Das Ziel dieser Masterarbeit war es, Erkenntnisse aus einer Arbeit von Matan Harel, Frank Mousset, Wojciech Samotij über des „Upper Tail Problem“ im Zufallsgraphen Gn,p auf den Gm,n zu übertragen. Hierbei konnte gezeigt werden, dass sich für eine analoge Parameterwahl die Wahrscheinlichkeit des Upper Tail Event auch im Gm,n sehr gut durch das Optimierungsproblem fX(d) = minfeG log(Nm) : G Kn and EG[X] (1 + d)E[X]g approximieren lässt. Für ein besseres Verständnis wird die Methodik zunächst ausführlich für komplette Graphen auf drei Knoten gezeigt, im Anschluss folgt der Beweis für allgemeine komplette Graphen. Dieser ist an analogen Stellen entsprechend knapp gehalten und konzentriert sich auf diejenigen Beweisteile, die im allgemeinen Fall einen elaborierteren Ansatz erfordern.

Magnetic particle imaging (MPI) is a novel tracer-based technique for medical imaging. The technique measures the response of the nanoparticles inside patients' blood stream in response to an external oscillating magnetic field. Based on this, the imaging process constructs the particles' spatial-dependent concentration, yielding a map of the blood vessels. Our aim is to determine a reliable model for the system function, a prerequisite for the imaging process. To avoid slow calibration, we use the nonlinear PDE Landau-Lifshitz-Gilbert equation, and study parameter identification in it. The inverse problem of parameter identification is investigated in two settings: a classical reduced version, and a new all-at-once version.

In this talk, we provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups.

Model-based simulations are an important tool for predicting physical phenomena. In the presence of uncertain physical parameters, which describe the system of interest, accurately assessing confidence in these results, e.g., performing Bayesian inference, can be computationally expensive, particularly when the uncertain physical parameters are spatially varying quantities. Not only are the corresponding high-fidelity (fine grid) simulations expensive, but commonly used statistical approaches, such as Markov chain Monte Carlo (MCMC), require an exceedingly high number of simulations. In this talk I will present a new scalable sampling method which complements the algorithmic framework of multilevel MCMC, where coarse grid simulations are used to inform the fine level proposal distribution, thereby accelerating MCMC. Specifically, using tools from algebraic multigrid, we form a (scalable) fine grid Gaussian random field realization from a fine grid proposal by combining Gaussian random fields sampled across multiple levels of discretization. In this talk, I will describe this new approach, corresponding theory, and numerical results when applied to a 3D subsurface flow application.

Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct distribution-free yet consistent tests of independence between random vectors. This is the problem addressed in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular examples. In a unified study, we derive a general asymptotic representation of center-outward rank-based test statistics under independence, extending to the multivariate setting the classical Hájek asymptotic representation results. This representation permits direct calculation of limiting null distributions and facilitates a local power analysis that provides strong support for the center-outward approach by establishing, for the first time, the nontrivial power of center-outward rank-based tests over root-n neighborhoods within the class of quadratic mean differentiable alternatives.