## ZIB-Report

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- 17-27
- A Combinatorial Proof of a König-type Theorem for Unimodular Hypergraphs (2017)
- We state purely combinatorial proofs for König- and Hall-type theorems for a wide class of combinatorial optimization problems. Our methods rely on relaxations of the matching and vertex cover problem and, moreover, on the strong coloring properties admitted by bipartite graphs and their generalizations.

- 17-26
- Adapting trabecular structures for 3D printing: an image processing approach based on µCT data (2017)
- Materials with a trabecular structure notably combine advantages such as lightweight, reasonable strength, and permeability for fluids. This combination of advantages is especially interesting for tissue engineering in trauma surgery and orthopedics. Bone-substituting scaffolds for instance are designed with a trabecular structure in order to allow cell migration for bone ingrowth and vascularization. An emerging and recently very popular technology to produce such complex, porous structures is 3D printing. However, several technological aspects regarding the scaffold architecture, the printable resolution, and the feature size have to be considered when fabricating scaffolds for bone tissue replacement and regeneration. Here, we present a strategy to assess and prepare realistic trabecular structures for 3D printing using image analysis with the aim of preserving the structural elements. We discuss critical conditions of the printing system and present a 3-stage approach to adapt a trabecular structure from $\mu$CT data while incorporating knowledge about the printing system. In the first stage, an image-based extraction of solid and void structures is performed, which results in voxel- and graph-based representations of the extracted structures. These representations not only allow us to quantify geometrical properties such as pore size or strut geometry and length. But, since the graph represents the geometry and the topology of the initial structure, it can be used in the second stage to modify and adjust feature size, volume and sample size in an easy and consistent way. In the final reconstruction stage, the graph is then converted into a voxel representation preserving the topology of the initial structure. This stage generates a model with respect to the printing conditions to ensure a stable and controlled voxel placement during the printing process.

- 17-25
- Lossy data compression reduces communication time in hybrid time-parallel integrators (2017)
- Parallel in time methods for solving initial value problems are a means to increase the parallelism of numerical simulations. Hybrid parareal schemes interleaving the parallel in time iteration with an iterative solution of the individual time steps are among the most efficient methods for general nonlinear problems. Despite the hiding of communication time behind computation, communication has in certain situations a significant impact on the total runtime. Here we present strict, yet no sharp, error bounds for hybrid parareal methods with inexact communication due to lossy data compression, and derive theoretical estimates of the impact of compression on parallel efficiency of the algorithms. These and some computational experiments suggest that compression is a viable method to make hybrid parareal schemes robust with respect to low bandwidth setups.

- 17-24
- A Propagation Approach to Acyclic Rolling Stock Rotation Optimization (2017)
- The rolling stock, i.e., railway vehicles, are one of the key ingredients of a running railway system. As it is well known, the offer of a railway company to their customers, i.e., the railway timetable, changes from time to time. Typical reasons for that are different timetables associated with different seasons, maintenance periods or holidays. Therefore, the regular lifetime of a timetable is split into (more or less) irregular periods where parts of the timetable are changed. In order to operate a railway timetable most railway companies set up sequences that define the operation of timetabled trips by a single physical railway vehicle called (rolling stock) rotations. Not surprisingly, the individual parts of a timetable also affect the rotations. More precisely, each of the parts brings up an acyclic rolling stock rotation problem with start and end conditions associated with the beginning and ending of the corresponding period. In this paper, we propose a propagation approach to deal with large planning horizons that are composed of many timetables with shorter individual lifetimes. The approach is based on an integer linear programming formulation that propagates rolling stock rotations through the irregular parts of the timetable while taking a large variety of operational requirements into account. This approach is implemented within the rolling stock rotation optimization framework ROTOR used by DB Fernverkehr AG, one of the leading railway operators in Europe. Computational results for real world scenarios are presented to evaluate the approach.

- 17-21
- Follicular competition: the selection of dominant follicles as a synergistic effect (2017)
- The estrous cycle of mono-ovulatory species such as cows or humans, is known to show two or more waves of follicular growth and decline between two successive ovulations. Within each wave, there is one dominant follicle escorted by subordinate follicles of varying number. Under the surge of the luteinizing hormone a growing dominant follicle ovulates. Rarely the number of ovulating follicles exceeds one. In the biological literature, the change of hormonal concentrations and individually varying numbers of follicular receptors are made responsible for the selection of exactly one dominant follicle, yet a clear cause has not been identified. In this paper, we suggest a synergistic explanation based on competition, formulated by a parsimoniously defined system of ordinary differential equations (ODEs) that quantifies the time evolution of multiple follicles and their competitive interaction during one wave. Not discriminating between follicles, growth and decline are given by fixed rates. Competition is introduced via a growth-suppressing term, equally supported by all follicles. We prove that the number of dominant follicles is determined exclusively by the ratio of follicular growth and competition. This number turns out to be independent of the number of subordinate follicles. The asymptotic behavior of the corresponding dynamical system is investigated rigorously, where we demonstrate that the omega-limit set only contains fixed points. When also including follicular decline, our ODEs perfectly resemble ultrasound data of bovine follicles. Implications for the involved but not explicitly modeled hormones are discussed.

