We study the mean-field limit of a particle-based system modeling the behavior of many indistinguishable pedestrians as their number increases. The base model is a modified version of Helbing's social force model. In the mean-field limit, the time-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank-Nicholson method. The advection part is solved using a Lax-Wendroff-Leveque method or an upwind Backward Euler method depending on the advection speed. Moreover, we use multilevel Monte Carlo to estimate observables from the particle-based system. We discuss these numerical methods, and present numerical results showing the convergence of observables that were calculated using the particle-based model as the number of pedestrians increases to those calculated using the probability density function satisfying the Fokker-Planck equation.

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The mean-field game theory is the study of strategic decision making in very large populations of weakly interacting individuals. Mean-field games have been an active area of research in the last decade due to its increased significance in many scientific fields. The foundations of mean-field theory go back to the theory of statistical and quantum physics. One may describe mean-field games as a type of stochastic differential game for which the interaction between the players is of mean-field type, i.e the players are coupled via their empirical measure. It was proposed by Larsy and Lions and independently by Huang, Malhame, and Caines. Since then, the mean-field games have become a rapidly growing area of research and has been studied by many researchers. However, most of these studies were dedicated to diffusion-type games. The main purpose of this thesis is to extend the theory of mean-field games to jump case in both discrete and continuous state space. Jump processes are a very important tool in many areas of applications. Specifically, when modeling abrupt events appearing in real life. For instance, financial modeling (option pricing and risk management), networks (electricity and Banks) and statistics (for modeling and analyzing spatial data). The thesis consists of two papers and one technical report which will be submitted soon: In the first publication, we study the mean-field game in a finite state space where the dynamics of the indistinguishable agents is governed by a controlled continuous time Markov chain. We have studied the control problem for a representative agent in the linear quadratic setting. A dynamic programming approach has been used to drive the Hamilton Jacobi Bellman equation, consequently, the optimal strategy has been achieved. The main result is to show that the individual optimal strategies for the mean-field game system represent 1/N-Nash equilibrium for the approximating system of N agents. As a second article, we generalize the previous results to agents driven by a non-linear pure jump Markov processes in Euclidean space. Mathematically, this means working with linear operators in Banach spaces adapted to the integro-differential operators of jump type and with non-linear partial differential equations instead of working with linear transformations in Euclidean spaces as in the first work. As a by-product, a generalization for the Koopman operator has been presented. In this setting, we studied the control problem in a more general sense, i.e. the cost function is not necessarily of linear quadratic form. We showed that the resulting unique optimal control is of Lipschitz type. Furthermore, a fixed point argument is presented in order to construct the approximate Nash Equilibrium. In addition, we show that the rate of convergence will be of special order as a result of utilizing a non-linear pure jump Markov process. In a third paper, we develop our approach to treat a more realistic case from a modelling perspective. In this step, we assume that all players are subject to an additional common noise of Brownian type. We especially study the well-posedness and the regularity for a jump version of the stochastic kinetic equation. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash Equilibrium. This paper, unfortunately, has not been completely finished and it is still in preprint form. Hence, we have decided not to enclose it in the thesis. However, an outlook about the paper will be included.

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The mesopause region monthly mean winds and semidiurnal tidal amplitudes and phases over Central Europe in the height range between 85-105 km have been measured at Collm Observatory continuously since September 1982. The regular annual cycle of the semidiurnal tidal amplitudes show in all cases maximum values during late August and September. In contrast to that, in autumn 2002 no enhancement of the tidal amplitudes was measured, while the autumn tidal phase transition occurred unusually early. The unexpected behaviour of the semidiurnal tides seems to be connected with a very early autumn transition of the zonal prevailing winds. This suggests that in 2002 the zonal mean winds influence the tidal propagation in a different way than usual. / Die monatlich gemittelten Grundwinde und halbtägigen Gezeiten werden am Collm seit September 1982 im Höhenbereich zwischen 85-105 km gemessen. Der normale Jahresgang der halbtägigen Gezeiten zeigt maximale Amplituden im Winter und vor allem im Spätsommer/Herbst. Dieses Maximum ist im Jahre 2002 nicht zu verzeichnen. Dies ist begleitet von einer ungewöhnlich frühen Phasenänderung von der Sommer- zur Winterposition. Das Verhalten der Gezeiten scheint mit einer sehr frühen Änderung des zonalen Grundwindes vom Sommer- zum Winterregime verbunden zu sein, so dass die Anomalie der mittleren Zirkulation für die Gezeitenanomalie verantwortlich zu sein scheint.

