In this thesis report, we investigate the difference in using physics-compatible elements and standard elements in the context of fluid flow through porous media. For this, we compare the H(div)-conforming Raviart-Thomas elements to Lagrange elements. Our model involves a channel flow over a porous media, resulting in a fluid and a porous subdomain. We distinguish between three approaches, as based on a research paper. The first is the one-domain approach called PE, which sets one equation on the entire domain. The other two approaches are two-domain approaches, which use different equations in the two subdomains, with interface conditions to couple them. The first of these is NSD, which couples Navier-Stokes with the Darcy equation. The second is NSF, which does the same but also incorporates the Forchheimer term, which models inertial effects.

In general, we were not able to make a completely successful model, as the PE case did not match the paper, and the results found when using Raviart-Thomas elements had some unexplained anomalies. We were also unable to carry out an accurate error analysis. We were able to see that Raviart-Thomas produced an exactly divergence-free solution, as expected from the theory, at the cost of more computing power and time needed.

Potato plant vitality is an important trait for good yield. Unfortunately, even with the best potato seed, vitality variation within a genotype is significant and it hurts both farmers and potato seed producers. HZPC and Averis Seeds hypothesize the existence of a link between the variation in the vitality of a plant and the field of production of the seed tuber. TU Delft, Utrecht University, HZPC and Averis Seeds joined forces to determine the existence and strength of this link in a multi-year
project funding this research. In this thesis we describe TU Delft’s contribution both in the measurement of plant vitality and the development of a model to predict plant vigor from chemical properties of the tuber...

The numerical solution of the Helmholtz equation presents significant challenges in computational mathematics and scientific computing, particularly for high-frequency problems in heterogeneous media. This dissertation addresses these challenges through the development of high-performance iterative methods, focusing on the critical balance between numerical efficiency and practical implementation on modern computing architectures.

The research is motivated by the growing computational demands in seismic imaging and other wave propagation applications, where increasing frequencies and larger domains necessitate more efficient solution strategies. Traditional approaches often struggle with the combined challenges of wavenumber-dependent convergence, pollution errors, and substantial memory requirements, particularly for three-dimensional problems in heterogeneous media.

This work presents a comprehensive framework for solving large-scale Helmholtz problems through matrix-free parallel implementations of preconditioned iterative methods. The framework combines Complex Shifted Laplace Preconditioner (CSLP) with advanced deflation techniques, implemented in a manner that eliminates the need for explicit matrix storage while maintaining computational efficiency. A key innovation is the development of matrix-free implementations for higher-order deflation methods combined with the CSLP preconditioner, achieved through carefully designed re-discretization schemes that preserve the advantages of Galerkin coarsening.

The methodology progresses from two-dimensional implementations to fully three-dimensional frameworks, incorporating increasingly sophisticated preconditioning techniques. A significant achievement is the development of a matrix-free parallel multilevel deflation preconditioner that exhibits near wavenumber-independent convergence while maintaining excellent parallel scalability. The implementation utilizes a hybrid MPI+OpenMP parallelization strategy, effectively addressing both computational and memory challenges in extreme-scale scenarios.

Extensive numerical experiments validate the effectiveness of these methods across a range of problem types, from academic test cases to industrial-scale applications. Notably, the framework successfully resolves a challenging seismic model, involving approximately 3.8 billion degrees of freedom, while achieving 86\% parallel efficiency when scaling to 2304 CPU cores. This demonstration of practical viability for large-scale heterogeneous problems represents a significant advance in computational capabilities for seismic imaging applications.

The research makes several fundamental contributions to the field of numerical analysis and scientific computing. First, it establishes new approaches for matrix-free implementation of state-of-the-art preconditioners, significantly reducing memory requirements while maintaining numerical efficiency. Second, it demonstrates the achievement of close-to wavenumber-independent convergence through carefully designed deflation strategies in a parallel computing environment. Third, it provides a comprehensive framework for solving extreme-scale Helmholtz problems that combines numerical robustness with practical applicability.

The methodologies developed in this work contribute to the broader field of scientific computing, demonstrating how careful algorithm design, combined with modern computing architectures, can address previously intractable problems in wave propagation modeling.

