Open Access

The algorithmic efficiency of Newton-based methods for Free Flight Trajectory Optimization is heavily influenced by the size of the domain of convergence. We provide numerical evidence that the convergence radius is much larger in practice than what the theoretical worst case bounds suggest. The algorithm can be further improved by a convergence-enhancing domain decomposition.

In many applications, geodesic hierarchical models are adequate for the study of temporal observations. We employ such a model derived for manifold-valued data to Kendall's shape space. In particular, instead of the Sasaki metric, we adapt a functional-based metric, which increases the computational efficiency and does not require the implementation of the curvature tensor. We propose the corresponding variational time discretization of geodesics and employ the approach for longitudinal analysis of 2D rat skulls shapes as well as 3D shapes derived from an imaging study on osteoarthritis. Particularly, we perform hypothesis test and estimate the mean trends.

In the context of gas transmission in decoupled entry-exit systems, many approaches to determine the network capacity are based on the evaluation of realistic and severe transport situations. In this paper, we review the Reference Point Method, which is an algorithm used in practice to generate a set of scenarios using the so-called transport moment as a measure for severity. We introduce a new algorithm for finding severe transport situations that considers an actual routing of the flow through the network and is designed to handle issues arising from cyclic structures in a more dynamical manner. Further, in order to better approximate the physics of gas, an alternative, potential based flow formulation is proposed. The report concludes with a case study based on data from the benchmark library GasLib.

We study System Dynamics models with several free parameters that can be altered by the user. We assume that the user's goal is to achieve a certain dynamic behavior of the model by varying these parameters. In order to the find best possible combination of parameter settings, several automatic parameter tuning methods are described in the literature and readily available within existing System Dynamic software packages. We give a survey on the available techniques in the market and describe their theoretical background. Some of these methods are already six decades old, and meanwhile newer and more powerful optimization methods have emerged in the mathematical literature. One major obstacle for their direct use are tabled data in System Dynamics models, which are usually interpreted as piecewise linear functions. However, modern optimization methods usually require smooth functions which are twice continuously differentiable. We overcome this problem by a smooth spline interpolation of the tabled data. We use a test set of three complex System Dynamic models from the literature, describe their individual transition into optimization problems, and demonstrate the applicability of modern optimization algorithms to these System Dynamics Optimization problems.

This thesis firstly presents a nonlinear extended deterministic model for the transmission dynamics of tuberculosis, based on realistic assumptions and data collected from the WHO. This model enables a comprehensive qualitative analysis of various aspects in the outbreak and control of tuberculosis in Sub-Saharan Africa countries and successfully reproduces the epidemiology of tuberculosis in Cameroon for the past (from 1994-2010). Some particular properties of the model and its solution have been presented using the comparison theorem applied to the theory of differential equations. The existence and the stability of a disease free equilibrium has been discussed using the Perron-Frobenius theorem and Metzler stable matrices. Furthermore, we computed the basic reproduction number, i.e. the number of cases that one case generates on average over the course of its infectious period. Rigorous qualitative analysis of the model reveals that, in contrast to the model without reinfections, the full model with reinfection exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the basic reproduction ratio (R0), is less than unity. The global stability of the disease-free equilibrium has been discussed using the concepts of Lyapunov stability and bifurcation theory. With the help of a sensitivity analysis using data of Cameroon, we identified the relevant parameters which play a key role for the transmission and the control of the disease. This was possible applying sophisticated numerical methods (POEM) developed at ZIB. Using advanced approaches for optimal control considering the costs for chemoprophylaxis, treatment and educational campaigns should provide a framework for designing realistic cost effective strategies with different intervention methods. The forward-backward sweep method has been used to solve the numerical optimal control problem. The numerical result of the optimal control problem reveals that combined effort in education and chemoprophylaxis may lead to a reduction of 80\% in the number of infected people in 10 years. The mathematical and numerical approaches developed in this thesis could be similarly applied in many other Sub-Saharan countries where TB is a public health problem.