Numerical simulation is necessary to understand natural phenomena, make assessments and predictions in various research and engineering fields, develop new technologies, etc. New algorithms are needed to take advantage of the increasing computational resources and utilize the emerging hardware and software infrastructure with maximum efficiency. Adaptive numerical discretization methods can accommodate problems with various physical, scale, and dynamic features by adjusting the resolution, order, and the type of method used to solve them. In applications that simulate real systems, the numerical accuracy of the solution is typically just one of the challenges. Measurements can be included in the simulation to constrain the numerical solution through a process called data assimilation in order to anchor the simulation in reality. In this thesis we investigate adaptive discretization methods and data assimilation approaches for large-scale numerical simulations. We develop and investigate novel multirate and implicit-explicit methods that are appropriate for multiscale and multiphysics numerical discretizations. We construct and explore data assimilation approaches for, but not restricted to, atmospheric chemistry applications. A generic approach for describing the structure of the uncertainty in initial conditions that can be applied to the most popular data assimilation approaches is also presented. We show that adaptive numerical methods can effectively address the discretization of large-scale problems. Data assimilation complements the adaptive numerical methods by correcting the numerical solution with real measurements. Test problems and large-scale numerical experiments validate the theoretical findings. Synergistic approaches that use adaptive numerical methods within a data assimilation framework need to be investigated in the future. / Ph. D.

Zugl.: Kairo, Univ., Diss., 2004

Ziel dieser Doktorarbeit ist ein mathematisches Modell zu entwickeln, um eine mögliche Ausbreitung des West-Nil-Virus (WNV) in Deutschland zu simulieren und zu bewerten. Das entwickelte Werkzeug soll auch auf eine weitere, durch Zecken übertragene Krankheit, dem Krim-Kongo-Hämorrhagischen Fieber (CCHFV) angewendet werden. Die durch den Klimawandel verursachte globalen Erwärmung unterstützt auch die Verbreitung und Entwicklung verschiedener Vektorpopulationen. Dabei hat eine Temperaturerhöhung einen positiven Einfluss auf den Lebenszyklus des Vektors und die Zunahme der Vektoraktivität. In dieser Arbeit haben wir ein Differentialgleichungsmodell (ODE) entwickelt, um den Einfluss eines regelmäßigen Eintrags von Infektionserregern auf die empfängliche Population unter Berücksichtigung des Temperatureinflusses zu verstehen. Als Ergebnis haben wir einen analytischen Ausdruck der Basisreproduktionszahl und deren Wechselwirkung mit der Temperatur gefunden. Eine Sensitivitätsanalyse zeigt, wie wichtig das Verhältnis der anfälligen Mücken zur lokalen Wirtspopulation ist. Als ein zentrales Ergebnis haben wir den zukünftigen Temperaturverlauf auf Basis der Modellergebnisse des IPCC in unser Modell integriert und Bedingungen gefunden, unter denen es zu einer dauerhaften Etablierung des West-Nil-Virus in Deutschland kommt. Darüber hinaus haben wir die entwickelten mathematischen Modelle verwendet, um verschiedene Szenarien zu untersuchen, unter denen sich CCHFV möglicherweise in einer naiven Population etablieren kann, und wir haben verschiedene Kontrollszenarien mathematisch abgeleitet, um die Belastung von einer Infektion durch Zecken zu bewältigen. / The objective of this thesis is to develop the necessary mathematical model to assess the potential spread of West Nile Virus (WNV) in Germany and employ the developed tool to analyse another tick-borne disease Crimean- Congo Hemorrhagic Fever (CCHFV). Given the backdrop of global warming and the climate change, increasing temperature has benefitted the vector population. The increase in the temperature has a positive influence in the life cycle of the vector and the increase in its activities. In this thesis, we have developed an Ordinary Differential Equation (ODE) model system to understand the influence of the periodic introduction of infectious agents into the local susceptible population while taking account of influence of temperature. As results, we have found an analytic expression of the basic reproduction number and its interplay with the temperature. The sensitivity analysis shows us the importance of the ratio between the susceptible mosquitoes to the local host population. As a central result we have extrapolated the temperature trend under different IPCC conditions and found the condition under which the circulation of West Nile Virus will be permanent in Germany. Furthermore, we have utilised the developed mathematical models to examine different scenarios under which CCHFV can potentially establish in a naive population along with we mathematically derived different control scenarios to manage the burden of tick infection.

