Masteruppsatser
Permanent URI for this collectionhttps://gupea-staging.ub.gu.se/handle/2077/28887
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Item Inversion Methods for Malign Melanoma Detection(2025-09-15) Kyhn, Georg; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThis thesis concerns the coefficient inverse problem from Maxwell’s equations, related to finding the electric permittivity and conductivity using backscattered time-dependent electrical fields. The aim is to reconstruct the permittivity and conductivity of Malignant Melanoma tissues, both in shape and in values. The forward problem is solved using a Domain Decomposition method, where both the Finite Element Method and the Finite Difference Method is used. The inverse problem is solved by the Conjugate Gradient Descent method, and numerical examples are performed in a homogeneous and non-homogeneous setting while adaptively refining a finite element mesh.Item The tip of the iceberg: using AI to identify toxic chemicals(2025-08-26) Edgren, Elin; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThis project was conducted to analyze the TRIDENT models described by Gustavsson et al. in the article Transformers enable accurate prediction of acute and chronic chemical toxicity in aquatic organisms [1]. The aims were to investigate the predictions made by the models, the relationship between the model’s chemical space, and the predictions they make. The methods used for the analyses were a combination of TRIDENT model predictions, modeling, and visualizations. The results of which were that there is a relationship between how accurately the TRIDENT models predict and the closeness the chemical has to the TRIDENT training data as well as the density of close neighbors in the training data. We also found that there are chemicals for which the TRIDENT models predict effective concentration values that are inconsistent with the measured value (label), possibly warranting further investigation of the chemical’s toxicity.Item Statistical Modeling for Prediction of Vehicle Usage(2025-06-10) Wiesner, Mathis; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperBattery electric vehicles (BEVs) are increasingly being adopted in all parts of the world. However, challenges such as maximizing the battery lifespan and efficiency, as well as the need for smart energy management systems remain. Newer generation BEVs have greater computational capabilities, enabling them to host smart functions that can address these challenges in new ways. A crucial component is understanding vehicle usage patterns, i.e., when the vehicle will be driven or charged. For example, accurate predictions of departure times allows to optimally perform battery thermal preconditioning, and predictions of arrival times can enable more intelligent decisions around household energy usage with new vehicle-to-grid capabilities. If a vehicle is expected to arrive during a period of high electricity prices, it may be advantageous to charge it using energy stored in a home battery. Knowing driving times also facilitates the scheduling of major software updates and the delivery of relevant traffic information. In this thesis, statistical models for predicting both single and recurrent driving events are developed using time-to-event methods. The models are based on the flexible Royston–Parmar framework, which allows us to obtain a parametric version of the famous Cox model and the Prentice-Williams- Peterson model that is used to model recurrent events. The models are trained and evaluated on historical driving data collected from a fleet of BEVs during the last three years, with predictive performance assessed by typical metrics such as the root mean squared error. The best-performing models achieve a mean absolute error of 60 minutes for departure time prediction and 100 minutes when predicting all usage events during the day. Furthermore, their performance is evaluated against that of traditional machine learning models.Item Minimising cycle time in an assembly line with human workers(2025-06-10) Lindmossen, Hannah; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperEfficient coordination is essential in automotive assembly lines, where multiple workers collaborate on a single product. This study focuses on optimising the cycle time of assembly processes involving human workers assigned specific tasks. A simulation routine was developed by enhancing an existing crowd-simulation algorithm to calculate and evaluate cycle time. Three optimisation models with various configurations were then implemented and assessed. The algorithms containing each of these three models utilise the simulation routine to identify performance bottlenecks and iteratively refine the solution. The simulation routine also generates the initial input data for the models. Using only this input with no iterations results in an estimated cycle time that serves as a lower bound on the optimal simulated cycle time. The simulation proved critical in accounting for small time disturbances, such as prolonged operations or collision avoidance, which significantly impact the cycle time. For the tested instance, all the solutions generated using the three algorithms achieved realistic cycle times, being approximately 11% above the derived initial lower bound. This is to be compared with the initial solutions, before