- 17-23
- Conflict-Free Railway Track Assignment at Depots (2017)
- Managing rolling stock with no passengers aboard is a critical component of railway operations. In particular, one problem is to park the rolling stock on a given set of tracks at the end of a day or service. Depending on the parking assignment, shunting may be required in order for a parked train to depart or for an incoming train to park. Given a collection of tracks M and a collection of trains T with fixed arrival-departure timetable, the train assignment problem (TAP) is to determine the maximum number of trains from T that can be parked on M according to the timetable and without the use of shunting. Hence, efficiently solving the TAP allows to quickly compute feasible parking schedules that do not require further shunting adjustments. In this paper, we present two integer programming models for solving the TAP. To our knowledge, this is the first integrated approach that considers track lengths along with the three most common types of parking tracks. We compare these models on a theoretical level. We also prove that a decision version of the TAP is NP-complete, justifying the use of integer programming techniques. Using stochastic and robust modelling techniques, both models produce parking assignments that are optimized and robust according to random train delays. We conclude with computational results for both models, observing that they perform well on real timetables.

- 17-22
- Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics (2017)
- We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.

- 17-18
- Uncertainty in Temperature-Based Determination of Time of Death (2017)
- Temperature-based estimation of time of death (ToD) can be per- formed either with the help of simple phenomenological models of corpse cooling or with detailed mechanistic (thermodynamic) heat transfer mod- els. The latter are much more complex, but allow a higher accuracy of ToD estimation as in principle all relevant cooling mechanisms can be taken into account. The potentially higher accuracy depends on the accuracy of tissue and environmental parameters as well as on the geometric resolution. We in- vestigate the impact of parameter variations and geometry representation on the estimated ToD based on a highly detailed 3D corpse model, that has been segmented and geometrically reconstructed from a computed to- mography (CT) data set, differentiating various organs and tissue types. From that we identify the most crucial parameters to measure or estimate, and obtain a local uncertainty quantifcation for the ToD.

- 17-20
- Field Heterogeneities and their Impact on Photocatalysis: Combining optical and kinetic Monte Carlo Simulations on the Nanoscale (2017)
- Gaining insights into the working principles of photocatalysts on an atomic scale is a challenging task. The obviously high complexity of the reaction mechanism involving photo-excited electrons and holes is one reason. Another complicating aspect is that the electromagnetic field, driving photocatalysis, is not homogeneous on a nanoscale level for particle based catalysts as it is influenced by the particle’s shape and size. We present a simple model, inspired by the CO2 reduction on titania anatase, which addresses the impact of these heterogeneities on the photocatalytic kinetics by combining kinetic Monte Carlo with electromagnetic wave simulations. We find that average activity and especially efficiency might differ significantly between different particles. Moreover, we find sizable variation of the catalytic activity on a single facet of a nanocrystal. Besides this quantitative heterogeneity, the coverage situation in general changes laterally on this facet and we observe a concomitant change of the rate-determining steps. This heterogeneity on all levels of photocatalytic activity is masked in experimental studies, where only the spatially averaged activity can be addressed. Microkinetic models based on experimental findings might therefore not represent the true micro- scopic behavior, and mechanistic conclusion drawn from these need to be handled with care.

- 17-19
- Reliable approximation of long relaxation timescales in molecular dynamics (2017)
- Many interesting rare events in molecular systems like ligand association, protein folding or con- formational changes happen on timescales that often are not accessible by direct numerical simulation. Therefore rare event approximation approaches like interface sampling, Markov state model building or advanced reaction coordinate based free energy estimation have attracted huge attention recently. In this article we analyze the reliability of such approaches: How precise is an estimate of long relaxation timescales of molecular systems resulting from various forms of rare event approximation methods? Our results give a theoretical answer to this question by relating it with the transfer operator approach to molecular dynamics. By doing so they also allow for understanding deep connections between the different approaches.