This master's thesis provides information on the subject of option pricing theory. Moreover, this topic is linked with sustainable Finance, which is essential in the battle against climate change. Green financing is a growing phenomenon, and a green bond is only a relatively new example of a green financial derivative. A three-dimensional PDE is at hand to determine the price of a green bond (also called a coupon value), but this PDE can only be solved numerically. This master thesis aims to determine the stability of certain numerical schemes that can be used to find a solution to the PDE. We will use both forward and backward differences to derive the numerical schemes. After the derivation process, a Von Neumann analysis is performed to conclude whether or not the schemes are stable. The research reveals that most of the numerical schemes are not stable. Amongst others, this is caused by the positive value of the interest rate $r$, the value of the carbon price, and the absence of damping factors. In some numerical schemes, the amplification factor is larger than one, but not by much. In other schemes, we can reduce the amplification factor to increase the stability. This means that the numerical schemes can still be used to find a reasonable value for a green bond.

Understanding multiphase flows is critical in nuclear engineering, particularly for processes such as coolant dynamics in nuclear reactors and safety scenario analyses involving different fluid phases. Numerical simulations are a valuable tool for studying these phenomena, especially when experimental approaches are impractical due to cost or safety concerns. While direct numerical simulations (DNS) offer detailed insights, their computational expense makes them impractical for turbulent flows, necessitating the use of turbulence models for efficiency.

This thesis introduces a novel machine learning framework designed to improve Reynolds-averaged Navier-Stokes models in turbulent stratified gas-liquid flows while employing the Boussinesq approximation. The framework encompasses two methods for turbulent viscosity field inversion and introduces correction terms in the turbulence model equations to ensure an accurate prediction of the turbulent viscosity field. Through sparse symbolic regression, the framework consistently discovers models that improve the accuracy of the baseline RANS model, even in untrained flow scenarios, though further testing is needed for varied flow regimes.

Key findings include the superior performance of sparse symbolic regression models over neural network (NN) models in improving the baseline RANS model accuracy. Notably, LASSO and elastic net techniques yielded the most successful models, significantly reducing baseline errors. However, these models did not surpass the Egorov damping approach in terms of accuracy, indicating the need for further refinement.

The developed models were numerically stable and robust, which is important for practical use. However, a main limitation is that the models' accuracy during training did not always correlate with the results when coupled with the RANS equations. Moreover, data from more varied flow conditions is needed to properly assess the generalizability of the models.

Overall, this research highlights the potential of data-driven turbulence modelling to enhance two-phase flow simulations, marking a significant step forward while also identifying areas for future improvement and exploration.

This report focuses on the process of aberration coefficient retrieval for measurements of a
Scanning Electron Microscope (SEM). The main goal is the verification of a modified aberration coefficient retrieval method described in an article by Uno et al. (2005) [1]. This method
first extracts spot profiles, represented by Point Spread Functions (PSFs), from a through-focus
series of SEM images using Fourier analysis. The size and shape of a PSF characterize the
performance of a SEM. Afterward, it determines characteristics from these PSFs to obtain
aberration coefficients.
The process consisted of three parts: probe shape extraction, PSF analysis, and aberration coefficient calculation, which included a consistency test. The original method was changed to
include clarification for the use of complex conjugates of functions in the Fourier domain, the
proper statistical definition of standard deviation as characteristic of the width of a PSF, the
curvature definition used in the method, and contour tracing around a PSF to exclude points
outside the contour. These adaptations were made to clarify the methods described in the reference article.
This modified method was tested against generated PSFs that used the wave-optical description
of five different aberrations. By increasing the value of these individual simulated aberration
coefficients C
′
n with factors s ∈ [1, S], S ∈ N, the consistency of this retrieval method could be
determined by calculating the ratio Q between simulated and retrieved aberration coefficients
as a function of the factor s.
The spot shape extraction method was implemented and assessed successfully, as the shapes of
the generated PSFs were successfully retrieved, and the use of complex conjugates of functions
in the Fourier domain resulted in rotated spot shapes.
Contour tracing around the retrieved PSFs had mixed results. For all aberrations, it presented
a good measure of the spot shape. The implementation of an algorithm that excluded points
pj outside the contour was not accurate for one test aberration: two-fold astigmatism A2. The
contour tracing algorithm excludes important parts of the shapes of the PSFs. From research
into the characteristics asymmetry µ, width σ, and curvature ρ it could be inferred that the first
two characteristics µ and σ represent the mean and standard deviation of discrete probability
mass functions. Curvature ρ has no such direct statistical equivalent, it was concluded however
that this characteristic presented a good measure for the effect of third-order spherical aberration in a PSF.
Results could not indicate that this wave-optical description of aberrations is consistent with
all of the coefficients retrieved using the modified method of aberration coefficient retrieval
as the ratio Q remained consistent for defocus Cdf , three-fold astigmatism A2 and third-order
spherical aberration Cs as a function of the factors s.
Continued research into this method of aberration retrieval is recommended, specifically in the
definitions of the characteristic curvature ρ, more advanced contour tracing, and the digitizations of aberration coefficients. More assessment and modification of this method of aberration
coefficient retrieval can allow for a more comprehensive understanding of the method, which
will eventually contribute to an automatic aberration corrector that could speed up SEM measurements significantly.

