The FFT has been a staple in the field of mathematics and computer science for almost 60 years. And yet, it still endures as an efficient algorithm in a multitude of fields. However, as significant technical advances has been made since its inception the demand on methods constantly get higher and higher, and the FFT is seldom enough to solve the problems of this day and age on its own. So how does the FFT perform on its own with today’s standards? This thesis aims to use the FFT to create two algorithms, an Image Denoising algorithm and a Digital Watermarking algorithm respectively, and analyse the efficacy of the algorithms with today’s standards. The results showed that the FFT on its own competently tackles problems well, however with increased demands on the algorithms, the limitations of the FFT became apparent. This underscores the prevalent trend of integrating the FFT with other specializedmethods, ensuring its continued relevance in an era of continuously advancing technologicaldemands.

Computing the Fermi-Dirac operator is done through recursive polynomial expansions, using the SP2 and SP2 Acc algorithms. The Gateaux derivative is computed for both schemes by mapping the zeroth and first order matrices onto a block upper triangular matrix, which is implemented in Python using Numpy arrays to store full matrices and in C++ by first constructing a basic matrix library to use as blocks in a later created block upper triangular matrix library which only directly references two blocks in a 2 x 2 block matrix. Computations of the Fermi-Dirac operator were carried out on artificially created Hamiltonians to verify correct implementations, as well on real life examples of Fock matrices resulting from ergo calculations on water clusters(http://ergoscf.org/). It was found that the relative error in the first order response in the density matrix was not different when using SP2 Acc compared to when using SP2.

The thesis proposes to model the flow of gases in a room by solving the incompressible Navier-Stokes equations in three dimensions. The transportationof scalars such as temperature and concentration of CO2 is modeled by solving the advection-diffusion equation, and the effects of temperature on thevelocity is accounted for by employing the Boussinesq approximation. Theequations are solved numerically in Matlab by using the finite volume methodon a cubic domain with Cartesian coordinates. The code is verified by comparison with the well studied case of a lid-driven cavity. Different boundaryconditions are discussed and boundary conditions to model inlets and outlets along with radiators and windows are proposed. A simplified model of aroom is created and is shown to provide reasonable results for low Reynoldsnumbers (Re ≈ 200). It is also shown that the model is not efficient enoughto run simulations with higher Reynolds numbers (Re ≈ 20000) that resultfrom a realistic choice of parameters.Teknisk-naturvetenskapliga fakult

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This project aims to identify a sustainable way to construct and train machine learning models. A crucial factor in creating effective machine learning models lies in having access to vast amounts of data. However, this can pose a challenge due to the confidentiality and dispersion of data across various entities. Collecting all the data can thus become a security concern, as transmitting it to a centralized computing location may expose the data to security risks. One solution to this issue is federated learning, which utilizes locally trained AI models. Instead of transmitting data to a centralized computing location, this approach entails sending locally trained AI models and combining them into a global model. In recent years, a method called Model Inversion Attacks has emerged, revealing their potential risk in the context of extracting training data from trained AI models. This methodology potentially heightens the vulnerability of sending models instead of data, posing a security risk. In this project, various Model Inversion Attack methodologies will be examined to further understand the risk of sending models instead of data. The papers examined showed some results of extracting data from trained AI models, although they do not raise significant concerns. Nonetheless, future research in MIA may create security concerns when sending models between parties. Sending parts of the locally trained models to the global model effectively neutralizes the effectiveness of all the examined Model Inversion Attack studies. However, from the results presented in this project, it is evident that challenges persist when only sending parts of a trained model. The challenge was to construct a usable federated learning model while only sending parts of a trained model. To achieve a good federated learning model, several adjustments had to be made to the algorithm, which showed some promising results for the future of federated learning.

Workplaces operating around the clock often use rotating shift schedules that repeat throughout the year, except for the summer schedule, which is non-cyclic due to vacations resulting in reduced staff availability. This thesis presents the development of an integer programming model designed to combine the formulation of a year-long rotating workforce schedule with special adjustments for the summer months. Both hard and soft constraints, the latter represented as objective functions, are integrated into the model to ensure schedule feasibility and quality. A case study tests various objective functions with different prioritising orders and analyses their impacts on each other. This reveals that the optimisation order, whether prioritising the summer or rotating schedule first, does not significantly affect the outcome. Comparing the generated schedule to a manually created one demonstrates that the model produces a high-quality schedule, successfully meeting the annual rotating requirements and the special adjustments needed for the summer period. Additionally, the model offers significant time savings and increased efficiency in schedule creation.

This thesis studies the logistical problem of serving customer demands by electric vehicles, a currently important problem given the growing emphasis on sustainability in industry. In optimization, this is called the electric vehicle routing problem (E-VRP), and it is generally to computationally demanding to solve with a single comprehensive model. The method of this thesis is therefore a heuristic approach, in which the routing and the charging of the vehicles are done separately. The routing disregards the energy considerations, and the scheduling assigns routes and charging to a fleet of electric vehicles. When considering the charging separately, it can be model more realistically than what is usually done in E-VRP. A bibliographical review of both E-VRP and of electrical scheduling modeling is conducted, with a focus on which aspects are normally prioritized. Then two different models for the routing, and four different models for the scheduling are described. This routing-scheduling framework is subsequently tested on both artificial trial instances and a real-world problem. Promising results are observed in terms of runtime, with the ability to obtain good solutions on large instances within seconds. The real-world scenario was also solved quickly. Additionally, the results suggest a versatility of the routing-scheduling framework across various E-VRP variations.

