TY - THES A1 - Massing, André T1 - 1D-Reduktion thermal signifikanter Aderstränge in der Hyperthermie-Modellierung T1 - 1D-Reduction of Thermal Significant Blood Vessels in the Context of Hyperthermia-Modelling N2 - In dieser Arbeit wird ein neuer Ansatz zur Modellierung von thermal signifikanten Gefäßsträngen im Hyperthermie-Kontext betrachtet. Ausgehend von einer Konvektions-Diffusions-Gleichung wird durch Reskalierung des Massenflussterms eine Reduktion des Adergebietes auf eine 1D-Struktur erreicht. Nach numerischen Vorbetrachtungen wird die Grenzgleichung innerhalb einer verallgemeinerten Sobolev-Algebra formuliert. Die Untersuchung der Lösungsfamilie in klassischen Funktionenräumen zeigt, dass deren schwacher Grenzwert die Lösung der korrespondierenden Diffusions-Gleichung ist. Die Diskretisierung einer formalen Grenzgleichung mit Linienstromanteil stellt jedoch eine gute Approximation an die Diskretisierung des ursprünglichen Problems dar, wenn man die lokale Maschenweite an die Gefäßradien koppelt und bei erhöhtem Genauigkeitsbedarf auf ein vollständiges 3D-Modell umschaltet. KW - Finite Elemente Methode KW - Mehrskalen-Modellierung KW - Bio-Heat-Transer-Equation KW - Asymptotische Analysis KW - Finite Element Method KW - Multiscale-Modelling KW - Bio-Heat-Transer-Equation KW - Asymptotic Analysis Y1 - 2009 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11407 ER - TY - THES A1 - Helfmann, Luzie T1 - Stochastic Modeling of Interacting Agent Systems Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-71887 ER - TY - THES A1 - Niemann, Jan-Hendrik T1 - A Dynamic Memory Approach for Consensus Problems with Packet Loss Y1 - 2019 ER - TY - THES A1 - Sali, Adrian T1 - Coupling of Monodomain and Eikonal Models for Cardiac Electrophysiology N2 - The primary goal of this paper is to study the coupling of monodomain and eikonal models for the numerical simulation of cardiac electrophysiology. Eikonal models are nonlinear elliptic equations describing the excitation time of the cardiac tissue. They are often used as very fast approximations for monodomain or bidomain models - parabolic reaction-diffusion systems describing the excitation wavefront in terms of ionic currents. The excitation front is a thin region with high gradients, whereas excitation times vary over larger domains. Hence, eikonal equations can be solved on much coarser grids than monodomain equations. Moreover, as eikonal models are not time-dependent, no time integration is needed. Eikonal models are derived from monodomain models making additional assumptions and using certain approximations. While generally the approximation is rather good, several specific situations are not well captured by eikonal models. We consider coupling the two models, i.e. using the monodomain model in regions where more accurate results or the shape of the wavefront are needed, and the eikonal model in the remaining parts of the domain, where the excitation time is sufficient. Restricting the monodomain simulation to a small subdomain reduces the computational effort considerably. Numerical methods for the simulation of the individual models are presented, with the finite element method as the main ingredient. Coupling conditions as well as algorithms for implementing the coupling are explained. The approximation quality and efficiency of the coupled model is illustrated on simple geometries using an Aliev-Panfilov membrane model. Y1 - 2016 ER - TY - THES A1 - Gürler, A. T1 - Selection and flexible optimization of binding modes from conformation ensembles. Y1 - 2006 ER - TY - THES A1 - Marwitz, Ch. T1 - Numerische Berechnung von modifizierten Blochmoden in der Bandlücke periodischer Strukturen Y1 - 2005 UR - http://www.zib.de/Numerik/NanoOptics/publications/PubText/Marwitz05.pdf ER - TY - THES A1 - Massing, Andre T1 - 1D-Reduktion thermal signifikanter Aderstränge in der Hyperthermie-Modellierung Y1 - 2007 ER - TY - THES A1 - Kettner, Benjamin T1 - Ein Algorithmus zur prismatoidalen Diskretisierung von