In this thesis, we study a numerical tool named multi-mesh method within the framework of the adaptive finite element method. The aim of this method is to minimize the size of the linear system to get the optimal performance of simulations. Multi-mesh methods are typically used in multi-physics problems, where more than one component is involved in the system. During the discretization of the weak formulation of partial differential equations, a finite-dimensional space associated with an independently refined mesh is assigned to each component respectively. The usage of independently refined meshes leads less degrees of freedom from a global point of view. To our best knowledge, the first multi-mesh method was presented at the beginning of the 21st Century. Similar techniques were announced by different mathematics researchers afterwards. But, due to some common restrictions, this method is not widely used in the field of numerical simulations. On one hand, only the case of two-mesh is taken into scientists\' consideration. But more than two components are common in multi-physics problems. Each is, in principle, allowed to be defined on an independent mesh. Besides that, the multi-mesh methods presented so far omit the possibility that coefficient function spaces live on the different meshes from the trial and test function spaces. As a ubiquitous numerical tool, the multi-mesh method should comprise the above circumstances. On the other hand, users are accustomed to improving the performance by taking the advantage of parallel resources rather than running simulations with the multi-mesh approach on one single processor, so it would be a pity if such an efficient method was only available in sequential. The multi-mesh method is actually used within local assembling process, which should not be conflict with parallelization. In this thesis, we present a general multi-mesh method without the limitation of the number of meshes used in the system, and it can be applied to parallel environments as well. Chapter 1 introduces the background knowledge of the adaptive finite element method and the pioneering work, on which this thesis is based. Then, the main idea of the multi-mesh method is formally derived and the detailed implementation is discussed in Chapter 2 and 3. In Chapter 4, applications, e.g. the multi-phase flow problem and the dendritic growth, are shown to prove that our method is superior in contrast to the standard single-mesh finite element method in terms of performance, while accuracy is not reduced.

Solutions were developed to solve the special photon transport problems. A respective Monte Carlo code were implemented. The photon transport calculations were made for simulation of light distribution in tissues, for 3D estimation of dosis brachytherapy source, for determination of scattering coincidences in PET and for solving of one special problem for ambient dosimetry. The developed calculation methods are based on a purely statistical approach and can therefore universal applied. The efficiency in respect of precision, numerical effectiveness, as well as memory requirement were optimised and verified by the calculations done. The modification and enhancement can be easy realised thanks to modular, object orientated implementation of the program and enables development of new application fields in physics and medicine. / Zur Lösung speziellen Probleme des Photonentransports wurden grundlegende Ansätze gewonnen und in einem Transportprogramm umgesetzt. Die Photonen-Transportrechnungen zur Simulation des Lichttransports in trüben Medien, für die dreidimensionale Dosisberechnung für interstitielle Brachytherapiequellen, für die Ermittlung der Streustrahlungsverteilungen in PET-Scannern und für die Lösung eines speziellen Problems der Umgebungsdosimetrie wurden durchgeführt. Die entwickelten Berechnungsmethoden basieren auf einer rein probabilistischen Herangehensweise und lassen sich deshalb universell verwenden. Ihre Leistungsfähigkeit hinsichtlich der Genauigkeit, der numerischen Effektivität sowie des Bedarfs an Rechenresourcen wurde optimiert und konnte durch die durchgeführten Berechnungen bestätigt werden. Durch den modularen, objektorientierten Programmaufbau sind Modifikationen und Erweiterungen relativ einfach durchzuführen. Das ermöglicht eine Erschließung von neuen Anwendungsgebieten in der Physik und Medizin.

Zur schnellen und sicheren Gepäckabfertigung ist eine Automatisierung der Gepäckprüfung zweckmäßig. Es wird eine mehrstufige Prüfung des Gepäcks durchgeführt. In der ersten Stufe werden vom Gepäckstück mehrere Röntgendurchstrahlungbilder angefertigt. Bei einem Teil der Gepäckstücke wird ein kritischer Bereich festgestellt, welcher mit einer zweiten Maschine genauer untersucht werden muss. Aus diesem Grund muss durch die zweite Maschine anhand nur eines weiteren Röntgendurchstrahlungsbildes erkannt werden, wo sich der kritische Bereich nach dem Transport durch das Förderband befindet. Zur Lösung dieses als Positionswiederfindung bezeichneten Problems wurde ein Verfahren entwickelt, welches auf einer Beziehung zwischen den Momenten der Grauwertverteilung des Durchstrahlungsbildes und den Momenten der Schwächungskoeffizentenverteilung des Objektes beruht. Das Verfahren wurde anhand von simulierten Beispielen getestet. / Automatic explosive detection is advantageous for fast and secure baggage control. The baggage is controlled by a multi level system. In level 1 the baggage will be inspected by a multi view x-ray radiography machine. The machine divides the baggage into secure and rejected bags and finds in the rejected bags a critical region for further inspection by a level 2 machine. For this reason the level 2 machine has to identify by only one additional x-ray view the new position of the critical region of the bag after the transport to the level 2 machine. To solve this problem a method was developed, which uses a relation between the moments of the grey-value of x-ray scan and the moments of the x-ray attenuation coefficient of the inspected object. The method was tested with simulated scans.

