scheduling problem are both NP-hard, so is the corresponding ScheLoc
Problem. Therefore, we propose an integer programming formulation and
different versions of clustering heuristics, where jobs are split into clusters
and each cluster is assigned to one of the possible machine locations. Since
the IP formulation can only be solved for small scale instances we propose
several lower bounds to measure the quality of the clustering heuristics. Ex-
tensive computational tests show the efficiency of the heuristics.

We consider storage loading problems where items with uncertain weights have
to be loaded into a storage area, taking into account stacking and
payload constraints. Following the robust optimization paradigm, we propose
strict and adjustable optimization models for finite and interval-based
uncertainties. To solve these problems, exact decomposition and heuristic
solution algorithms are developed.
For strict robustness, we also present a compact formulation based
on a characterization of worst-case scenarios.
Computational results show that computation times and algorithm
gaps are reasonable for practical applications.
Furthermore, we find that the robustness concepts show different
potential depending on the type of data being used.

This work presents a framework for the computation of complex geometries containing intersections of multiple patches with Reissner-Mindlin shell elements. The main objective is to provide an isogeometric finite element implementation which neither requires drilling rotation stabilization, nor user interaction to quantify the number of rotational degrees of freedom for every node. For this purpose, the following set of methods is presented. Control points with corresponding physical location are assigned to one common node for the finite element solution. A nodal basis system in every control point is defined, which ensures an exact interpolation of the director vector throughout the whole domain. A distinction criterion for the automatic quantification of rotational degrees of freedom for every node is presented. An isogeometric Reissner-Mindlin shell formulation is enhanced to handle geometries with kinks and allowing for arbitrary intersections of patches. The parametrization of adjacent patches along the interface has to be conforming. The shell formulation is derived from the continuum theory and uses a rotational update scheme for the current director vector. The nonlinear kinematic allows the computation of large deformations and large rotations. Two concepts for the description of rotations are presented. The first one uses an interpolation which is commonly used in standard Lagrange-based shell element formulations. The second scheme uses a more elaborate concept proposed by the authors in prior work, which increases the accuracy for arbitrary curved geometries. Numerical examples show the high accuracy and robustness of both concepts. The applicability of the proposed framework is demonstrated.

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.

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.

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.

In this paper we propose a procedure to extend classical numerical schemes for
hyperbolic conservation laws to networks of hyperbolic conservation laws. At the
junctions of the network we solve the given coupling conditions and minimize the
contributions of the outgoing numerical waves. This flexible procedure allows
us to also use central schemes at the junctions. Several numerical examples are
considered to investigate the performance of this new approach compared to the
common Godunov solver and exact solutions.

Glioma is a common type of primary brain tumor, with a strongly invasive potential, often exhibiting nonuniform, highly irregular growth. This makes it difficult to assess
the degree of extent of the tumor, hence bringing about a supplementary challenge for the treatment. It is therefore necessary to understand the
migratory behavior of glioma in greater detail.
In this paper we propose a multiscale model for glioma growth and migration. Our model couples the microscale dynamics (reduced to the binding of surface receptors to the
surrounding tissue) with a kinetic transport equation for the cell density on the mesoscopic level of individual cells. On the latter scale we also include the
proliferation of tumor cells via effects of interaction with the tissue. An adequate parabolic scaling yields a convection-diffusion-reaction equation, for which the coefficients
can be explicitly determined from the information about the tissue obtained by diffusion tensor imaging. Numerical simulations relying on DTI measurements confirm the biological
findings that glioma spreads
along white matter tracts.

We argue that the concepts of resilience in engineering science and robustness in mathematical optimization are strongly related. Using evacuation planning as an example application, we demonstrate optimization techniques to improve solution resilience. These include a direct modelling of the uncertainty for stochastic or robust optimization, as well as taking multiple objective functions into account.

We consider an uncertain traveling salesman problem, where distances between nodes are not known exactly, but may stem from an uncertainty set of possible scenarios. This uncertainty set is given as intervals with an additional bound on the number of distances that may deviate from their expected, nominal value.
A recoverable robust model is proposed, that allows a tour to change a bounded number of edges once a scenario becomes known. As the model contains an exponential number of constraints and variables, an iterative algorithm is proposed, in which tours and scenarios are computed alternately.
While this approach is able to find a provably optimal solution to the robust model, it also needs to solve increasingly complex subproblems. Therefore, we also consider heuristic solution procedures based on local search moves using a heuristic estimate of the actual objective function. In computational experiments, these approaches are compared.
Finally, an alternative recovery model is discussed, where a second-stage recovery tour is not required to visit all nodes of the graph. We show that the previously NP-hard evaluation of a fixed solution now becomes solvable in polynomial time.

The ordered weighted averaging objective (OWA) is an aggregate function over multiple optimization criteria which received increasing attention by the research community over the last decade. Different to the ordered weighted sum, weights are attached to ordered objective functions (i.e., a weight for the largest value, a weight for the second-largest value and so on). As this contains max-min or worst-case optimization as a special case, OWA can also be considered as an alternative approach to robust optimization.
For linear programs with OWA objective, compact reformulations exist, which result in extended linear programs. We present new such reformulation models with reduced size. A computational comparison indicates that these formulations improve solution times.