[pt] Esta dissertação tem o objetivo de aplicar os conceitos e ferramentas da Análise Real e Álgebra Linear num estudo sobre a teoria de existência, unicidade e estabilidade de soluções de sistemas de equações diferenciais ordinárias, considerando sistemas gerais parametrizados, lineares e não-lineares. / [en] This dissertation aims to apply the concepts and tools of Real Analysis and Linear Algebra to the theory of existence, uniquiness and stability of solutions of ordinary differential equations systems, considering general parametric, linear and non-linear systems.

No description available.

Gearboxes are always subject of study in order to increase their efficiency. Energy losses in gear contacts are transformed into heat which is distributed among the gearbox components increasing their temperature. A thermal model of the gearbox brings the opportunity of a deeper understanding of the heat dissipated related to the power losses in the gear contact. A MATLAB program based on ordinary differential equations was developed in order to make a thermal model of an FZG test gearbox. The model is based on a thermal network where each node represents a machine element. The thermal network is composed by thermal resistances due to deformation in the gear contact, conduction, convection and radiation. With thermal resistances, power losses and thermal inertia of each element, the temperature evolution was obtained by applying the First Principle of Thermodynamics. Due to the temperature evolution, heat transfer between different elements was estimated. Additionally, experimental results from an FZG test rig were implemented in the model and also used to verify its accuracy. Furthermore, additional features to the model such as a cooling system and spray lubrication were also studied. Results show a wide capability and handling of the program in terms of thermal analysis: heat flux direction and magnitude, visual tools such as thermal network of the test gearbox, as well as the analysis of different operating conditions. With these tools, an approach to the minimum amount of lubricant necessary and other ways to quench overheating could then be reached. Keywords: Thermal network, FZG gear test rig, heat flow, temperature, MATLAB, ODE. / Växellådor är ständigt ett forskningsområde för att förbättra deras verkningsgrad. Energiförluster i kuggkontakter omvandlas till värme som sprids i växellådan som sedan värmer upp komponenterna. En termisk modell av växellådan gör det möjligt för djupare förståelse hur värmen sprids i förhållande till energiförlusterna i kuggkontakten. Ett MATLAB-program baserat på ordinära differential-ekvationer utvecklades för att göra en termisk modell av en växellåda i en kuggrigg från FZG. Modellen är baserad på ett termiskt nätverk där varje nod representerar en maskinkomponent. Det termiska nätverket består av resistanser som uppstår på grund av deformation i kuggkontakten, ledning, konvektion och strålning. Med termiska resistanser, energiförluster, termisk tröghet från komponenterna och genom att applicera termodynamikens första grundsats kunde temperatur-genereringen bestämmas. Från temperatur-genereringen kunde värme-ledningen mellan komponenter uppskattas. Testresultat från en FZG-kuggrigg användes för att verifiera modellens noggrannhet. Andra egenskaper till modellen, som ett annat kylsystem och spraysmörjning studerades för att undersöka möjligheteten att adderas till modellen. Resultat visar på en bred användning av modellen i avseende på termisk analys: värmeflödets storlek och riktning, ett visuellt redskap för växellådans temperatur och hur växellådans temperatur varierar under olika driftförhållanden. Med de här redskapen kan den minsta oljemängden som behövs för att smörja kuggkontakten undersökas och hur kylning av kugghjulen kan förbättras. Nyckelord: Termiskt nätverk, FZG kugghjuls-rigg, värmeflöde, temperatur, MATLAB, ODE

