Open Access

In this paper we describe methods to approximate functions and differential operators on adaptive sparse (dyadic) grids. We distinguish between several representations of a function on the sparse grid and we describe how finite difference (FD) operators can be applied to these representations. For general variable coefficient equations on sparse grids, genuine finite element (FE) discretizations are not feasible and FD operators allow an easier operator evaluation than the adapted FE operators. However, the structure of the FD operators is complex. With the aim to construct an efficient multigrid procedure, we analyze the structure of the discrete Laplacian in its hierarchical representation and show the relation between the full and the sparse grid case. The rather complex relations, that are expressed by scaling matrices for each separate coordinate direction, make us doubt about the possibility of constructing efficient preconditioners that show spectral equivalence. Hence, we question the possibility of constructing a natural multigrid algorithm with optimal O(N) efficiency. We conjecture that for the efficient solution of a general class of adaptive grid problems it is better to accept an additional condition for the dyadic grids (condition L) and to apply adaptive hp-discretization.

The paper presents a comprehensive model of a banking system that integrates network effects, bankruptcy costs, fire sales, and cross-holdings. For the integrated financial market we prove the existence of a price-payment equilibrium and design an algorithm for the computation of the greatest and the least equilibrium. The number of defaults corresponding to the greatest price-payment equilibrium is analyzed in several comparative case studies. These illustrate the individual and joint impact of interbank liabilities, bankruptcy costs, fire sales and cross-holdings on systemic risk. We study policy implications and regulatory instruments, including central bank guarantees and quantitative easing, the significance of last wills of financial institutions, and capital requirements.

Die Arbeit untersucht die Anwendung von maschinellem Lernen zur Erkennung von Aktivitäten von Schiffen anhand von AIS-Signalen. Das Automatic Identification System (AIS) wird von Schiffen genutzt, um Informationen über ihren Status in regelmäßigen Intervallen zu übertragen. Auf Basis der Daten wurden mithilfe von Machine Learning-Algorithmen aus der Gruppe der überwachten Klassifikationsalgorithmen Modelle gelernt, die in der Lage sind zu erkennen, welcher Aktivität ein Schiff zu einem Zeitpunkt nachgeht. Da das erfolgreiche Lernen eines Modells von einer sorgfältigen Datenvorbereitung abhängt, wurden verschiedene Verfahren zur Datenvorbereitung verwendet. Anschließend wurden verschiedene Algorithmen eingesetzt, darunter der Random Forest und k-NN, um Modelle zu lernen. Die Ergebnisse zeigen, dass die Aktivitäten mit einer Genauigkeit von bis zu 99% erkannt werden konnten, wenn in der Datenvorbereitung geeignete Verfahren gewählt wurden.

In the last years generative models have gained large public attention due to their high level of quality in generated images. In short, generative models learn a distribution from a finite number of samples and are able then to generate infinite other samples. This can be applied to image data. In the past generative models have not been able to generate realistic images, but nowadays the results are almost indistinguishable from real images. This work provides a comparative study of three generative models: Variational Autoencoder (VAE), Generative Adversarial Network (GAN) and Diffusion Models (DM). The goal is not to provide a definitive ranking indicating which one of them is the best, but to qualitatively and where possible quantitively decide which model is good with respect to a given criterion. Such criteria include realism, generalization and diversity, sampling, training difficulty, parameter efficiency, interpolating and inpainting capabilities, semantic editing as well as implementation difficulty. After a brief introduction of how each model works on the inside, they are compared against each other. The provided images help to see the differences among the models with respect to each criterion. To give a short outlook on the results of the comparison of the three models, DMs generate most realistic images. They seem to generalize best and have a high variation among the generated images. However, they are based on an iterative process, which makes them the slowest of the three models in terms of sample generation time. On the other hand, GANs and VAEs generate their samples using one single forward-pass. The images generated by GANs are comparable to the DM and the images from VAEs are blurry, which makes them less desirable in comparison to GANs or DMs. However, both the VAE and the GAN, stand out from the DMs with respect to the interpolations and semantic editing, as they have a latent space, which makes space-walks possible and the changes are not as chaotic as in the case of DMs. Furthermore, concept-vectors can be found, which transform a given image along a given feature while leaving other features and structures mostly unchanged, which is difficult to archive with DMs.

Im Rahmen dieser Bachelorarbeit werden zwei Skelettierungsalgorithmen für Baum-Punktwolken implementiert und miteinander verglichen. Darüber hinaus werden zwei Erweiterungen für einen der beiden Algorithmen präsentiert.