unbeschränkten Außenräumen in 2D und 3D unter Einhaltung von Nebenbedingungen Y1 - 2006 UR - http://www.zib.de/Numerik/NanoOptics/publications/PubText/Kettner06.pdf ER - TY - THES A1 - Pomplun, Jan T1 - Rigorous FEM-Simulation of Maxwell’s Equations for EUV-Lithography Y1 - 2006 UR - http://www.zib.de/Numerik/NanoOptics/publications/PubText/Pomplun06.pdf ER - TY - THES A1 - Pollok, T. T1 - Separabilität von Helmholtz- und Maxwellgleichungen über unbeschränkten heterogenen Gebieten Y1 - 2007 UR - http://www.zib.de/Numerik/NanoOptics/publications/PubText/Pollok07.pdf ER - TY - THES A1 - Schräer, Ingo T1 - Modellreduktion in der regionalen Hyperthermie mittels POD Y1 - 2008 CY - Fachbereich Mathematik ER - TY - THES A1 - Ruprecht, Daniel T1 - Konstruktion transparenter Randbedingungen für die Wellengleichung mit Hilfe der Polbedingung Y1 - 2005 ER - TY - THES A1 - Sim, I. T1 - Analysis und Numerik der Resonanzen des Schrödinger-Operators in 1D Y1 - 2005 ER - TY - THES A1 - Riemer, Alexander T1 - Accuracy, stability, convergence of rigorous thermodynamic sampling methods Y1 - 2006 ER - TY - THES A1 - Massing, Andre T1 - 1D-Reduktion thermal signifikanter Aderstränge in der Hyperthermie-Modellierung Y1 - 2007 CY - Fachbereich Mathematik ER - TY - THES A1 - Schräer, Ingo T1 - Modellreduktion in der Hyperthermie mittels POD Y1 - 2008 CY - Fachbereich Mathematik ER - TY - THES A1 - Scharkoi, O. T1 - Conformation Dynamics Design of Pain Relief Drugs Y1 - 2009 ER - TY - THES A1 - Durmaz, Vedat T1 - Theoretical Investigations on HBCD and PBCD Y1 - 2009 ER - TY - THES A1 - Brüggemann, Jo Andrea T1 - Numerical Modelling of the human menstrual cycle applied to ovulation prediction N2 - In this Master Thesis, the modelling process and the required biological background to adequately describe the hormonal dynamics of the human menstrual cycle are presented. The aim of this Master thesis project is to establish a mathematical routine that enables prediction of potentially fertile time windows. A system of four parameter dependent ordinary differential equations coupled with two algebraic equations is built, accounting for the main kinetic drivers of the human menstrual cycle { basal body temperature, estradiol, progesterone, luteinizing and follicle stimulating hormone. The key parameters are identifed using a nonlinear least squares approach to match the model to individual data. Aiming to solve the optimality problem arising, the Newton and Gauss-Newton method are examined and their local convergence theory is outlined. By the nature of the application, a globalised Gauss-Newton method seems more convenient. An adaptive trust region approach to determine the step lengths is chosen, which together with an linearly-implicit Euler extrapolation for the given differential algebraic equation system makes parameter estimation possible. The simulation results are discussed particularly in view of limitations and problems arising through the modelling approach. Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-46168 ER - TY - THES A1 - Knöchel, Jane T1 - Mathematische Modellierung metabolischer Einflussfaktoren auf den Milchkuhzyklus N2 - In den letzten Jahren ist die reproduktive Leistung von Milchkühen parallel zur steigenden Milchleistung drastisch gesunken. Um die zu Grunde liegenden biologischen Mechanismen besser zu verstehen, wurde in dieser Arbeit eine Modellerweiterung des vorhandenen Modells von Stötzel et al. (2011) um metabolische Einflussfaktoren angestrebt. Hierfür wurde zuerst aus veröffentlichten, experimentell gewonnen Erkenntnissen herausgearbeitet, wie die metabolischen Faktoren auf das periodische Zyklusverfahren der Milchkuh Einfluss nehmen. Daraus wurde ersichtlich, dass Glukose, Insulin und IGF-I die wichtigsten metabolischen Faktoren sind, aber lediglich IGF-I als dynamische Größe in das Modell eingeht. Zur Validierung des erweiterten Modells, wurden im Rahmen dieser Arbeit Szenarien aus der Literatur entwickelt, welche eine negative Energiebilanz