any refining iterations were performed, which were approximately 18%, 27%, and at least 136%, respectively, above the initial lower bound. While all three algorithms demonstrated a potential for solving the type of problem addressed, further testing across diverse scenarios is required to generalise our findings. Future work could refine the optimisation models, adding more aspects necessary in reality, like workload and ergonomics, to enhance productivity in realworld assembly lines.Item Statistical Modeling of Insect Trajectories Using a Kalman Filter(2025-05-19) Wang, Chengjie; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThis thesis investigates the foraging behavior of the beetle Scarabaeus zambesianus using state-space models (SSM) and the Kalman filter. The study focuses on tracking the beetle’s trajectory from video footage, employing an iterative algorithm that combines image processing techniques with the Kalman filter to estimate the beetle’s location and velocity while minimizing measurement noise. The methodology involves converting video frames into a sequence of images and applying a Kalman filter to estimate the beetle’s true state (location and velocity) while accounting for measurement noise. An iterative tracking algorithm is developed, combining video processing techniques with the Kalman filter to continuously update the beetle’s location as it moves across frames. The algorithm begins by detecting the beetle’s initial position using a sliding window approach, which evaluates intensity changes between consecutive frames. The Kalman filter is then employed to refine the trajectory estimates by integrating predictions from the state-space model with noisy observations. The results show that the Kalman filter significantly improves trajectory accuracy, though challenges emerge when the beetle moves into areas with similar background colors, such as a dung pile. Sensitivity and residual analyses are conducted to evaluate the algorithm’s robustness and performance. Furthermore, linear regression models are used to analyze the relationship between the beetle’s velocity and its distance from the burrow, revealing a weak but statistically significant negative correlation. This research demonstrates the effectiveness of statistical modeling in studying insect behavior and provides a foundation for future work on insect movement dynamics.Item Dilation theory and som of its applications(2025-05-19) Mehrabi, Mohammadhossein; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThe purpose of this thesis is to explain some results in dilation theory. In this regard, it has been given the principal results and ideas on positive maps and dilation theory such as Nagy’s theorem, von Neumann’s inequality and some different versions of it, Stinespring’s theorem, dilation for commuting and non-commuting sets of operators, and Shilov boundaries. Also, Nevanlinna-Pick’s interpolation and an application in control theory has been considered.Item Discretization of the Interior Neumann Problem using Lusin Wavelets(2025-05-19) Jonsson, Jakob; Timlin, Emil; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperWe prove the existence of Hilbert space frames for the complex Hardy subspaces of L2(T), consisting of simple rational functions whose poles are arranged according to a Whitney partition. We also present parts of the classical existence theory for the Dirichlet and Neumann problems based on layer potentials. We use our frame to construct new methods—the Casazza-Christensen method (CC method) and the Whitney method of fundamental solutions (WMFS)—for numerically solving Laplace’s equation with Neumann boundary data on the unit disk. Our goal is to resolve problems in computing the solution to high accuracy near the boundary, that is typical for the boundary integral equation method (BIE method). The methods are implemented in MATLAB and their performances are analyzed. Both methods converge exponentially when the exact solution is a polynomial, but when the exact solution is a rational function with a pole just outside of the boundary, the convergence is considerably slower. For polynomial data, the accuracy of the new methods near the boundary is much better than for a simple implementation of the BIE method. Some partial theoretical results related to the convergence and conditioning of the method of fundamental solutions and the WMFS are proved.Item Bisc, a biclustering extension to scregclust(2025-04-14) Franzén, Sebastian; Birve, Filip; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperItem Similarity Problems: Which Groups Are Unitarizable?