En anatomie computationnelle, on suppose que les formes d'organes sont issues des déformations d'un template commun. Les données peuvent être des images ou des surfaces d'organes, les déformations peuvent être des difféomorphismes. Pour estimer le template, on utilise souvent un algorithme appelé «max-max» qui minimise parmi tous les candidats, la somme des carrées des distances après recalage entre les données et le template candidat. Le recalage est l'étape de l'algorithme qui trouve la meilleure déformation pour passer d'une forme à une autre. Le but de cette thèse est d'étudier cet algorithme max-max d'un point de vue mathématique. En particulier, on prouve que cet algorithme est inconsistant à cause du bruit. Cela signifie que même avec un nombre infini de données et avec un algorithme de minimisation parfait, on estime le template original avec une erreur non nulle. Pour prouver l'inconsistance, on formalise l'estimation du template. On suppose que les déformations sont des éléments aléatoires d'un groupe qui agit sur l'espace des observations. L'algorithme étudié est interprété comme le calcul de la moyenne de Fréchet dans l'espace des observations quotienté par le groupe des déformations. Dans cette thèse, on prouve que l'inconsistance est dû à la contraction de la distance quotient par rapport à la distance dans l'espace des observations. De plus, on obtient un équivalent de biais de consistance en fonction du niveau de bruit. Ainsi, l'inconsistance est inévitable quand le niveau de bruit est suffisamment grand. / In computational anatomy, organ shapes are assumed to be deformation of a common template. The data can be organ images but also organ surfaces, and the deformations are often assumed to be diffeomorphisms. In order to estimate the template, one often uses the max-max algorithm which minimizes, among all the prospective templates, the sum of the squared distance after registration between the data and a prospective template. Registration is here the step of the algorithm which finds the best deformation between two shapes. The goal of this thesis is to study this template estimation method from a mathematically point of view. We prove in particular that this algorithm is inconsistent due to the noise. This means that even with an infinite number of data, and with a perfect minimization algorithm, one estimates the original template with an error. In order to prove inconsistency, we formalize the template estimation: deformations are assumed to be random elements of a group which acts on the space of observations. Besides, the studied algorithm is interpreted as the computation of the Fréchet mean in the space of observations quotiented by the group of deformations. In this thesis, we prove that the inconsistency comes from the contraction of the distance in the quotient space with respect to the distance in the space of observations. Besides, we obtained a Taylor expansion of the consistency bias with respect to the noise level. As a consequence, the inconsistency is unavoidable when the noise level is high.

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.

This thesis is devoted to the rigorous study of approximations for (multi-phase) mean curvature flow and related equations. We establish convergence towards weak solutions of the according geometric evolution equations in the BV-setting of finite perimeter sets. Our proofs are of variational nature in the sense that we use the gradient flow structure of (multi-phase) mean curvature flow. We study two classes of schemes, namely phase-field models and thresholding schemes. The starting point of our investigation is the fact that both, the Allen-Cahn Equation and the thresholding scheme, preserve this gradient flow structure. The Allen-Cahn Equation is a gradient flow itself, while the thresholding scheme is a minimizing movements scheme for an energy that Γ-converges to the total interfacial energy. In both cases we can incorporate external forces or a volume-constraint. In the spirit of the work of Luckhaus and Sturzenhecker (Calc. Var. Partial Differential Equations 3(2):253–271, 1995), our results are conditional in the sense that we assume the time-integrated energies to converge to those of the limit. Although this assumption is natural, it is not guaranteed by the a priori estimates at hand.

In dieser Arbeit wurde der Frage nachgegangen, inwieweit sich die Habitatansprüche von drei Brutvogelarten, der Offenlandart Feldlerche (Alauda arvensis), der Heckenart Neuntöter (Lanius collurio) und der Waldart Schwarzspecht (Dryocopus martius), mit Landschaftsstrukturmaßen darstellen lassen, und ob sich Landschaftsstrukturmaße für die Habitatmodellierung eignen. Basis für die Berechnung der Landschaftsstrukturmaße ist ein Flächenschema des IÖR-Monitors aus dem Jahr 2013, welches aus Daten des AFIS-ALKIS-ATKIS-Modells (AAA-Modells) aufgebaut wurde. Dieses Schema bietet redundanzfreie Flächennutzungsdaten für ganz Deutschland. Da es nur flächenhafte Elemente enthielt, wurde es um gepufferte linienhafte Elemente, genauer um Hecken, Baumreihen und Feldwege ergänzt. Die Artdaten stammen aus dem Monitoring häufiger Brutvögel (MhB), ebenfalls aus dem Jahr 2013. Die Berechnungen der Landschaftsstrukturmaße wurden mittels ArcGIS-Modellen durchgeführt. Für die Feldlerche und den Schwarzspecht wurden die Landschaftsstrukturmaße Mean Shape Index (MSI), Mean Patch Size (MPS), Anteil geeigneter Habitate (Percentage of Landscape, PLand), Total Core Area (TCA), Fläche geeigneter Biotope ohne anthropogene Störeinflüsse (Fl_ungest) und die Kantendichte der Landschaft (Edge Density, ED) berechnet. Für den Neuntöter sind es MSI, MPS, PLand, Fl_ungest, die Kantendichte und die Fläche geeigneter Gehölzbiotope und Hecken. Es wurde aufgezeigt, dass teilweise höchst signifikante lineare Zusammenhänge zwischen dem Vorkommen der drei Arten und den Landschaftsstrukturmaßen existieren. Die damit erklärten Streuungen der Brutpaarzahlen sind allerdings relativ gering. Das Bestimmtheitsmaß B oder R² der Regressionsgeraden beträgt für die Feldlerche maximal 0,285 bei der Fläche ungestörter Habitate, für den Schwarzspecht maximal 0,332 bei dem Anteil geeigneter Habitate und beim Neuntöter lediglich 0,038, ebenfalls für die Fläche ungestörter Habitate. Der Grund hierfür ist, dass die Arten Ansprüche an die Habitate stellen, die sich nicht mit Landschaftsstrukturmaßen erklären lassen. Die Modelle der multiplen linearen Regression sind ungeeignet, um Brutpaarzahlen der Arten vorherzusagen. Ohnehin war es nur für die Feldlerche möglich, ein solches Modell zu erstellen, das höhere Bestimmtheitsmaße aufweist als die einzelnen Landschaftsstrukturmaße. Deutlich bessere Ergebnisse wurden mit einem Modell erzielt, das die Eignung der Landschaft und ihrer Struktur als Habitat anhand einer Bewertungsmatrix beurteilt. Hier wurde bestimmt, wie hoch der Anteil besetzter Untersuchungsflächen an der Gesamtzahl von Untersuchungsflächen einer bestimmten Gesamtpunktzahl ist. Die Zusammenhänge zwischen Punktzahl und Anteil besetzter Flächen wurde mit teils nichtlinearen Regressionsfunktionen dargestellt. Der Anteil erklärter Abweichungen (R²) beträgt bei der Funktion der Feldlerche 97,1%, der des Schwarzspechts 88,5% und der des Neuntö-ters 49,3%.