This dissertation presents a series of studies related to the slurry mean residence time analysis and the pad-wafer contact characterization in Chemical Mechanical Planarization (CMP). The purpose of these studies is to further understand the fundamentals of CMP and to explore solutions to some of CMP's challenges. Mean residence time (MRT) is a widely used term that is mostly seen in classical chemical engineering reactor analysis. In a CMP process, the wafer-pad interface can be treated as a closed system reactor, and classical reactor theory can be applied to the slurry flow through the region. Slurry MRT represents the average time it takes for fresh incoming slurry to replace the existing slurry in the region bound between the pad and the wafer. Understanding the parameters that have an impact on MRT, and therefore removal rate, is critical to maintain tight specifications in the CMP process. In this dissertation, we proposed a novel slurry injection system (SIS) which efficiently introduced fresh slurry into the pad-wafer interface to reduce MRT. Results indicated that SIS exhibited lower slurry MRT and dispersion numbers but higher removal rates than the standard pad center slurry application by blocking the spent slurry and residual rinse water from re-entering the pad-wafer interface during polishing. Another study in this dissertation dealt with the effect of pad groove width on slurry MRT in the pad-wafer interface as well as slurry utilization efficiency (η). Three concentrically grooved pads with different groove widths were tested at different polishing pressures to experimentally determine the corresponding MRT using the residence time distribution (RTD) technique. Results showed that MRT and η increased significantly when the groove width increased from 300 to 600μm. On the other hand, when the groove width increased further to 900μm, MRT continued to increase while n remained constant. Results also indicated that MRT was reduced at a higher polishing pressure while η did not change significantly with pressure for all three pads. In the last study of this dissertation, the effect of pad surface micro-texture on removal rate during tungsten CMP was investigated. Two different conditioner discs ("Disc A" and "Disc B") were employed to generate different pad surface micro-textures during polishing. Results showed that "Disc B" generated consistently lower removal rates and coefficients of friction than "Disc A". To fundamentally elucidate the cause(s) of such differences, pad surface contact area and topography were analyzed using laser confocal microscopy. The comparison of the pad surface micro-texture analysis on pad surfaces conditioned by both discs indicated that "Disc A" generated a surface having a smaller abruptness (λ) and more solid contact area which resulted in a higher removal rate. In contrast, "Disc B" generated many large near-contact areas as a result of fractured and collapsed pore walls.

The basic concept to be used in studying the question of one-particle interpretations of relativistic wave equations is that of observables and operator representations that are different from the more usual classically motivated observables and representations. In particular, the concept of a mean-position observable will be used to determine to what extent the one-particle "problems" can be resolved.

Phonon transport analysis in nano- and micro-porous materials is critical to their energy-related applications. Assuming diffusive phonon scattering by pore edges, the lattice thermal conductivity can be predicted by modifying the bulk phonon mean free paths with the characteristic length of the nanoporous structure, i.e., the phonon mean free path (Lambda(pore)) for the pore-edge scattering of phonons. In previous studies (Jean et al., 2014), a Monte Carlo (MC) technique have been employed to extract geometry determined Lambda(pore) for nanoporous bulk materials with selected periods and porosities. In other studies (Minnich and Chen, 2007; Machrafi and Lebon, 2015), simple expressions have been proposed to compute Lambda(pore). However, some divergence can often be found between lattice thermal conductivities predicted by phonon MC simulations and by analytical models using Lambda(pore). In this work, the effective Lambda(pore) values are extracted by matching the frequency-dependent phonon MC simulations with the analytical model for nanoporous bulk Si. The obtained Lambda(pore) values are usually smaller than their analytical expressions. These new values are further confirmed by frequency-dependent phonon MC simulations on nano porous bulk Ge. By normalizing the volumetric surface area A and Lambda(pore) with the period length p, the same curve can be used for bulk materials with aligned cubic or spherical pores up to dimensionless p.A of 1.5. Available experimental data for nanoporous Si materials are further analyzed with new Lambda(pore) values. In practice, the proposed model can be employed for the thermal analysis of various nanoporous materials and thus replace the time-consuming phonon MC simulations.

Master of Science / Department of Statistics / Paul Nelson / Trimmed means are robust estimators of location for distributions having heavy tails. Theory and simulation indicate that little efficiency is lost under normality when using appropriately trimmed means and that their use with data from distributions with heavy tails can result in improved performance. This report uses the principle of equivariance applied to trimmed means sampled from a Cauchy distribution to form a discrepancy function of the data and parameters whose distribution is free of the unknown median and scale parameter. Quantiles of this discrepancy function are estimated via asymptotic normality and simulation and used to construct confidence intervals for the median of a Cauchy distribution. A nonparametric approach based on the distribution of order statistics is also used to construct confidence intervals. The performance of these intervals in terms of coverage rate and average length is investigated via simulation when the data are actually sampled from a Cauchy distribution and when sampling is from normal and logistic distributions. The intervals based on simulation estimation of the quantiles of the discrepancy function are shown to perform well across a range of sample sizes and trimming proportions when the data are actually sampled from a Cauchy distribution and to be relatively robust when sampling is from the normal and logistic distributions.

We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.