Be it in the weather forecast or while swimming in the Baltic Sea, in almost every aspect of every day life we are confronted with flow phenomena. A common model to describe the motion of viscous incompressible fluids are the Navier-Stokes equations. These equations are not only relevant in the field of physics, but they are also of great interest in a purely mathematical sense. One of the difficulties of the Navier-Stokes equations originates from a non-linear term. In this thesis, we consider the Oseen equations as a linearisation of the Navier-Stokes equations. We restrict ourselves to the two-dimensional case. Our domain will be the unit square. The aim of this thesis is to find a suitable numerical method to overcome known instabilities in discretising these equations. One instability arises due to layers of the analytical solution. Another instability comes from a divergence constraint, where one gets poor numerical accuracy when the irrotational part of the right-hand side of the equations is large. For the first cause, we investigate the layer behaviour of the analytical solution of the corresponding stream function of the problem. Assuming a solution decomposition into a smooth part and layer parts, we create layer-adapted meshes in Chapter 3. Using these meshes, we introduce a numerical method for equations whose solutions are of the assumed structure in Chapter 4. To reduce the instability caused by the divergence constraint, we add a grad-div stabilisation term to the standard Galerkin formulation. We consider Taylor-Hood elements and elements with a discontinous pressure space. We can show that there exists an error bound which is independent of our perturbation parameter and get information about the convergence rate of the method. Numerical experiments in Chapter 5 confirm our theoretical results.

Cell migration, the active translocation of cells is involved in various biological processes, e.g. development of tissues and organs, tumor invasion and wound healing. Cell migration behavior can be divided into two distinct classes: single cell migration and collective cell migration. Single cell migration describes the migration of cells without interaction with other cells in their environment. Collective cell migration is the joint, active movement of multiple cells, e.g. in the form of strands, cohorts or sheets which emerge as the result of individual cell-cell interactions. Collective cell migration can be observed during branching morphogenesis, vascular sprouting and embryogenesis. Experimental studies of single cell migration have been extensive. Collective cell migration is less well investigated due to more difficult experimental conditions than for single cell migration. Especially, experimentally identifying the impact of individual differences in cell phenotypes on individual cell migration behavior inside cell populations is challenging because the tracking of individual cell trajectories is required. In this thesis, a novel mathematical modeling approach, individual-based lattice-gas cellular automata (IB-LGCA), that allows to investigate the migratory behavior of individual cells inside migrating cell populations by enabling the tracking of individual cells is introduced. Additionally, stochastic differential equation (SDE) approximations of individual cell trajectories for IB-LGCA models are constructed. Such SDE approximations allow the analytical description of the trajectories of individual cells during single cell migration. For a complete analytical description of the trajectories of individual cell during collective cell migration the aforementioned SDE approximations alone are not sufficient. Analytical approximations of the time development of selected observables for the cell population have to be added. What observables have to be considered depends on the specific cell migration mechanisms that is to be modeled. Here, partial integro-differential equations (PIDE) that approximate the time evolution of the expected cell density distribution in IB-LGCA are constructed and coupled to SDE approximations of individual cell trajectories. Such coupled PIDE and SDE approximations provide an analytical description of the trajectories of individual cells in IB-LGCA with density-dependent cell-cell interactions. Finally, an IB-LGCA model and corresponding analytical approximations were applied to investigate the impact of changes in cell-cell and cell-ECM forces on the migration behavior of an individual, labeled cell inside a population of epithelial cells. Specifically, individual cell migration during the epithelial-mesenchymal transition (EMT) was considered. EMT is a change from epithelial to mesenchymal cell phenotype which is characterized by cells breaking adhesive bonds with surrounding epithelial cells and initiating individual migration along the extracellular matrix (ECM). During the EMT, a transition from collective to single cell migration occurs. EMT plays an important role during cancer progression, where it is believed to be linked to metastasis development. In the IB-LGCA model epithelial cells are characterized by balanced cell-cell and cell-ECM forces. The IB-LGCA model predicts that the balance between cell-cell and cell-ECM forces can be disturbed to some degree without being accompanied by a change in individual cell migration behavior. Only after the cell force balance has been strongly interrupted mesenchymal migration behavior is possible. The force threshold which separates epithelial and mesenchymal migration behavior in the IB-LGCA has been identified from the corresponding analytical approximation. The IB-LGCA model allows to obtain quantitative predictions about the role of cell forces during EMT which in the context of mathematical modeling of EMT is a novel approach.

We study cellular automata as discrete dynamical systems and in particular investigate under which conditions two cellular automata are topologically conjugate. Based on work of McKinsey, Tarski, Pierce and Head we introduce derivative algebras to study the topological structure of sofic shifts in dimension one. This allows us to classify periodic cellular automata on sofic shifts up to topological conjugacy based on the structure of their periodic points. We also get new conjugacy invariants in the general case. Based on a construction by Hanf and Halmos, we construct a pair of non-homeomorphic subshifts whose disjoint sums with themselves are homeomorphic. From this we can construct two cellular automata on homeomorphic state spaces for which all points have minimal period two, which are, however, not topologically conjugate. We apply our methods to classify the 256 elementary cellular automata with radius one over the binary alphabet up to topological conjugacy. By means of linear algebra over the field with two elements and identities between Fibonacci-polynomials we show that every conjugacy between rule 90 and rule 150 cannot have only a finite number of local rules. Finally, we look at the sequences of finite dynamical systems obtained by restricting cellular automata to spatially periodic points. If these sequences are termwise conjugate, we call the cellular automata conjugate on all tori. We then study the invariants under this notion of isomorphism. By means of an appropriately defined entropy, we can show that surjectivity is such an invariant.