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Algorithms that solve the shortest path problem can largely be split into the two categories of label setting and label correcting. The Multiobjective Shortest Path (MOSP) problem is a generalization of the classical shortest path problem in terms of the dimension of the cost function. We explore the differences of two similar MOSP label setting algorithms. Furthermore, we present and prove a general method of how to derive Fully Polynomial Time Approximation Schemes (FPTAS) for MOSP label setting algorithms. Finally, we explore two pruning techniques for the one to one variants of exact label setting MOSP algorithms and adapt them to their FPTAS variants.
Scheduling ist ein wichtiger Forschungsgegenstand im Bereich der diskreten Optimierung. Es geht darum, einen Schedule, d.h. einen Ablaufplan, für gegebene Ereignisse zu finden. Dieser soll optimal hinsichtlich einer Zielfunktion wie zum Beispiel minimaler Dauer oder Kosten sein. Dabei gibt es in der Regel Nebenbedingungen wie Vorrangbeziehungen zwischen den Ereignissen oder zeitliche Einschränkungen, die zu erfüllen sind. Falls die Ereignisse periodisch wiederkehren, spricht man von periodischem Scheduling. Beispiele sind das Erstellen von Zugfahrplänen, die Schaltungvon Ampelsignalen oder die Planung von Produktionsabläufen.
Mathematisch können diese Probleme mit dem Periodic Event Scheduling Problem (PESP) modelliert werden, das als gemischt-ganzzahliges Programm formuliert werden kann. In dieser Bachelorarbeit wird ein Ansatz zur Lösung des PESP mittels Zerlegung und Dualisierung entwickelt. In den Kapiteln 2 und 3 werden zunächst die notwendigen graphentheoretischen Grundlagen und das PESP eingeführt. In Kapitel 4 wird das PESP durch Fixierung der ganzzahligen Variablen in lineare Programme zerlegt. Dieses Unterproblem wird dualisiert und wieder in das PESP eingesetzt. Dafür ist eine weitere Nebenbedingung nötig.
Im fünften Kapitel behandeln wir die Lösung des teildualisierten PESP. Eine Möglichkeit ist es, sich auf eine Teilmenge der Nebenbedingungen zu beschränken. Eine weitere Möglichkeit ist ein Algorithmus, derähnlich wie BendersZerlegung die Nebenbedingungen dynamisch erzeugt. Dieser Algorithmus wird in Kapitel 6 implementiert und an vier Beispielen getestet.
Common models and solving approaches for the problem of periodic timetabling, that is, determining periodic arrival and departure times of a given public transportation network’s lines so that the total weighted travel time of all passengers is minimized, fail to take passenger behavior into account. Current research is attempting to resolve this issue and first results show a positive impact on solution quality. This thesis aims to give a brief overview of literature on the topic of timetabling and to then state a heuristic approach to Integrated Timetabling by blending passenger behavior into the Modulo Network Simplex algorithm, which is part of the research carried out by the author’s study group at the Zuse Institut Berlin in the framework of MATHEON’s research project MI-3 supported by the Einstein Foundation Berlin.
Prediction and Analysis of Drugs Inducing Liver Injury (DILI) using ProTox-II Computational-Tool
(2019)
In order to better understand the relationship between shape of the nasal cavity and to find objective classification for breathing obstruction, a population of 25 cases of healthy nasal cavity and 27 cases with diagnosed nasal airway obstruction (NAO) was examined for correlations between morphological, clinical and CFD parameters. For this purpose a workflow was implemented in Tcl to perform automatic measurements of morphological parameters of nasal cavity surfaces in Amira, which has as output a table with all estimated values. Furthermore, the statistical analysis was designed using Python to find the most probable subset of parameters that are predictors of nasal cavity pathology and consisted of correlation analysis, the selection of the best possible subset of parameters that could be used as predictors of clinically stated pathology of the nasal cavity by a logistic regression classifier. As a result, 10 most promising parameters were identified: mean distance between the two isthmuses, left isthmus contour, area ratio between the two isthmuses, left isthmus height, height ratio between the two isthmuses, left isthmus width, right isthmus width, right isthmus hydraulic diameter, mean distance of septal curvature between the septum enclosing walls of the nasal cavity, velocities volume average by expiration. As it turns out, most parameters refer to the isthmus region. This was to be expected since this region
plays an important role in the airflow system of the nasal cavity.
In this thesis, I want to analyze how neural networks trained to resemble a given function based on particular values react to additional knowledge about the function in terms of a differential equation that the function satisfies.
In the first chapter, I will recall the definition of initial value problems and discuss the existence and uniqueness of their solutions, as well as touch on a method to numerically approximate them. Furthermore, I will give a mathematical introduction to neural networks.
Then, I will consider a system of two first-order differential equations and test how their solution will be approximated by a neural net. In the next step, I want to see how added noise to the training data and the influence of the differential equation on the loss functional of the neural net will affect the error of the predicted solution against the exact data. Furthermore, I will test if this influence can help reduce the amount of necessary data points in order to reach similar degrees of accuracy.