oder den Effekt von Diätrestriktionen auf den Milchkuhzyklus simulieren sollten. Ferner wurden Messdaten für IGF, E2 und P4 von Kawashima et al. (2007) und Cerri et al. (2011) bereit gestellt. Das erweiterte Modell umfasst lediglich 8 Parameter mehr als das Ursprungsmodell, wobei gezeigt werden konnte, dass weder die Bedingung für die Lösbarkeit noch die Stabilität in den Anfangswerten eingebüßt wurde. Die Modellkurven konnten mittels des NLSCON erfolgreich an die bereitgestellten Messdaten für IGF, E2 und P4 angepasst werden. Anhand der Ergebnisse aus den Szenarien ließ sich zeigen, dass es teilweise gelingt über die metabolischen Einflussfaktoren eine negative Energiebilanz für unterernährte Kühe darzustellen. Die Rückführung in das periodische Zyklusverhalten verlief jedoch im Vergleich zu in vivo Beobachtungen zu schnell ab. Eine mögliche Erklärung hierfür ist ein zu geringer Einfluss von IGF-I auf das Modell. Darüber hinaus kam es trotz (stark) reduzierter LH-Konzentration zur Ovulation. Auf diesen Erkenntnissen aufbauend könnte in Zukunft die Follikelmodellierung modifiziert oder die Hillfunktion, die die Ovulation darstellt, überarbeitet werden. N2 - During the last few years there has been a drastic decline in the fertility of dairy cows. Therefore the goal of this work is to identify the underlying mechanisms for this problem by extending the model from Stötzel et al. (2011) to include the influence of metabolic factors. For this purpose the influence of the metabolic factors was determined based on published, experimentally obtained findings. The important metabolic factors were glucose, insulin and IGF-I, but only IGF-I was shown to be dynamic over the course of a cycle and therefore implemented into the model. To validate the chosen model extension we developed scenarios from the literature in order to simulate the impact of negative energy balance as well as the effect of diet restrictions on the bovine oestrous cycle. Furthermore we got measurement data from Kawashima et al. (2007) and Cerri et al. (2011) for IGF, E2 and P4. The new model has just 8 parameters more, while maintaining the same solvability and stability in the initial values. It was possible to fit the model with the use of NLSCON to the measurement data for IGF, E2 and P4. In addition the scenarios allowed us to show that the model is able to simulate a negative energy balance in parts. Merely the return into the cyclic behaviour was faster than observed in vivo. One possible explanation would be, that the influence of IGF-I on the model is not strong enough. The model showed a deficit while simulating the scenarios since a marked reduction of the LH concentration still led to ovulation. Thus future research could be done in this direction, where it would probably be necessary to modify the follicle function or the hill function responsible for ovulation. KW - Systembiologie Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-46170 N1 - Diplomarbeit ER - TY - THES A1 - Berg, Mascha T1 - Laufzeitoptimierung der Robusten Perron Cluster Analyse (PCCA+) Y1 - 2012 ER - TY - THES A1 - Lubkoll, Lars T1 - Optimal Control in Implant Shape Design KW - optimal control Y1 - 2011 ER - TY - THES A1 - Lehmann, Felix T1 - Inexaktheit in Newton-Lagrange-Verfahren für Optimierungsprobleme mit Elliptischen PDGL-Nebenbedingungen N2 - Bei der numerischen Lösung von Optimalsteuerungsproblemen mit elliptischen partiellen Differentialgleichungen als Nebenbedingung treten unvermeidlich Diskretisierungs- und Iterationsfehler auf. Man ist aus Aufwandsgründen daran interessiert die dabei entstehenden Fehler nicht sehr klein wählen zu müssen. In der Folge werden die linearisierten Nebenbedingungen in einem Composite-Step-Verfahren nicht exakt erfüllt. In dieser Arbeit wird der Einfluss dieser Ungenauigkeit auf das Konvergenzverhalten von Newton-Lagrange-Verfahren untersucht. Dabei sollen mehrere einschlägige lokale Konvergenzresultate diskutiert werden. Anschließend wird ein konkretes Composite-Step-Verfahren