The dynamics of demand responsive ride pooling (DRRP) systems is considered in a mean-field framework. The relevant dimensionless quantities determining the performance and viability of the system are identified. In the presence of an already established dominant market participant with comparable service quality (like, e.g., the private car), the mutual interaction of the actors (i.e., the customers sharing rides) by virtue of the route assignment algorithm gives rise to a discontinuous transition between two strongly different modes of operation. One of them represents the typical (unfavorable) performance of current ride pooling systems, while the other represents a new mode of operation in which virtually all customers use DRRP.

In practice, when one has many candidate variables as explanatory variables in multiple regression, there is always the possibility that variables that are important determinants of the response variable might be omitted from the model, while unimportant variables might be included. Both types of errors are important, and in this dissertation it is attempted to quantify the probabilities of these errors. A simulation study is reported in this dissertation. Different numbers of variables, i.e. p= 4 to 20 are assumed, and different sample sizes, i.e. n=0.5p, p, 2p, 4p. For each p the underlying model assumes that roughly half of the independent variables are actually correlated with the dependant variable and the other half not. The noise is ε~ N(0, σ2, where σ2, is set fixed. The data was simulated 10000 times for each combination of n and p using known underlying models and ε randomly selected from of a normal distribution. For this investigation the full model and forward selection regression are compared. The mean squared error of the estimated coefficient β(p) is determined from the true β of each n and p set. A full discussion, as well as graphs, is presented. / Dissertation (MSc)--University of Pretoria, 2011. / Statistics / unrestricted

Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statis- tical information on the population’s distribution. In this dissertation, we study the homogenization of a stationary first-order MFG and seek to find a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove existence and uniqueness of the solution to these limit problems. Next, we notice that the homogenized problem resembles the problem involving effective Hamiltoni- ans and Mather measures, which arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, and Aubry–Mather theory. Thus, we develop algorithms to solve the homogenized problem, the effective Hamil- tonians, and Mather measures. To do that, we construct the Hessian Riemannian flow. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather mea- sures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

This dissertation studies quantum decoherence in anharmonic oscillator systems to monitor and understand the way the systems evolve. It also explores methods to control the systems' evolution, and the effects of decoherence when applicable. We primarily do this by finding the time evolution of the systems using their Lie algebraic structures. We solve for a generalized Caldirola-Kanai Hamiltonian, and propose a general way to produce a desired evolution of the system. We apply the analysis to the effects of Dirac delta fluctuations in mass and frequency, both separately and simultaneously. We also numerically demonstrate control of the generalized Caldirola-Kanai system for the case of timed Gaussian fluctuations in the mass term. This is done in a way that can be applied to any system that is made up of a Lie algebra. We also explore the evolution of an optomechanical coupled mirror-laser system while maintaining a second order coupling. This system creates anharmonic effects that can produce cat states which can be used for quantum computing. We find that the decoherence in this system causes a rotational smearing effect in the Husimi function which, with the second order term added, causes rotational smearing after a squeezing effect. Finally, we also address the dynamic evolution and decoherence of an anharmonic oscillator with infinite coupling using the Born-Markov master equation. This is done by using the Lie algebraic structure of the Born-Markov master equation's superoperators when applying a strategic mean field approximation to maintain dynamic flexibility. The system is compared to the Born-Markov master equation for the harmonic oscillator, the regular anharmonic oscillator, and the dynamic double anharmonic oscillator. Throughout, Husimi plots are provided to visualize the dynamic decoherence of these systems.

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As demand for wind power continues to grow, it is becoming increasingly important to minimize the risk, characterized by the variance, that is associated with long-term power forecasts. This thesis investigated variance reduction in power forecasts from wind farm layout optimization.The problem was formulated as a multi-objective optimization one of maximizing mean-plant-power and minimizing variance. The ε−constraint method was used to solve the bi-objectiveproblem in a two-step optimization framework where two sequential optimizations are performed. The first is maximizing mean wind farm power alone and the second, minimizing variance with a constraint on the mean power which is the value from the first optimization. The results show that the variance in power estimates can be reduced by up to 30%, without sacrificing mean-plant-power for the different farm sizes and wind conditions studied. This reduction is attributed to the multi-modality of the design space which allows for unique solutions of high mean plant power at different power variances. Thus, wind farms can be designed to maximize power capture with greater confidence.