(2024-08-20) Westlund, Tim; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThis thesis covers some theory on similarity of group representations to unitary representations. We discuss the notion of amenability and give some classes of groups that are amenable. We then prove the Dixmier-Day theorem, that states that a locally compact group G is unitarizable if it is amenable. We also investigate the converse of this statement, which is still an open problem. We will give some statements where we make some assumptions on the similarity that are equivalent to amenability. We will also investigate when bounded algebra homomorphism A → B(H), where A is a C∗-algebra, are similar to a *-homomorphism. We will present connections between the unitarizability of groups and unitarizability of group C∗-algebras, and this will be useful for some results about the converse of the Dixmier-Day theorem. We will also investigate the notions of completely positive and completely bounded maps and prove Stinespring’s theorem for completely positive maps followed by Wittstock’s theorem for completely bounded maps. We then prove Haagerup’s theorem that states that unitarizability of homomorphisms is equivalent to the property of being completely bounded.Item Efficient Implementation of the 3D Helmholtz equation in C++/PETSc(2024-08-12) Köhle, René; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThe paper describes the comparison of different preconditioners for the solution of the Helmholtz equation with Krylov subspace methods in three dimensions. The solution of this equation has applications in microwave imaging and microwave hyperthermia for cancer detection and treatment. Due to the challenging nature of the Helmholtz equation, we employ a frequency and convergence analysis in two and three dimensions. We examine the sensitivity of the equation to various parameters and determine the effectiveness of various preconditioners. The use of finite difference approximation and preconditioned Krylov subspace methods allows for a convergence order of 2 to be achieved. The numerical results provide support for the aforementioned statement. Provided that there are no issues with resonant frequencies the desired convergence is achieved. This is applicable to the results obtained for different parameter functions and frequencies, as well as for two- and three-dimensional problems.Item Prime number races(2024-08-12) Elofsson, Carl; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperIn this thesis we investigate the behaviour of primes in arithmetic progressions, with a focus on the phenomenon known as Chebyshev’s bias. Under the assumption of the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, we prove that there is a bias towards quadratic non-residues. Additionally we extend the investigation to the setting of function fields. In the function field setting, we investigate the behaviour of prime polynomials in residue classes modulo a fixed monic polynomial. Moreover, we prove that for an irreducible polynomial m there is a bias towards quadratic non-residues modulo m.Item Point Process Learning estimation of bandwidth selection for the pair correlation function.(2024-08-12) Östling, Jens; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperIn this master thesis I use the newly developed method of Point Process Learning (PPL) to select the bandwidth used when estimating the Pair correlation function with kernel estimation. The selection of bandwidth when using kernel estimation is non trivial as different bandwidths give different results. Cronie et. al showed in their paper that the PPL method out performed the current state of the art when selecting the bandwidth for the intensity function. In this thesis I try the method when selecting the bandwidth for the pair correlation function. The results are promising but the scale of this pilot study is not large enough to give any conclusive results.Item Tensor products of Leavitt algebras(2024-08-12) Tinghammar Jönsson, Siri; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperWe outline the construction of the Leavitt algebras Ln, and prove that L2 ⊗L2 is not isomorphic to L2 by using Hochschild homology.Item Entropic Proximal Gradient Method for Generalized Optimal Transport Problems(2024-08-12) NIlsson, Sara; Gustav, Svensson; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperOptimal transport, a fundamental problem in applied mathematics, involves finding the most efficient way to move mass from multiple sources to multiple destinations. Previously known approaches employ entropic regularization combined with the Sinkhorn iterations, a technique known for its efficiency in solving large-scale optimal transport problems. This thesis presents a new method for solving generalized optimal transport problems using the entropic proximal gradient method. The method breaks down the complex problem into a sequence of standard optimal transport problems, solved by the Sinkhorn iterations. We provide theoretical foundations, including proof of convergence and termination criteria, along with a detailed implementation and numerical experiments showing the algorithm’s applicability. The results of this thesis may offer improvements in computational performance for generalized optimal transport problems, making it a valuable tool for applications in economics, machine learning, and other fields where optimal transport is utilized.Item Optimisation and Sampling Strategies for Bayesian Mixed-Effects Models: A comparative study(2024-08-12) Larsson, Vilhelm; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperIn this thesis we investigate, and compare, the efficiency of both stochastic optimisation using stochastic automatic differentiation (SAD) and Bayesian sampling using Metropolis- Hastings algorithms for parameter estimation of the Ornstein-Uhlenbeck process. We present the necessary, non-trivial, theory and methods required to complete such investigation. The results obtained would indicate that in most cases we tested the Bayesian sampling is preferable due to several reasons. Firstly it is a less time