formuliert, in dem die Genauigkeit der inneren Iterationsverfahren adaptiv gesteuert werden kann. Am Ende der Arbeit wird an zwei Musterproblemen die hohe Übereinstimmung der analytischen Voraussagen und der tatsächlichen Performanz der dargestellten Methoden demonstriert. KW - PDE KW - adaptive Newton KW - inexact Newton-Lagrange Y1 - 2013 UR - urn:nbn:de:0297-zib-41972 ER - TY - THES A1 - Meisel, Kristin T1 - Mathematische Modellierung der Knochenremodellierung Y1 - 2012 ER - TY - THES A1 - Nielsen, Adam T1 - Von Femtosekunden zu Minuten Y1 - 2012 UR - http://www.zib.de/nielsen/Masterarbeit.pdf ER - TY - THES A1 - Zwingel, Oliver T1 - Die Cross-Entropy-Methode für stochastische Optimierungsprobleme Y1 - 2015 ER - TY - THES A1 - Lange, Christoph T1 - Approximating and Clustering Markov Chains via Kullback-Leibler-Divergence Y1 - 2015 ER - TY - THES A1 - Schöpke, Jessica T1 - Korrelation für lineare und nichtlineare Zusammenhänge Y1 - 2015 ER - TY - THES A1 - Breitbach, Paula T1 - Molecular Binding Kinetics of CYP P450 by Using the Infinitesimal Generator Approach Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-69076 ER - TY - THES A1 - von Lindheim, Johannes T1 - On Intrinsic Dimension Estimation and Minimal Diffusion Maps Embeddings of Point Clouds Y1 - 2018 UR - http://www.zib.de/ext-data/manifold-learning/ ER - TY - THES A1 - Bender, Wilhelm T1 - On a Novel Approach for Global Optimization of Non-Convex Problems N2 - In this work new approaches for global optimization are developed. We move from theoretical connections in complexity theory to very practical results in protein-ligand docking. In regard to the efficiency of the suggested optimizer, continuity methods are motivated, that yield an improved optimization method. Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-77074 ER - TY - THES A1 - Wulkow, Hanna T1 - Regularization of Elliptic Partial Differential Equations using Neural Networks N2 - This thesis presents a method for interpolating data using a neural network. The data is sparse and perturbed and is used as training data for a small neural network. For severely perturbed data, the network does not manage to find a smooth interpolation. But as the data resembles the solution to the one-dimensional and time-independent heat equation, the weak form of this PDE and subsequently its functional can be written down. If the functional is minimized, a solution to the weak form of the heat equation is found. The functional is now added to the traditional loss function of a neural network, the mean squared error between the network prediction and the given data, in order to smooth out fluctuations and interpolate between distanced grid points. This way, the network minimizes both the mean squared error and the functional, resulting in a smoother curve that can be used to predict u(x) for any grid point x. KW - elliptic partial differential equations KW - neural networks Y1 - 2020 ER - TY - THES A1 - Gaedke-Merzhäuser, Lisa T1 - Adaptive Multigrid Solvers for Space-Time Discretisations Y1 - 2019 ER - TY - THES A1 - Ribera Borrell, Enric T1 - From ergodic infinite-time to finite-time transition path theory Y1 - 2019 ER - TY - THES A1 - Baumann, Felix T1 - Impact of Contact Surfaces on the Death Estimation N2 - The determination of time of death is one of the central tasks in forensic medicine. A standard method of time of death estimation elies on matching temperature measurements of the corpse with a post-mortem cooling model. In addition to widely used empirical post-mortem models, modelling based on a precise mathematical simulation of the cooling process have been gaining popularity. The simulation based cooling models and the resulting time of death estimates dependon a large variety of parameters. These include hermal properties for different body tissue types, environmental conditions such as temperature and air flow, and the presence of clothing and coverings. In this thesis we focus on a specific arameter - the contact