Jedes Jahr infizieren sich mehr als 200 Millionen Menschen mit Malaria. Eine halbe Millionen von ihnen verstirbt. Die Mehrzahl der Krankheits- und Todesfälle wird durch den Parasiten Plasmodium falciparum verursacht, einen von sechs Stämmen von Malariaparasiten, der Menschen infizieren kann. Der P. falciparum-Parasit hat in unterschiedlichem Maße Resistenzen gegen die meisten derzeit verwendeten Malariamittel entwickelt, und es besteht ein ständiger Bedarf an der Entwicklung neuer Malariamedikamente. Zwei Wirkstoffe, die sich derzeit in der klinischen Erprobung gegen Malaria befinden, zielen auf ’Ionenpumpen’ in der Oberflächenmembran des Malariaparasiten ab. Die Ionenregulation im Parasiten P. falciparum war in den letzten Jahrzehnten Gegenstand umfangreicher Forschung, welche zu einem allgemeinen Verständnis darüber geführt, wie der Parasit seine interne Ionenhaushalt reguliert. Es wurde jedoch noch nicht versucht, diese Erkenntnisse in ein quantitatives Modell zu integrieren. In dieser Arbeit habe ich ein mathematisches Modell für die Ionenhomöostase im asexuellen, intra-erythrozytären Stadium des Parasiten P. falciparum entwickelt. Das Modell bietet neue Einblicke in bisher unerklärte, experimentelle Beobachtungen und sagt die Wechselwirkungen von Ionentransport-Inhibitoren voraus. Das neu entwickelte Modell der Ionenregulation im Parasiten wurde in ein bereits bestehendes mathematisches Modell der Ionenregulation im Wirtserythrozyten integriert, um ein vorläufiges "kombiniertes Modell" des parasiteninfizierten Erythrozyten als Ganzes zu erstellen. Die Ergebnisse dieses kombinierten Modells wurden mit den Ergebnissen einer begrenzten Anzahl von Experimenten verglichen, die im Rahmen dieser Arbeit durchgeführt wurden. In diesen Experimenten wurde die Veränderung der infizierten Erythrozyten nach verschiedenen osmotischen Störungen gemessen. Die im Rahmen dieser Arbeit durchgeführte mathematische Modellierung trägt zum Verständnis der gegenseitigen Abhängigkeiten bei, die bei der Ionenregulierung des Malariaparasiten eine Rolle spielen, und bietet einen Rahmen für das Verständnis der Auswirkungen von "Ionentransport-hemmenden" Malariamitteln. / Malaria is currently responsible for more than 200 million estimated cases and half a million deaths annually, with the majority of cases and deaths attributable to Plasmodium falciparum, one of six strains of malaria parasite able to infect humans. The P. falciparum parasite has developed varying degrees of resistance against most, if not all, of the antimalarial drugs currently available and there is an ongoing need to develop new antimalarial agents. Two compounds, which are currently in clinical trials against malaria target an ’ion pump’ on the surface membrane of the malaria parasite. Ion regulation in the P. falciparum parasite has been the subject of extensive studies over recent decades. This research has led to a general understanding of how the parasite regulates its internal ionic composition. However, there has not yet been any attempt to integrate these findings into a quantitative model. In the work presented in this thesis, I have developed a mathematical model for ion homeostasis in the asexual intra-erythrocytic blood-stage of the P. falciparum parasite. The model provides new insights into formerly unexplained in vitro observations and predicts interactions of ion transport inhibitors. The newly formulated model of ion regulation in the parasite was integrated with a pre-existing mathematical model for ion regulation in the host erythrocyte to generate a preliminary ’combined model’ of the parasite-infected erythrocyte as a whole. Outputs from this combined model were compared to the results from a limited number of experiments conducted in the course of this thesis. These experiments entailed measuring the change of infected erythrocytes following different osmotic perturbations. The mathematical modelling conducted in the course of this work adds to the understanding of the interdependencies involved in malaria parasite ion regulation and provides a framework to help understand the effects of ’ion-transport-inhibiting’ antimalarial agents.

ITC/USA 2006 Conference Proceedings / The Forty-Second Annual International Telemetering Conference and Technical Exhibition / October 23-26, 2006 / Town and Country Resort & Convention Center, San Diego, California / This paper describes an adaptive data management architecture capable of supporting order-of-magnitude data volume increases without a priori knowledge of data structures. The architecture allows users to generate and maintain data in optimal legacy formats while managing and extracting information with common analysis tools. This paper shows how an object-oriented data management system can manage both data and the knowledge imparted to the data by users.