Achtergrond: Migraine is een veel voorkomende hoofdpijnaandoening waar neuropeptiden een belangrijke rol lijken te spelen. Dit onderzoek richt zich op het begrijpen van de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming bij vrouwen met en zonder migraine. Hiervoor worden LSCI en LDPI metingen getransformeerd met een Fourier transformatie. Doel: Het doel van dit onderzoek is om verschillen te identificeren in de invloed van zenuwen en gladde spiercellen op de bloeddoorstroming tussen vrouwen met en zonder migraine. Methode: LSCI metingen van de VASCULAR studie worden gebruikt. Fourier analyse wordt toegepast om de invloed van zenuwen en gladde spiercellen te vinden in verschillende fasen: baseline, piek en plateau. Vergelijkingen worden gemaakt tussen vrouwen met migraine (met en zonder aura) en controlepersonen zonder migraine. Resultaten: In de baseline fase zijn minimale verschillen in de invloed van de zenuwen en de gladde spiercellen gevonden tussen vrouwen met en zonder migraine. Tijdens de piek fase lijkt er een verhoogde invloed van de zenuwen en de gladde spiercellen op de bloeddoorstroming bij vrouwen zonder migraine. Conclusie: In ons onderzoek zijn geen evidente verschillen gevonden. Wel lijken er enkele trends in de data te zitten. Verdere studies zijn nodig om deze bevindingen te bevestigen en uit te breiden.

District heating leverages centralised, high efficiency combined heat and power (CHP) systems. It uses waste heat to lower energy consumption and reduce greenhouse emissions. The system also supports renewable energy sources like geothermal and biomass, providing a sustainable heating alternative.

This report examines a nonlinear network of pressure and flow challenges. It focuses on enhancing the Newton-Raphson method and refining direct solving techniques for a single time-step. As net- works grow in complexity, efficient and effective solutions become crucial. Various strategies to speed up the Newton-Raphson algorithm in Gradyent’s heat network solver are discussed, where derivative calculations are straightforward. Both direct and iterative methods to improve the algorithm’s steps are explored. The effectiveness of these enhancements is tested and evaluated across networks of different sizes.sc

Implied volatility is critical in financial markets, especially for option pricing. Traditional methods for its calculation sometimes are not well suited to some scenarios. Recent developments in neural networks have provided more efficient alternatives.

Leveraging advances in quantum computing, our research introduces quantum neural networks for computing implied volatility, assessing the feasibility and characteristics of this novel approach. We focus on two quantum neural network architectures: Dissipative Quantum Neural Networks (DQNN) and Parameterized Variational Quantum Circuits (PVQNN). DQNN, similar to classical neural networks in structure and training ease, faces challenges with quantum state outputs and data decoding, impacting performance negatively. Besides, limited by the reliance on network output states at each layer, DQNN faces challenges in implementation with the current state of quantum hardware.

In contrast, PVQNN offers a more promising solution. Compared to DQNN, PVQNN requires fewer qubits, can apply traditional optimizers to train the model, and can run on NISQ devices. This research thoroughly examines various aspects influencing PVQNN's performance, including training data characteristics, data re-uploading technology, network size, data encoding methods, and quantum circuit design. The selected PVQNN model can achieve high accuracy in implied volatility computation with $R^2$ of approximately 0.999. In addition, we find that the PVQNN can obtain satisfactory results even with limited training data, setting it apart from traditional neural networks.

This thesis not only adopts a new model to compute implied volatility but also deepens the understanding of quantum neural networks in financial modeling. However, due to resource constraints, our experiments are conducted in simulations on traditional computers, and thus our study focuses mainly on the expressive power of QNNs rather than their operational efficiency.

