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Publication Type: search.filters.UBSTyp.Abschlussarbeit (Master)Start date: 2022Subject: search.filters.subject.510

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Now showing 1 - 8 of 8
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    Adjoint functors between crossed squares and [2,0]-simplicial groups
    (2025) Asiki, Natalia-Maria
    We consider the category [2,0]-SimpGrp of [2,0]-simplicial groups, the category CrSq of crossed squares and the category 2-CrMod of 2-crossed modules. Porter constructed a functor Sq from [2,0]-SimpGrp to CrSq. The category CrSq carries a transposition functor Tr. Conduché has constructed a total 2-crossed module functor To from CrSq to 2-CrMod and a reconstruction equivalence Rec from 2-CrMod to [2,0]-SimpGrp. We show that Sq is left-adjoint to the composite of Tr, To and Rec.
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    The resolution equivalence for n-complexes
    (2022) Klein, Veronika
    An injective resolution equivalence is constructed which generalizes the construction from classical homological algebra to n-complexes. The construction proceeds by showing that a functor in the converse direction is an equivalence. The injective resolution functor is defined to be its inverse.
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    Approximating solutions to non-linear partial differential equations with kernel methods
    (2026) Bacchetta, Milan
    This work studies a generalized interpolation approach for approximating nonlinear partial differential equations. It translates existing Gaussian-process literature on this method into the language of kernel methods and proves convergence of the resulting framework in a stronger topology and under milder assumptions on the RKHS. Further, a residual-greedy algorithm is developed and analyzed, in which interpolation points are chosen sequentially using an error indicator, and its convergence is established. The work concludes with numerical experiments, including a parameter study and a comparison between the residual-greedy strategy and a priori selected interpolation points.
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    Universelle Konstruktionen für Relationen
    (2024) Bechtel, Jonas
    In der Arbeit wird die Kategorie der Mengen und Relationen betrachtet, deren Objekte Mengen und deren Morphismen Relationen sind. Gegeben seien zwei Relationen, welche als Ursprungsmenge die gleiche Menge aufweisen. Für diese Relationen können je nach Situation die Eigenschaften linkstotal, rechtstotal, linkseindeutig und rechtseindeutig gefordert sein. Für die sich hieraus ergebenden Ausgangssituationen wurde untersucht, wann stets ein Pushout konstruiert werden kann und für welche Situationen es Gegenbeispiele zur Existenz eines Pushouts gibt. Existiert ein Pushout, so ist in der Arbeit auch die Pushout-Konstruktion angegeben. Des Weiteren wurde die Faktorisierung einer Relation über ihren Graphen betrachtet und hierfür eine universelle Konstruktion angegeben.
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    Simulating stochastic processes with variational quantum circuits
    (2022) Fink, Daniel
    Simulating future outcomes based on past observations is a key task in predictive modeling and has found application in many areas ranging from neuroscience to the modeling of financial markets. The classical provably optimal models for stationary stochastic processes are so-called ϵ-machines, which have the structure of a unifilar hidden Markov model and offer a minimal set of internal states. However, these models are not optimal in the quantum setting, i.e., when the models have access to quantum devices. The methods proposed so far for quantum predictive models rely either on the knowledge of an ϵ-machine, or on learning a classical representation thereof, which is memory inefficient since it requires exponentially many resources in the Markov order. Meanwhile, variational quantum algorithms (VQAs) are a promising approach for using near-term quantum devices to tackle problems arising from many different areas in science and technology. Within this work, we propose a VQA for learning quantum predictive models directly from data on a quantum computer. The learning algorithm is inspired by recent developments in the area of implicit generative modeling, where a kernel-based two-sample-test, called maximum mean discrepancy (MMD), is used as a cost function. A major challenge of learning predictive models is to ensure that arbitrarily many time steps can be simulated accurately. For this purpose, we propose a quantum post-processing step that yields a regularization term for the cost function and penalizes models with a large set of internal states. As a proof of concept, we apply the algorithm to a stationary stochastic process and show that the regularization leads to a small set of internal states and a constantly good simulation performance over multiple future time steps, measured in the Kullback-Leibler divergence and the total variation distance.
