Modern decision making in the digital age is highly driven by the massive amount of data collected from different technologies and thus affects both individuals as well as economic businesses. The benefit of using these data and turning them into knowledge requires appropriate statistical models that describe the underlying observations well. Imposing a certain parametric statistical model goes along with the need of finding optimal parameters such that the model describes the data best. This often results in challenging mathematical optimization problems with respect to the model’s parameters which potentially involve covariance matrices. Positive definiteness of covariance matrices is required for many advanced statistical models and these constraints must be imposed for standard Euclidean nonlinear optimization methods which often results in a high computational effort. As Riemannian optimization techniques proved efficient to handle difficult matrix-valued geometric constraints, we consider optimization over the manifold of positive definite matrices to estimate parameters of statistical models. The statistical models treated in this thesis assume that the underlying data sets used for parameter fitting have a clustering structure which results in complex optimization problems. This motivates to use the intrinsic geometric structure of the parameter space. In this thesis, we analyze the appropriateness of Riemannian optimization over the manifold of positive definite matrices on two advanced statistical models. We establish important problem- specific Riemannian characteristics of the two problems and demonstrate the importance of exploiting the Riemannian geometry of covariance matrices based on numerical studies.

This study scrutinizes press photographs published during the first 6 weeks of the Russian War in Ukraine, beginning February 24th, 2022. Its objective is to shed light on the emotions evoked in Internet-savvy audiences. This empirical research aims to contribute to the understanding of emotional media effects that shape attitudes and actions of ordinary citizens. Main research questions are: What kind of empathic reactions are observed during the Q-sort study? Which visual patterns are relevant for which emotional evaluations and attributions? The assumption is that the evaluations and attributions of empathy are not random, but follow specific patterns. The empathic reactions are based on visual patterns which, in turn, influence the type of empathic reaction. The identification of specific categories for visual and emotional reaction patterns are arrived at in different methodological processes. Visual pattern categories were developed inductively, using the art history method of iconography-iconology to identify six distinct types of visual motifs in a final sample of 33 war photographs. The overarching categories for empathic reactions—empty empathy, vicarious traumatization and witnessing—were applied deductively, building on E. Ann Kaplan's pivotal distinctions. The main result of this research are three novel categories that combine visual patterns with empathic reaction patterns. The labels for these categories are a direct result of the Q-factorial analysis, interpreted through the lense of iconography-iconology. An exploratory nine-scale forced-choice Q-sort study (Nstimuli = 33) was implemented, followed by self-report interviews with a total of 25 participants [F = 16 (64%), M = 9 (36%), Mage = 26.4 years]. Results from this exploratory research include motivational statements on the meanings of war photography from semi-structured post-sort-interviews. The major result of this study are three types of visual patterns (“factors”) that govern distinct empathic reactions in participants: Factor 1 is “veiled empathy” with highest empathy being attributed to photos showing victims whose corpses or faces were veiled. Additional features of “veiled empathy” are a strong anti-politician bias and a heightened awareness of potential visual manipulation. Factor 2 is “mirrored empathy” with highest empathy attributions to photos displaying human suffering openly. Factor 3 focused on the context. It showed a proclivity for documentary style photography. This pattern ranked photos without clear contextualization lower in empathy than those photos displaying the fully contextualized setting. To the best of our knowledge, no study has tested empathic reactions to war photography empirically. In this respect, the study is novel, but also exploratory. Findings like the three patterns of visual empathy might be helpful for photo selection processes in journalism, for political decision-making, for the promotion of relief efforts, and for coping strategies in civil society to deal with the potentially numbing or traumatizing visual legacy of the War in Ukraine.

Coastal erosion describes the displacement of land caused by destructive sea waves, currents or tides. Due to the global climate change and associated phenomena such as melting polar ice caps and changing current patterns of the oceans, which result in rising sea levels or increased current velocities, the need for countermeasures is continuously increasing. Today, major efforts have been made to mitigate these effects using groins, breakwaters and various other structures. This thesis will find a novel approach to address this problem by applying shape optimization on the obstacles. Due to this reason, results of this thesis always contain the following three distinct aspects: The selected wave propagation model, i.e. the modeling of wave propagation towards the coastline, using various wave formulations, ranging from steady to unsteady descriptions, described from the Lagrangian or Eulerian viewpoint with all its specialties. More precisely, in the Eulerian setting is first a steady Helmholtz equation in the form of a scattering problem investigated and followed subsequently by shallow water equations, in classical form, equipped with porosity, sediment portability and further subtleties. Secondly, in a Lagrangian framework the Lagrangian shallow water equations form the center of interest. The chosen discretization, i.e. dependent on the nature and peculiarity of the constraining partial differential equation, we choose between finite elements in conjunction with a continuous Galerkin and discontinuous Galerkin method for investigations in the Eulerian description. In addition, the Lagrangian viewpoint offers itself for mesh-free, particle-based discretizations, where smoothed particle hydrodynamics are used. The method for shape optimization w.r.t. the obstacle’s shape over an appropriate cost function, constrained by the solution of the selected wave-propagation model. In this sense, we rely on a differentiate-then-discretize approach for free-form shape optimization in the Eulerian set-up, and reverse the order in Lagrangian computations.