This thesis presents a novel approach to the pricing of green bonds, a growing segment in financial markets with an emphasis on environmental sustainability. Unlike traditional financial instruments, green bonds uniquely incorporate environmental considerations, particularly carbon price (c_t), along with traditional factors like the short rate (r_t), into their valuation. This integration is increasingly relevant in today’s economy, reflecting a shift towards sustainable finance. The core of this research involves applying advanced numerical methods, including the Finite Difference Method, Crank-Nicolson discretization, GMRES and Bi-CGSTAB, in order to develop and analyze pricing models for both green and conventional bonds. The study aims to assess how environmental factors impact the efficiency of these numerical techniques and to compare the outcomes with conventional bond models. The research reveals that green bonds, compared to conventional bonds, present unique numerical challenges, notably requiring more iterations for convergence in iterative methods GMRES and Bi-CGSTAB because of the high carbon price volatility (σc) and the ’Greenium’ phenomenon. Moreover, the comparative analysis showed that while Bi-CGSTAB outperforms GMRES in the green bond model, the opposite is true for conventional bonds. This study not only contributes to the theoretical understanding of green bond pricing but also offers practical insights for financial analysts and investors navigating this evolving market.

Keywords: Green Bonds, Bond Pricing, Zero-Coupon Bond, Short Rate Modeling, Numerical Methods, Crank-Nicolson, GMRES, BiCGSTAB, Environmental Finance, Sustainable Investing, Comparative Analysis, Financial Modeling.

Implied volatility surfaces (IVS) are essential for option pricing and risk management. Recently, generative deep learning models, such as the Denoising Diffusion Probabilistic Model (DDPM), have gained popularity for generating IVS. However, these machine-generated surfaces do not necessarily meet arbitrage-free constraints, e.g., calendar spread and butterfly arbitrage.
This thesis addresses this limitation by integrating arbitrage-free constraints into the DDPM generation process. We introduce a classifier specifically designed to detect violations of arbitrage-free conditions. During the DDPM’s iterative generation process, the classifier guides the formation of the IVS at each step, effectively minimizing potential arbitrage opportunities. To evaluate the effectiveness of our approach, we compare the extent of arbitrage violations in IVS generated by the standard DDPM with those produced by our enhanced DDPM-Classifier model. The results demonstrate that the DDPM-Classifier significantly reduces arbitrage opportunities and improves the quality of IVS.

Counterparty Credit Risk (CCR) refers to the risk that a counterpary involved in a financial contract will default before the final settlement of the contract, resulting in unrealized financial gains. One risk measure for managing counterparty credit risk is the Potential Future Exposure (PFE). The PFE is defined as the 97.5%-quantile of the exposure distribution. The traditional numerical method for computing the PFE is the Monte Carlo simulation method. Recent research has produced a semi-analytical method based on Fourier-cosine series expansion which produce PFE estimates with at least five times the accuracy of the Monte Carlo simulation in one-tenth of the CPU time. In this thesis this COS-PFE framework is combined with Monte Carlo simulation with the goal of reducing the variance of the PFE estimates.
In this thesis three methods are developed. Firstly, the Control Variate method that uses the COS-PFE framework to retrieve the CDF of the portfolio's exposure from which control variates are sampled. The second method developed uses Adaptive Importance Sampling. This method uses the COS-PFE framework to estimate the PFE and expected exposure (EE) of the portfolio. Using the risk factor that has the highest correlation with the exposure of the portfolio a shift is found such that the new joint probability distribution, from which the risk factors are sampled, has an EE that coincides with the portfolio's PFE. The third method developed in this thesis uses Adaptive Importance Sampling based on the Cross-Entropy method. This method aims to find an auxiliary probability distribution which minimizes the Kullbeck-Leibler divergence between itself and the theoretical zero-variance estimator.
The methods were tested using portfolios containing 100, 1000 and 10000 derivatives, both with and without collaterals. Testing showed that the control variate method was not able to produce a variance reduction compared to the straight forward Monte Carlo simulation. However, it did demonstrate the ability to produce a variance reduction for the EE.
The method that finds an optimal shift was successful in producing a variance reduction. The variance of the PFE estimates produced by this method was, on average, 3.5 times lower than the variance found using the straight forward Monte Carlo simulation. However, the algorithm required between 96 and 1773 seconds to produce a PFE estimate.
The adaptive importance sampling method using the cross-entropy approach was tested on all portfolio, both with and without collateral. This method was shown to produce PFE estimates with a significantly lower variance than the straight forward Monte Carlo simulation. For the uncollateralized portfolios containing 100, 1000 and 10000 derivatives the variance of the PFE estimates were on average 35.4, 38.6 and 37.2 times smaller compared to straight forward Monte Carlo simulation. For the portfolios with collateral this performance remains. We conclude that this method produces more accurate PFE estimations with significant lower variance than straight forward Monte Carlo simulation within the same CPU time.