Infantile nystagmus is a condition that causes involuntary, bilateral and conjugate oscillations of the eyes, which are predominately restricted to the horizontal plane. In order to investigate the cause of nystagmus, computational models and nonlinear dynamics techniques have been used to model and analyse the oculomotor system. Computational models are important in making predictions and creating a quantitative framework for the analysis of the oculomotor system. Parameter estimation is a critical step in the construction and analysis of these models. A preliminary parameter estimation of a nonlinear dynamics model proposed by Broomhead et al. [1] has been shown to be able to simulate both normal rapid eye movements (i.e. saccades) and nystagmus oscillations. The application of nonlinear analysis to experimental jerk nystagmus recordings, has shown that the local dimensions number of the oscillation varies across the phase angle of the nystagmus cycle. It has been hypothesised that this is due to the impact of signal dependent noise (SDN) on the neural commands in the oculomotor system. The main aims of this study were: (i) to develop parameter estimation methods for the Broomhead et al. [1] model in order to explore its predictive capacity by fitting it to experimental recordings of nystagmus waveforms and saccades; (ii) to develop a stochastic oculomotor model and examine the hypothesis that noise on the neural commands could be the cause of the behavioural characteristics measured from experimental nystagmus time series using nonlinear analysis techniques. In this work, two parameter estimation methods were developed, one for fitting the model to the experimental nystagmus waveforms and one to saccades. By using the former method, we successfully fitted the model to experimental nystagmus waveforms. This fit allowed to find the specific parameter values that set the model to generate these waveforms. The types of the waveforms that we successfully fitted were asymmetric pseudo-cycloid, jerk and jerk with extended foveation. The fit of other types of nystagmus waveforms were not examined in this work. Moreover, the results showed which waveforms the model can generate almost perfectly and the waveform characteristics of a number of jerk waveforms which it cannot exactly generate. These characteristics were on a specific type of jerk nystagmus waveforms with a very extreme fast phase. The latter parameter estimation method allowed us to explore whether the model can generate horizontal saccades of different amplitudes with the same behaviour as observed experimentally. The results suggest that the model can generate the experimental saccadic velocity profiles of different saccadic amplitudes. However, the results show that best fittings of the model to the experimental data are when different model parameter values were used for different saccadic amplitude. Our parameter estimation methods are based on multi-objective genetic algorithms (MOGA), which have the advantage of optimising biological models with a multi-objective, high-dimensional and complex search space. However, the integration of these models, for a wide range of parameter combinations, is very computationally intensive for a single central processing unit (CPU). To overcome this obstacle, we accelerated the parameter estimation method by utilising the parallel capabilities of a graphics processing unit (GPU). Depending of the GPU model, this could provide a speedup of 30 compared to a midrange CPU. The stochastic model that we developed is based on the Broomhead et al. [1] model, with signal dependent noise (SDN) and constant noise (CN) added to the neural commands. We fitted the stochastic model to saccades and jerk nystagmus waveforms. It was found that SDN and CN can cause similar variability to the local dimensions number of the oscillation as found in the experimental jerk nystagmus waveforms and in the case of saccade generation the saccadic variability recorded experimentally. However, there are small differences in the simulated behaviour compared to the nystagmus experimental data. We hypothesise that these could be caused by the inability of the model to simulate exactly key jerk waveform characteristics. Moreover, the differences between the simulations and the experimental nystagmus waveforms indicate that the proposed model requires further expansion, and this could include other oculomotor subsystem(s).

In models for networks of regulatory interactions in biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has lead to the use of continuous time 'switching' ordinary differential equations (ODEs), sometimes referred to as Glass networks. These Glass networks are the result of a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties in certain cases due to the discontinuities of the system, such as non-uniqueness. In this thesis we explore the use of 'ramp' functions as an alternative approximation to the sigmoid, which restores continuity to the ODE and removes the assumption of infinitely fast switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for producing a ramp system from a certain Glass network is given. Solutions are explored in two dimensions, and then in higher dimensions under two different restrictions. Periodic behaviour is explored in both cases using mappings between threshold boundaries. Limitations in these methods are explored, and a general proof of the existence of periodic solutions in negative feedback loops is given. / Graduate