High-dimensional data imputation is a critical challenge in semiconductor metrology, where secondary measurements are often purposely omitted to optimize throughput. This thesis examines the Missing By Design (MBD) framework—an industrially motivated scenario in which data are systematically uncollected to reduce measurement overhead—and investigates a range of imputation solutions tailored to the particular complexities of wafer reflectivity and overlay. After establishing the physical, rank-deficient nature of wafer metrology data through singular-value decompositions and principal component analyses, we explore several classes of methods: linear regressions and matrix-completion techniques for baseline comparisons; deep neural-network regression (MLP) to capture nonlinearities; a contrastive-learning adaptation of CLIP for pairwise matching of primary–secondary measurements; and novel Bridge Models that refine coarse CLIP estimates with localized residual translations. Additionally, we integrate overlay-based domain constraints into CLIP via domain-guided neural network regularization (DG), ensuring physically coherent tool-to-tool (T2T) predictions. Comprehensive experiments on proprietary wafer datasets confirm that linear approaches including regressions and matrix completion methods, despite capturing the low rank structure of the data, underperform in downstream overlay and T2T prediction due to subtle nonlinear relationships. Deep neural networks offer strong reconstruction accuracy, yet demand extensive hyperparameter tuning and deeper network structures than contrastive alternatives such as CLIP-like approaches, which yield architecturally efficient, instance-based retrievals, but can lack the precision needed for rigorous overlay alignment. DG regularization, as an extension of the CLIP framework, considerably enhances T2T consistency and reduces raw reconstruction error. Meanwhile, the Bridge Model combines a CLIP-derived coarse imputation with a smaller learnable residual map between encoder domains, bridging global pairwise alignment and localized corrections for improved reconstruction and downstream tasks. Overall, this thesis presents a flexible suite of tools that advance high-dimensional MBD imputation in wafer metrology, offering valuable insights and a robust methodological foundation for future industrial applications.

Dutch pension funds are under the supervision by De Nederlandsche Bank (DNB) and they must adhere to the Financial Assessment Framework (FTK), which outlines the methods for calculating liabilities, buffer reserves, and risk factors. As part of the FTK, a feasibility test must be performed, which is a scenario-based analysis of pension funds' investment strategies and their pension policies based on economic scenario sets projected 100 years into future. Every quarter, DNB publishes these economic scenarios (P-scenarios) and equivalent risk-neutral scenarios (Q-scenarios) are used to calculate the effects of the pension fund reform in the Netherlands. These scenarios are generated by a stochastic model, which is revised every 5 years by a commission appointed by the Dutch government, called the Commission Parameters. The model currently in use is based on Commission Parameters 2022 and is referred to as CP2022.

The CP2022 model is an affine and arbitrage-free model framework for correlated interest rate, inflation rate, stock index and consumer price index, each being driven by Heston's stochastic volatility process. To generate realistic economic scenarios, the model is calibrated quarterly using bond and option market data. Currently, as part of the calibration process, option prices are computed under the dynamics stated by CP2022 using a Monte Carlo-based approach. As an alternative, we resort to a Fourier method called the COS method, which relies on the availability of the characteristic function of the state factors. We propose an efficient option valuation scheme for pricing European call and put options, and cliquet options under CP2022. Compared to the Monte Carlo method, our scheme demonstrates superior accuracy and computational efficiency.

This thesis addresses a timetabling problem known as Strategic Passenger-Oriented Timetabling (SPOT). SPOT is a timetabling problem that involves assigning departure and arrival times to train services. Unlike classical timetabling, which typically emphasizes infrastructural feasibility, SPOT shifts the focus toward enhancing the passenger experience without infrastructural constraints. The timetables generated by SPOT provide information regarding important connections for passengers and can later be adjusted to accommodate infrastructural constraints. SPOT evaluates a timetable based on the perceived travel time consisting of drive, dwell and transfer times, along with additional penalties for inconveniences by transfers and initial waiting times of passengers.

An initial research focused on using a Mixed-Integer Linear Programming (MILP) approach to solve the SPOT problem. However, this MILP formulation has multiple problems. The MILP tends to perform slow for small problem instances and becomes significantly worse as the problem size increases. Additionally, the effectiveness of the timetables generated by the MILP are often unreliable, because there is a substantial gap between the perceived travel time of the best-found timetable and the best-found lower bound.