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    Quantum support vector machines of high-dimensional data for image classification problems
    (2023) Vikas Singh, Rajput
    This thesis presents a comprehensive investigation into the efficient utilization of Quantum Support Vector Machines (QSVMs) for image classification on high-dimensional data. The primary focus is on analyzing the standard MNIST dataset and the high-dimensional dataset provided by TRUMPF SE + Co. KG. To evaluate the performance of QSVMs against classical Support Vector Machines (SVMs) for high-dimensional data, a benchmarking framework is proposed. In the current Noisy Intermediate Scale Quantum (NISQ) era, classical preprocessing of the data is a crucial step to prepare the data for classification tasks using NISQ machines. Various dimensionality reduction techniques, such as principal component analysis (PCA), t-distributed stochastic neighbor embedding (tSNE), and convolutional autoencoders, are explored to preprocess the image datasets. Convolutional autoencoders are found to outperform other methods when calculating quantum kernels on a small dataset. Furthermore, the benchmarking framework systematically analyzes different quantum feature maps by varying hyperparameters, such as the number of qubits, the use of parameterized gates, the number of features encoded per qubit line, and the use of entanglement. Quantum feature maps demonstrate higher accuracy compared to classical feature maps for both TRUMPF and MNIST data. Among the feature maps, one using 𝑅𝑧 and 𝑅𝑦 gates with two features per qubit, without entanglement, achieves the highest accuracy. The study also reveals that increasing the number of qubits leads to improved accuracy for the real-world TRUMPF dataset. Additionally, the choice of the quantum kernel function significantly impacts classification results, with the projected type quantum kernel outperforming the fidelity type quantum kernel. Subsequently, the study examines the Kernel Target Alignment (KTA) optimization method to improve the pipeline. However, for the chosen feature map and dataset, KTA does not provide significant benefits. In summary, the results highlight the potential for achieving quantum advantage by optimizing all components of the quantum classifier framework. Selecting appropriate dimensionality reduction techniques, quantum feature maps, and quantum kernel methods is crucial for enhancing classification accuracy. Further research is needed to address challenges related to kernel optimization and fully leverage the capabilities of quantum computing in machine learning applications.
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    Portability of irregular randomized simulations to GPUs
    (2025) Melchinger, Andreas J.
    Graphics Processing Units (GPUs) have become fundamental to high-performance computing, offering substantial acceleration for data-parallel workloads. However, their architectural design - optimized for regular, uniform computations - poses significant challenges when applied to irregular randomized simulations, which are prevalent in fields such as particle physics, fluid dynamics, and molecular modeling. This thesis investigates the GPU-based acceleration of the Direct Simulation Monte Carlo (DSMC) algorithm, a stochastic method used to simulate rarefied gas dynamics and a representative case of irregular computation. Focusing on the DSMC implementation in the PICLas software framework, the work identifies key GPU challenges such as thread divergence, irregular memory access, and control flow variability. To address these, an abstraction model was developed and translated into a CUDA-based GPU prototype, incorporating optimization strategies including memory coalescence, hierarchical sorting, and divergence-aware sampling via Metropolis-Hastings. Experimental evaluations demonstrate significant performance improvements over both a naive GPU and sequential CPU baseline, particularly in complex simulation scenarios. The optimized GPU implementation achieved speed-up factors of up to four orders of magnitude for large-scale simulations, while preserving algorithmic correctness and scalability. These results confirm that, with carefully designed abstractions and architecture-aware optimizations, irregular randomized simulations like DSMC can be effectively ported to GPUs. This thesis contributes a validated prototype, generalizable optimization strategies, and insights into the performance boundaries of GPU-based stochastic simulation, paving the way for broader application in high-performance scientific computing.
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    On the Bisson-Tsemo model category of graphs
    (2022) Hess, Jannik
    Gezeigt wird, auf Grundlage von Bisson und Tsemo, dass die Kategorie der Graphen eine Modellkategorie darstellt. Es wird ein hinreichendes Kriterium dafür gezeigt, dass ein Graphmorphismus ein Quasiisomorphismus ist. Des Weiteren werden einige Beispiele und Gegenbeispiele konstruiert.