We price continuously monitored barrier options under GBM and SABR through a newly developed neural network, the COS-CPD network, based on the COS-CPD method. With the pricing PDE and the COS method, we transform the problem of pricing the barrier options into a problem of finding the survival characteristic function through an initial boundary value problem. By applying trigonometric expansion, derived by integrating out the Fourier expansion on the time derivative of the unknown function, to the survival characteristic function, we find an approximation that can be inserted into the IBVP. Without CPD, this results in a linear system which can be solved to find the expansion coefficients, but this leads to the curse of dimensionality. If we include CPD to remove this curse of dimensionality, resulting in the COS-CPD network, the Alternating Least Squares method must be used to find the factor matrices that replace the original expansion coefficient tensor.

Geomechanical simulations can give essential insights into subsurface processes, but typically require solving large, ill-conditioned linear systems. An important method for solving these linear systems is the Conjugate Gradient method, but applying this method to ill-conditioned matrices can result in slow convergence. To improve the convergence of the Conjugate Gradient method, the iterative solver is preconditioned using the Algebraic Multigrid method. In Algebraic Multigrid methods, a hierarchy of matrices of different sizes is derived. When applying these methods as a preconditioner to the Conjugate Gradient method, on each level of the multigrid hierarchy fast convergence is observed in particular components of the residual. This leads to much fewer iterations being required in the Conjugate Gradient method, at the cost of the iterations being computationally more expensive. These Algebraic Multigrid methods do require a more problem-specific setup configuration than more simple preconditioners like the Jacobi preconditioner. In this research, the Conjugate Gradient method preconditioned with various Algebraic Multigrid methods is studied and compared with the Jacobi preconditioned Conjugate Gradient method. For this, the Conjugate Gradient method, preconditioned with both the Jacobi and Algebraic Multigrid-based methods, is applied to linear problems derived from geomechanical simulations. Using Algebraic Multigrid preconditioners can reduce the number of iterations required for convergence of the Conjugate Gradient method by a factor of 80. While a single iteration with an Algebraic Multigrid preconditioner is more time-expensive than an iteration with a Jacobi preconditioner, significant reductions, of up to five times, are observed in the runtimes of the linear solver. This comes at the cost of a higher peak memory requirement in the application of the linear solver. The vast reduction of the runtime of the linear solvers makes the studied geomechanical simulations significantly faster. This makes the Algebraic Multigrid preconditioners a valuable addition to these simulations.

The demand for sustainable and clean energy sources has become increasingly vital in addressing the challenges of climate change and energy security. Hydrogen, with its high energy density and potential for carbon-free energy conversion, has emerged as a promising candidate for future energy systems. Efficient storage and retrieval of hydrogen are crucial for its widespread utilization, for which a promising approach is underground hydrogen storage in geological porous media. This thesis aims to explore and advance the understanding of hydrogen storage in geological porous media, specifically focusing on pore-scale modeling and contact angle analysis.

This research aims to overcome the limitations of current hydrogen storagemethods and develop more efficient energy storage systems. Porous materials like sandstones have special characteristics that make them suitable for storing hydrogen underground. To design and operate underground hydrogen storage on a large scale, it is important to understand how fluids move through these materials. The way hydrogen is stored and released is influenced by complex processes happening at a very small scale (μm). To accurately simulate these processes, we need to study how fluids move in the pores, including factors like capillary pressure (the pressure difference between nonwetting and wetting phases, which is one of the main forces acting at pore scale transport) and relative permeability (how easily fluids flow through the pores where other fluids are also present).