between corpse and underground - and investigate its influence on the time of death estimation. Resulting we aim to answer the question whether it is necessary to consider contact mechanics in the underlying mathematical cooling model. KW - time of death estimation KW - contact mechanics KW - heat equation KW - inverse problems Y1 - 2020 ER - TY - THES A1 - Semler, Phillip T1 - Mixed finite elements in cloth simulation N2 - This master thesis investigates the use and behaviour of a mixed finite element formulation for the simulation of garments. The garment is modelled as an isotropic shell and is related to its mid-surface by energetic degeneration. Based on this, an energy functional is constructed, which contains the deformation and the mid-surface vector as degree of freedom. It is then shown why this problem does not correspond to a saddle point problem, but to a non-convex energy minimization. The implementation of the energy minimization takes place with the ZIB-internal FE framework Kaskade7.4, whereby a geometric linear and different geometric non-linear problems are examined, whereby for a selected, non-linear example a comparison is made with an existing implementation on basis of Morley elements. The further evaluations include the analysis of the quantitative and qualitative results, the used solution method, the behaviour of the system energy as well as the used CPU time. KW - cloth simulation KW - shell elements KW - mixed finite elements Y1 - 2020 ER - TY - THES A1 - Kostré, Margarita T1 - Hybrid Models and Simulations of Reaction-Diffusion Processes N2 - In this thesis,we are interested in multiscale models for particle-based reaction diffusion (PBRD) simulations,where we focus on coupling particle-based simulations to macroscopic chemical reservoirs. These reservoirs are given by a mean concentration of chemical species that can vary in time and space. We model these reservoirs as reaction-diffusion partial differential equations (PDEs). The goal of this work is to achieve a mathematically consistent coupling between the PBRD simulations and the reaction-diffusion PDEs. Y1 - 2019 ER - TY - THES A1 - Zumbansen, Max T1 - Parameter estimation of multivalent processes on ITC datasets Y1 - 2019 ER - TY - THES A1 - Hoffmann, Lara T1 - Automating PCCA+ applied to Eye-Tracking Data Y1 - 2019 ER - TY - THES A1 - Schotte, Merlind T1 - Finite-Elemente-Techniken zur mechanischen Stabilitätsanalyse der Hyomandibula von Rochen N2 - Haie und Rochen sind die einzigen Wirbeltiere, die sich durch ein Knorpelskelett auszeichnen, das mit winzigen mineralisierten Elementen bedeckt ist, die sogenannten Tesserae. Obwohl dieser mineralisierte Knorpel seit hunderten von Millionen Jahren ein charakteristisches Merkmal dieser Tiere ist, ist der funktionelle Vorteil dieses Gewebestruktur bisher unklar geblieben. In dieser Masterarbeit wird diese Struktur mithilfe der Finite-Elemente-Methode(FEM) untersucht, wobei biologische Informationen zu den Muskeln und der geometrischen Gegebenheiten aus einem hochaufgelösten, kontrastgefärbten μCT-Scan einer Stechrochenhyomandibula verwendet werden. Für die Analyse werden Computer-Aided-Design(CAD)-Modelle und FE-Modelle konstruiert sowie vorhandene Methoden erweitert, um bei den Berechnungen die Charakteristika des biologischen Objektes bestmöglich zu berücksichtigen. Die Modelle bauen auf einer Segmentierung der Hyomandibula auf, wobei die mehreren tausend Tesserae in diesem Datensatz bereits voneinander isoliert sind. Für die Modellierung der Hyomandibula wird als erstes ein CAD-Modell konstruiert. Bei dieser Konstruktion wird eine Tesserae-Mittelfläche erstellt und im Anschluss nach innen und außen verschoben. Der nächste Schritt basiert auf dem Einbau von Zwischenwänden, die die Tesserae voneinander separieren. Auf Grundlage dieses Modells wird ein Tetraedergitter generiert, das die Basis für den zweiten Teil dieser Arbeit darstellt - die Berechnungen mit der FEM. Die wesentliche Aufgabe des zweiten Teils beinhaltet die