The cause of unintentional initiations of energetic materials is an important area of study due to the risks that comes with storing energetic materials such as high explosives. The current models used to simulate the process of heating energetic materials by a hot metal fragment do not give reliable predictions. The objective of this thesis is to study the current models in order to get a better understanding of how to improve the accuracy of the simulations. The heat transfer in the fragment and energetic material is modeled using the heat equation and the reaction rates in the chemical decomposition of the energetic material are modeled using Arrhenius equations. This study shows the importance of accurately implementing the contact area and heat transfer coefficient between the fragment and the energetic material. The thermal conductivity has a significantly smaller affect on the initiation time compared to the heat transfer coefficient. Furthermore, the dimensions of the fragment affect the resulting simulations greatly, while the dimensions of the energetic material only does so for sufficiently small dimensions. / Orsaken till oavsiktliga initieringar av energetiska material är ett viktigt studieområde på grund av riskerna som följer med att lagra energiskt material, så som sprängämnen. De nuvarande modellerna som används för att simulera uppvärmningsprocessen av energetiska material med ett hett metallfragment ger inte tillförlitliga förutsägelser. Syftet med denna uppsats är att studera de nuvarande modellerna för att få en bättre förståelse för hur man kan förbättra noggrannheten i simuleringarna. Värmeöverföringen i fragmentet och det energetiska materialet modelleras med hjälp av värmeledningsekvationen och reaktionshastigheterna i den kemiska nedbrytningen av det energetiska materialet modelleras med hjälp av Arrhenius-ekvationer. Denna studie visar vikten av att korrekt implementera kontaktytan och värmeöverföringskoefficient mellan fragmentet och det energetiska materialet. Den termiska konduktiviteten har en betydligt mindre effekt på initieringstiden jämfört med värmeöverförings- koefficienten. Vidare så påverkar fragmentets dimensioner de resulterande simuleringarna i hög grad, medan dimensionerna av det energetiska materialet gör så endast för tillräckligt små dimensioner.

Underwater explosions are often modeled with Arbitrary Lagrangian-Eulerian (ALE) Finite Element Methods. The objective of this thesis is to validate the simulation method, with respect to the propagating shock wave. A two-dimensional axisymmetric model of a spherical TNT charge submerged in water is simulated using LS-DYNA. The explosive is modeled with the Burn Fraction technique and the Jones-Wilkins-Lee equation of state. Water is modeled as a non-viscous fluid, with the Grüneisen equation of state. The convergence for different mesh resolutions, the effect of different advection methods, and varied constants in the artificial viscosity are examined. Generally, the simulations agree well with empirical results, but the maximum pressure diminishes more rapidly with distance compared to experiments. The excessive dampening is most notable in the early stages of the propagation. Also, unexpected oscillations are observed near the discontinuity. The choice of advection scheme and constants in the artificial viscosity do not resolve the issues suggesting that other numerical techniques for treating the discontinuity should be considered. / Undervattensexplosioner simuleras ofta med ALE-baserade finita elementmetoder. Detta examensarbete avser att validera simuleringsmetoden med hänsyn till stötvågens utbredning i vattnet. En tvådimensionell axisymmetrisk modell av en sfärisk TNT-laddning nedsänkt i vatten simuleras i LS-DYNA. Laddningen modelleras med hjälp av brinnfraktioner och Jones-Wilkins-Lee tillståndsekvation. Vattnet modelleras som en inviskös fluid tillsammans med Grüneisens tillståndsekvation. Nätkonvergens, val av advektionsmetod och ändring av konstanter i den artificiella viskositeten studeras. Övergripande resultat stämmer väl överens med empirisk data, men stötvågens topptryck avtar fortare än väntat. Denna dämpning är tydligast i utredningens tidiga skeden. Dessutom observeras oväntade oscillationer kring stötvågens diskontinuerliga tryckprofil. Val av advektionsmetod och konstanter tillhörande artificiella viskositeten verkar ha liten betydelse för resultaten. En alternativ numerisk metod för behandling av stötvågens diskontinuitet bör implementeras.

Perfectly matched layer (PML) is a commonly used method of absorbing waves at a computational boundary for partial differential equation (PDE) problems. In this thesis, methods for improving the usability of implementations in Comsol Multiphysics is addressed. The study looks at complex coordinate stretching PMLs in the context of Helmholtz equation using the finite element method (FEM). For a PML to work it has to be set up properly with parameters that takes into account the properties of the problem. It is not always straight forward. Some theory behind PMLs is presented and experimentation on PML properties performed. Methods for PML optimization and adaption is presented. Currently, the way PMLs is applied in COMSOL Multiphysics requires the user to perform many tasks; setting up a geometry, meshing and choosing a suitable complex coordinate stretching. Using a so-called extra-dimension implementation it is possible to attach PMLs as boundary conditions in COMSOL Multiphysics. This simplifies for the user since the geometry and mesh is handled by the software. / Perfectly matched layer (PML) är en metod som ofta används för vågabsorbering vid randen för problem med partiella differentialekvationer (PDE). I det här examensarbetet presenteras metoder som förenklar användingen av PMLer i COMSOL Multiphysics. Studien kollar på PMLer baserade på komplex-koordinatsträckning med fokus på Helmholtz ekvation och finita elementmetoden (FEM). För att en PML ska fungera måste den sättas upp på rätt sätt med parametrar anpassade efter det givna problemet. Att göra detta är inte alltid enkelt. Teori presenteras och experiment på PMLer görs. Flera metoder för optimisering och adaption av PMLer presenteras. I nuläget kräver appliceringen av PMLer i COMSOL Multiphysics att användaren sätter upp en geometri, ett beräkningsnät och väljer den komplexa koordinatsträckningen. Genom att använda COMSOLs implementation av extra dimensioner är det möjligt att applicera PMLer som randvilkor. I en sådan implementation kan geometri och beräkningsnät skötas av mjukvaran vilket underlättar för användaren.