Pore-scale modeling is a useful tool for simulating and understanding how hydrogen behaves in the tiny pore spaces of porous materials. These models help us see how hydrogen moves, spreads out, and interacts with the pore walls at a very small level. Another important aspect is studying the contact angles in the system of hydrogen, water, and porous material. These angles tell us about the way these substances interact at the interfaces between solids, liquids, and gases. By studying these processes and measuring contact angles, we can gain a better understanding of how hydrogen is stored and released, considering factors like pressure, temperature, the type of material, and how easily fluids flow through the pores. This knowledge will help us design better systems for storing hydrogen energy in porous materials on a larger scale.

The primary objectives of this thesis are as follows: To develop pore-scale models for simulating and understanding underground hydrogen storage in geological porousmedia. To investigate the contact angle between hydrogen, brine, and sandstone systems and their influence on storage and release mechanisms. To analyze the contact angle for a mixture of hydrogen-methane in the brine/sandstone system and assess its implications for hydrogen storage. To develop a dynamic pore network model to capture the dynamic behavior of hydrogen in geological porous media. To draw conclusions from the findings and propose future research directions in the field of hydrogen energy storage.

Integrated electrical power flow simulations are concerned with solving the steady-state load flow problem on integrated transmission and distribution electricity networks. We have developed a framework to run these simulations efficiently, whilst keeping in mind the differences between these network types and accommodating the practical considerations of system operators. We need such a framework to analyse the interaction that these systems might have as a result of the energy transition.

To develop a framework to run integrated power flow simulations, we have worked in two stages. Firstly, we have studied how we can model an integrated network. We have found two ways of modelling an integrated network: using a homogeneous configuration in which both networks are modelled using three phases and using a hybrid network configuration in which both networks keep their original configuration but in which the coupling substation takes care of the phase dimension mismatch between the two sides. Next to that, we have found two ways of solving an integrated system: either by coupling them into one system and solving that as a whole (we call this the unified approach) or by keeping two separate systems and iterating between these networks (we call this the Manager‐Fellow Splitting (MFS) method).

We have concluded that the unified methods are generally faster than MFS methods and that a hybrid network configuration leads to faster results, making the interconnected method the most efficient.

In the second stage, we have focused on the efficiency of these simulations. During every Newton‐Raphson iteration in power flow simulations, a linear system is solved. We have therefore studied several Krylov subspace and preconditioning techniques that can solve this linear system efficiently. We have applied Krylov and preconditioning combinations to integrated network simulations to check again the performances of the simulations on large test cases . During this stage, we applied them to networks up to a size of 800,000 buses as we were interested in efficient scaling of the methods that were originally the object of study.

In the second stage, we saw that the MFS methods were performing better than unified methods. Furthermore, preconditioned Krylov subspace methods had a similar performance to direct methods. t is difficult to judge why this happened. A reason could be that the library in which we performed these simulations, PETSc, is optimised for parallel computations in which multiple smaller blocks are solved at the same time whilst we were doing only sequential computations.

Finally, we have striven to incorporate operational convenience for Transmission and Distribution System Operators (TSOs and DSOs) during the development of this integration framework, by considering their computational and privacy concerns. The way that this framework is built, can take away some of their concerns.

To summarise, we have created an open‐source framework to run efficient steady-state power flow simulations on integrated transmission and distribution networks. This framework is tested on simplified test cases but shows potential for large system simulations. Moreover, it takes into account the considerations of system operators and can be utilised in other applications besides integrated analysis.

Computational Fluid Dynamics (CFD) offers numerous benefits, notably the ability to study flows that are challenging or costly to investigate using experiments. A central challenge in CFD lies in simulating fluid flow around complex geometries. Additionally, the governing equations follow conservation laws. This thesis aims to establish the foundation for constructing Immersed divergence-conforming finite element spaces that address both challenges. To tackle the first issue, an immersed method is considered. Instead of generating a mesh that conforms to the object where the flow occurs, the approach involves placing the object within a predefined mesh. However, this introduces new challenges, the most significant of which is the ill-conditioning of the associated linear system. The condition number depends on the location of the immersed object and can lead to an extremely large condition number. The second challenge is resolved by discretizing the problem in a way that preserves essential topological and homological structures at a discrete level, utilizing the principles of finite element exterior calculus. Within this thesis, a structure-preserving subcomplex is developed for the de Rham complex in 1D, and its accuracy is validated through numerical experimentation. Optimal convergence is achieved, the discrete inf-sup test is satisfied, and the resulting linear system's conditioning remains unaffected by the placement of the immersed object. However, the constructed structure-preserving subcomplex for the de Rham complex in 2D, known as the immersed divergence-conforming finite element spaces, does not exhibit convergence. For future research, I recommend focusing on simpler vertical and skewed cuts before exploring more intricate immersed shapes. These simpler cuts involve straightforward choices for edge basis functions/1-form basis functions that are relevant. Additionally, if the outcomes remain unsatisfactory, considering a global approach instead of a local approach could be worthwhile.