Konstruktion eines FE-Modells, das in die lineare FEM einzuordnen ist. In dieses fließen die Randbedingungen und die Materialeigenschaften des Skelettelements ein. Mithilfe der FEM lassen sich die durch Krafteinwirkung auftretenden Verschiebungen und Spannungen der Hyomandibula ermitteln. Die Ergebnisse geben Rückschlüsse über das Verhalten des Materials, die Besonderheiten der Geometrie und damit über die Stabilität des Objektes. Der dritte Teil der Abschlussarbeit beschäftigt sich mit zusätzlichen Anpassungen und Änderungen der Modelle (CAD-Modell und FE-Modell), um diese Resultate mit dem Modell zu vergleichen, das der Biologie am nächsten kommt. Diese Anpassungen beinhalten das Variieren der Dicke der mineralisierten Schicht, die Konstellation des Materials und den Abstand zwischen den Tesserae. Die Ergebnisse zeigen, dass sowohl die Tesserae-Struktur als auch die zwischen den Tesserae liegenden Kollagenfasern vermutlich keinen Einfluss auf die Mechanik und dementsprechend die Stabilität haben. Jedoch lassen die unterschiedlichen Resultate im Fall verschieden gewählter Tesserae-Dicken annehmen, dass diese sehr wohl einen Einfluss besitzen. N2 - Sharks and rays are the only vertebrates characterized by a peculiar cartilage skeleton covered with minute mineralized tiles called tesserae. Although this tessellated cartilage has been a defining feature of this lineage for hundreds of millions of years, the functional advantage of this tissue design has remained obscure. In this Master’s thesis, the finite element method (FEM) is used to investigate the role of tiling in the skeleton, using biological information on the muscles and shape from a high-resolution, contrast-stained μCT-scan of a stingray hyomandibula. For this purpose, computer aided design (CAD) models and FE models are constructed and existing methods will be extended to take the best possible account of the characteristics of the biological object. The analysis builds from a segmentation of the hyomandibula, with the dataset’s several thousand tesserae pre-isolated from one another. For the modeling of the hyomandibula a CAD model is constructed. In this construction, a medial surface is created using the tesserae which is subsequently moved to the inside and outside. The next step is to integrate partition walls separating the tesserae elements. Based on this model, a tetrahedral grid is then generated, which is the content of the second part of this thesis, the calculation with the FEM. A finite element model is constructed including boundary conditions and material properties of the skeletal element to compute the results using the linear FEM theory. The FEM is used to determine the displacements and stresses resulting from biological forces acting on the hyomandibula. The results allow conclusions of the material behavior, geometric characteristics (tesselation) and therefore the object stability. Important contributions of this thesis are additional adjustments and changes to the models (CAD model and FE model) to make it more appropriate to study the biological data at hand. The tesserae thickness, the length of the collagenious fibres and the constellation of the material are the parameters, which are changed. In contrast to the tesserae thicknesses, the results suggest that both the arrangement of the tesserae and its interposed collagen fibers have no substantial influence on the mechanics. Y1 - 2018 ER - TY - THES A1 - Betz, Fridtjof T1 - Investigation and application of contour integral methods for resonance expansion in nanophotonics Y1 - 2020 ER - TY - THES A1 - Wulkow, Niklas T1 - Modelling the Spread of Innovations by a Markov Process in a Bayesian Framework Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-66869 ER - TY - THES A1 - Binkowski, Felix T1 - On the convergence behavior of spectral deferred correction methods for convection-diffusion equations N2 - Spectral deferred correction (SDC) Methoden, vorgestellt von Dutt, Greengard und Rokhlin in [1], sind iterative