consuming procedure in our tests, secondly the estimates are often closer to the true parameters and finally that the variance of the method is in many cases smaller than the variance of the SAD procedure. However in some cases, for instance in cases of multiple mixed effects, the SAD methods yields preferable results. However we are unable to conclusively recommend any method due to the settings of our tests not being broad enough. Moreover these results can later be used to guide eventual further investigations into these methods, some avenues of which are suggested.Item Dilaton Theory and some of its Applications(2024-07-04) Mehrabi, Mohammadhossein; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThe purpose of this thesis is to explain some results in dilation theory. In this regard, it has been given the principal results and ideas on positive maps and dilation theory such as Nagy’s theorem, von Neumann’s inequality and some different versions of it, Stinespring’s theorem, dilation for commuting and non-commuting sets of operators, and Shilov boundaries. Also, Nevanlinna-Pick’s interpolation and an application in control theory has been considered.Item Quantum Graphs: Different Perspectives(2024-07-04) Stancevic, Dejan; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperIn this thesis, we investigate two different notions of quantum graphs. The first approach is through quantum adjacency matrices, while, the second approach is through bimodules over finite-dimensional algebras. We establish the equivalence between these approaches, following the work by M. Daws [Daw24]. Along the way, we explore the role of quantum graphs as operator systems within the context of quantum information theory, serving as an extension of confusability graphs in classical information theory. Furthermore, we explore the concept of quantum isomorphism between quantum graphs. We use the standard definition of quantum isomorphism, defined via the quantum adjacency matrix approach, to introduce an equivalent notion in the bimodules approach (employing the equivalence between the approaches). The goal of this work is to present and contribute to the growing knowledge of quantum graphs and isomorphisms between them.Item Data Augmentation for Point Process Learning(2024-07-03) Rost, Mathis; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperThis thesis introduces and evaluates ideas for the use of data augmentation in the area of Point Process Learning. Motivated by the regularizing effect of training with augmented data sets, we create a follow-up work to the paper ”A cross-validation-based statistical theory for point processes” by Cronie et al. [2023]. We develop methods for applying data augmentation to point process data. We discuss the possibilities of augmenting the existing process with additional data points generated by a noise process or by moving the already existing points in space. The developed methods are applied to common point process models, like the hard-core process and the area interaction process. The augmented data is then used for inference. The performed simulation study, where different discussed options are applied, shows promising results. The regularizing effect of data augmentation can be observed and thus motivates further investigation in to this topic.Item Point Process Learning estimation of bandwidth selection for the pair correlation function(2024-06-25) Östling, Jens; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperIn this master thesis I use the newly developed method of Point Process Learning (PPL) to select the bandwidth used when estimating the Pair correlation function with kernel estimation. The selection of bandwidth when using kernel estimation is non trivial as different bandwidths give different results. Cronie et. al showed in their paper that the PPL method out performed the current state of the art when selecting the bandwidth for the intensity function. In this thesis I try the method when selecting the bandwidth for the pair correlation function. The results are promising but the scale of this pilot study is not large enough to give any conclusive results.Item Low-lying zeroes of L-functions attached to modular forms(2024-03-22) Söderberg, Alf; University of Gothenburg/Department of Mathematical Science; Göteborgs universitet/Institutionen för matematiska vetenskaperWe study the family of L-functions attached to Hecke newforms of weight k and level N and their low-lying zeroes. First, we recall the Density Conjecture of Katz and Sarnak and how it predicts the behaviour of the low-lying zeroes of any natural family of L-functions. Then, we review some basic theory of modular forms as an appropriate background to the subsequent investigations. Next, we follow the article [ILS00] by Iwaniec, Luo and Sarnak in their treatment of the 1-level density of our family at hand. From them we recover that the Density Conjecture holds for bounded support of ϕ when kN --> ∞ and N is squarefree, conditional on the Generalized Riemann Hypothesis. Also, following Miller [Mil09] we find a term of lower order when k is fixed and N --> ∞ through the primes. Lastly, we study the 1-level density through the Ratios Conjecture. The prediction of the Ratios Conjecture allows any compact support of ϕ, as well as agreeing with the explicit calculations down to a power-saving error term.