Erasmus MC is conducting a study on the increased risk of cardiovascular disease in women who have migraine. For this purpose, 50 healthy women and 50 women with migraine in age category of 40-60 years were compared using measurements of their dermal blood flow.
These measurements are done using two different imaging devices: Laser Speckle Imaging and Laser Doppler Imaging. The result is a recording of the fluctuations in the dermal blood flow over a time period influenced by circumstances and various mechanisms and consists of signal like curves. Erasmus MC have processed these measurements and obtained results. However, they are interested in whether applying Fourier Transform could offer additional insights or new (perhaps faster) methods to process the measurements.
The signal like curves can possibly be decomposed into sine and cosine waves using Fourier Transform. The result of the Fourier Transform consists of a visualisation of the intensity of the various components of such a waveform/curve. Based on that, it is possible to estimate the contribution of the mechanisms to blood perfusion. Existing literature and previous studies provide frequency intervals related to several mechanisms that influence the dermal blood flow.
This thesis will research to what extent the mechanisms contribute to the dermal blood flow across the different phases and how their behaviour differs when the circumstances for the measurement are adjusted. Perhaps there are behaviours that are instantly recognizable across all patients. An interesting question to answer would be whether it is possible to directly visualise the risk groups.

Although energy commodity price forecasting has been around for quite some time, up until recently, especially in Europe, it mostly concerned other energy commodities than gas. That is why in this work, a forecasting model is presented for the single day forecast of the daily VWAP price of the TTF month-ahead gas contract from 2020/01/02 until 2023/08/03. A model combining machine learning and data preprocessing is proposed. First a decomposition of the gas price is produced by a combination of Variational Mode Decomposition (VMD) and Independent Component Analysis (ICA). With these decompositions an initial volatility regime split is chosen because of the unusual characteristics of the dataset. During the forecasting period the Russo-Ukrainian war started, and the economy was recovering from the COVID-19 crisis, causing the gas price to surge to levels previously unknown. Using the decompositions as input, a final price prediction has been made using a Gated Recurrent Unit Neural Network (GRUNN) and Support Vector Regression (SVR), amongst others. The proposed model is compared to a selection of benchmarks, one of which is the naive forecast. To conclude, a selection of exogenous variables is added to the model to improve the performance. Gas storage and the UK NBP gas price are chosen for their specific characteristics. The best performance the model exhibits is between 0.38% MAPE for the low volatility regime, and 1.4% MAPE for the high volatility regime.

This thesis investigates the behaviour of one-dimensional blood flow in the human arterial tree. It focuses on modeling the change in velocity, cross-sectional area, and concentration. The equations governing this flow are derived using the physical laws of conservation of mass, conservation of momentum, and the
advection-diffusion equation. To solve this set of equations numerically, the Finite Volume Method in combination with a flux difference splitting approach is employed. The study begins with an examination of a single artery, revealing wave-like behavior in pressure and concentration. Pressure propagates along the vessel, while concentration gradually dilutes over time. This
behaviour is also visible when expanding to the 55-artery model of the entire human body. The results of this model were found to be very comparable to the literature. Additionally, the propagation of a chemical
species injected into an artery near the heart is investigated. It takes around 25 seconds to propagate to
the legs but the amount of concentration is greatly reduced. The strength of this wave upon reaching the
legs is investigated for different values of the diffusion coefficient. It is found that when
D = 0.02 cm2/s, the strength is around the 80% of its original value while a value of D = 2 cm2/s reduces
it to 40%. Further research should focus on expanding this model and improving its accuracy. Especially
the exact concentration waves in single arteries are very prone to spurious oscillations. By optimizing the
numerical code, the accuracy can be greatly increased. Moreover, more realistic additions can be made to
this model. Such as expanding to a two-way model enabling to inject at various points around the arterial
tree. Nevertheless, this model provides insight into the propagation of blood and concentration in the
human circulatory system, highlighting the need for refinement.