Verfahren zur numerischen Lösung von Anfangswertproblemen für gewöhnliche Differentialgleichungen. Wenn diese Methoden konvergieren, dann wird unter Verwendung von Zeitschrittverfahren niedriger Ordnung eine Kollokationslösung berechnet. Die Lösung von steifen Anfangswertproblemen ist eine relevante Problemstellung in der numerischen Mathematik. SDC-Methoden, speziell für steife Probleme, werden von Martin Weiser in [2] konstruiert. Die Theorie und die Experimente beziehen sich dabei auf Probleme, die aus räumlich semidiskretisierten Reaktions-Diffusions-Gleichungen entstehen. In dieser Arbeit werden die Ansätze aus [2] auf Konvektions-Diffusions-Gleichungen angewendet und das resultierende Konvergenzverhalten von SDC-Methoden untersucht. Basierend auf einem einfachen Konvektions-Diffusions-Operator, dessen spektrale Eigenschaften umfassend studiert werden, wird ein Schema zur Verbesserung dieses Verhaltens entwickelt. Numerische Experimente zeigen, dass eine Verbesserung der in [1] eingeführten SDC-Methoden möglich ist. Die Untersuchungen ergeben weiterhin, dass das auch für komplexere Konvektions-Diffusions-Probleme gilt. [1] Alok Dutt, Leslie Greengard, und Vladimir Rokhlin. “Spectral deferred correction methods for ordinary differential equations.” In: BIT 40.2 (2000), pp. 241–266. [2] Martin Weiser. “Faster SDC convergence on non-equidistant grids by DIRK sweeps.” In: BIT 55.4 (2015), pp. 1219–1241. Y1 - 2017 ER - TY - THES A1 - Sikorski, Alexander T1 - An Information-Theoretic Empirical Bayes Method and its Application to a Systems Biology Model N2 - This thesis covers the development and application of an empirical Bayes method to the problem of parameter estimation in systems biology. The goal was to provide a general and practical solution to the Bayesian inverse problem in the case of high dimensional parameter spaces making use of present cohort-data. We show that the maximum penalized likelihood estimator (MPLE) with information penalty is based on natural, information-theoretic considerations and admits the desirable property of transformation invariance. Due to its concavity, the objective function is computationally feasible and its mesh-free Monte-Carlo approximation enables its application to high-dimensional problems eluding the curse of dimensionality. We furthermore show how to apply the developed methods to a real world problem by the means of Markov chain Monte-Carlo sampling (MCMC), affirming its proficiency in a practical scenario. Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-65098 ER - TY - THES A1 - Gutsche, Philipp T1 - Convergence Study of the Fourier Modal Method for Nano-optical Scattering Problems in Comparison with the Finite Element Method N2 - Nano-optical scattering problems play an important role in our modern, technologically driven society. Computers, smartphones and all kinds of electronic devices are manufactured by the semiconductor industry which relies on production using photomasks as well as optical process control. The digital world, e.g. the world wide web, is based on optical interconnects and so-called quantum computers based on optics are supposed to be next generation computers. Moreover, global economic progress demands new and sustainable energy resources and one option is to make use of the power stored in optical radiation from the sun. Additionally, understanding fundamental physics such as the optical properties of asymmetric, or chiral, structures could promote future innovations in engineering. In order to understand and manipulate these kinds of processes, physics provides a well established model: the so-called Maxwell’s equations. Stated by James Clerk Maxwell in 1862, this description of the interaction of light and matter still provides a profound basis for the analysis of electromagnetic phenomena. However, real world problems cannot be calculated using simple mathematics. Rather, computer simulations are needed to obtain solutions of the physical model. Finding suitable methods to solve these problems opens up a wide variety of possibilities. On the one hand, there are methods which require long computing times. On the other hand, some algorithms depend on high memory usage. That is why the field of numerics deals with the question which method is optimally suited for specific problems. The aim of this work is to investigate the applicability of the so-called Fourier Modal Method (FMM) to nano-optical scattering problems in general. Since simple analytical solutions are non-existent for most recent physical problems, we use the Finite Element Method (FEM) to double-check performance of the FMM. Mathematics provide reliable procedures to control the errors of numerics using the FEM. Yet up to now it has not been possible to rigorously classify the quality of the Fourier Modal Method’s results. It is not fully understood whether the process of investing more and more computing resources yields more accurate results. So, we have to ask ourselves: does the numerical method invariably converge? In spite of this uncertainty when using the FMM, it is a well established method dating back to the 1980s. This numerical method has recently been used to optimize performance of solar cells [19] as well as to improve the optical properties of so-called single-photon sources [41] which are essential for quantum cryptography. The latter is a promising candidate to increase digital security and revolutionise cryptography techniques. Furthermore, with the help of the Fourier Modal Method an important issue in optics has been partly resolved: angular filtering of light was made possible by using a mirror which becomes transparent at a certain viewing angle [77]. In addition, an improved numerical technique to design so-called Photonic Crystal waveguides based on the FMM was developed recently [15]. Photonic Crystals are used in the fields of optical bio-sensing and for the construction of novel semiconductor devices. Moreover, approaches to link the FMM and the FEM try to combine advantages of both methods to obtain fast and accurate results [81]. These ideas are closely linked to the well-known concept of Domain Decomposition within the FEM [88]. Here, one possibility to couple domains is to use the scattering matrix formalism as it is done in the FMM. In the scope of this convergence study, we state Maxwell’s equations, particularly for periodic geometries. We describe two physical phenomena of nano-optics, namely chirality and opto-electrical coupling, and define the errors of our simulations. Afterwards, the two investigated methods are analysed with respect to their general properties and a way to unify modelling physics when using both algorithms is presented. With the help of various numerical experiments, we explore convergence characteristics of the FMM and draw conclusions about the ability of this approach to provide accurate results and, consequently, its potential for research on technological innovations. KW - Nano-optics KW - Fourier Modal Method (FMM) KW - Rigorous Coupled Wave Analysis (RCWA) KW - Finite Element Method (FEM) Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-56084 ER - TY - THES A1 - Pommer, Daniel T1 - Konditionsverbesserung von Markov-Modellen Y1 - 2013 ER - TY - THES A1 - Riesland, Mark Daniel T1 - Anwendbarkeit von Thermostaten in der moleküldynamischen Simulation Y1 - 2012 ER - TY - THES A1 - Josten, Jan-Martin T1 - Östrogenitäts-Approximation mittels physikochemischem Oberflächenvergleich Y1 - 2012 ER - TY - THES A1 - Lie, Han Cheng T1 - A Meshless Method for Conformational Analysis Y1 - 2012 ER - TY - THES A1 - Haack, Fiete T1 - Representative Spectral Clustering for Large Data Sets applied to Gene Expression Data Y1 - 2009 ER - TY - THES A1 - Günther, Matthias T1 - Implementation and verification of methods for the validation of knee joint kinematics KW - Biomechanics Y1 - 2014 ER - TY - THES A1 - Schuh, Hannah T1 - Identification of Anisotropies in Welding Seams Using Ultrasound Measurements Y1 - 2016 ER -