This thesis explores a heuristic approach to solving SPOT. Specifically, two heuristics are employed: Local Search and Simulated Annealing. Both operate by exploring a neighborhood of possible timetables around a current timetable to identify better alternatives. The absence of infrastructural constraints in SPOT simplifies the creation of these neighborhoods, as different train services do not influence each other. Additionally, multiple lower bounds are calculated for the perceived travel time, providing benchmarks to evaluate the quality of the solutions. Overall, these heuristics are effective in finding good timetables within a short period, with the lower bounds being reasonably close to the perceived travel time of the generated timetables.

This thesis concerns the modelling, risk analysis and deposit rate optimization for Non-Maturing Deposits (NMDs), applied on aggregate data for the Dutch banking sector. The final NMD model consists of three parts, where there is a clear separation between the model for the term structure, the deposit rate and the deposit volume. The final model is calibrated on public data from 2004 to 2023.

For the term structure, we calibrate a shifted CIR model using a genetic algorithm on historical bond rates that were obtained via a Svensson model. The deposit volume is modelled via the change in the logarithm of the volume, which is regressed using Ordinary Least Squares on market and deposit rate parameters. The model is made autoregressive to remove serial correlation in the residuals.

Additionally, we describe and conduct a deposit rate model optimization for several classes of deposit rate models. We define two properties, the profit and the shortage, and maximize over a weighted sum of the two. For the optimization, we use a Robbins-Monro type model for its ability to deal with expectations, without the requirement of approximating the expectation at every step.

With the calibrated model and optimized deposit rate models, we introduce a way of risk analysis, by looking at the quantiles of the deposit volume, conditioned on the minimum of the market rate being in the lowest quantiles. For our model, this risk analysis indicates that a low interest rate environment is not problematic for the final, maximum and minimum volumes over a period of at least 10 years. On the contrary, the quantiles of the volume are higher when conditioned on rates being very low, indicating that the worst scenario for the deposit volume is not a scenario with low interest rates.

For simple models, we find that the procedure converges to an optimum value within an acceptable timeframe; for more complicated models, we find that the convergence is either slow, or that the these parameters do not influence the criterion that is optimized, compared to, for instance, the random component in the volume evolution.

When the volume is considered as an input for the deposit rate model, an optimal model is obtained that outperforms models only considering the market rate significantly on the two considered properties. This suggests that taking the volume into account is beneficial for maintaining a strong NMD portfolio.

Our methods are applied to public, aggregate data. As a result, we are unable to observe the effects of consumers switching banks within the Dutch system. We suspect that our approach works better when single bank data is available to the experimenter. All the methods that were described in this thesis are applicable to such data, yielding a possibly interesting way of conducting optimization, as well as risk analysis within a bank.

This thesis adds to the literature by introducing a way of risk analysis that is incorporated in the model itself. For this analysis quantiles of the distribution of the dependent variable are calculated for both unconditioned and conditioned scenarios. Concretely for this thesis, quantiles of the deposit volume, conditioned on the market rate minimum over the whole period being in the lowest quantile, are calculated.

Secondly, the thesis adds to the literature by introducing a method for optimizing the model for setting the deposit rate. Optimal models obtained via this approach can be used to challenge current ways that banks set their deposit rate.

This thesis aims to contribute to the understanding of how waves interact with soil. It is crucial for various applications in Civil Engineering to analyze the behaviour of soil and to understand the physics behind it. This master thesis contributes to this understanding via studying the impact of the boundary conditions on the model results with the aim of being able to model interaction between waves and soil.
We assume a media that is poroelastic and fully-saturated, unless stated otherwise. We also assume that the porous media consists of incompressible soil particles and pore water particles that may either be compressible or incompressible. The main goals of this thesis are (1) to describe the response of porous media to transient hydraulic loads using numerical methods like the Finite-Element Method, and (2) to apply it to a one-dimensional case whereby a sandbed is subjected to waves. Currently, it is common to predict the changes in pore water pressures in porous media subjected to transient hydraulic loads using Biot’s model, which often assumes compressible pore water, assumes zero effective stresses on the surface of the seabed, and assumes that the wave load is completely carried by the pore water pressure only. Recently, a new model is proposed by Van Damme and Den Ouden-Van der Horst suggesting that transient hydraulic loads acting on a porous medium affect both the pore water pressures and effective stresses in soils. Note that this makes sure that the momentum balance equations are satisfied throughout the computational domain and its boundaries. The boundary conditions in this case do not satisfy Terzaghi’s effective stress principle, whereas the standard has been to impose Terzaghi’s effective stress principle when solving Biot’s equations. Terzaghi’s principle states that the sum of the effective stresses and pore water pressures must equal the hydraulic loads, whereas Biot’s model is in line with this principle.
The model of Biot and the new model of Van Damme and Den Ouden-Van der Horst describe the physics differently which can have a large impact on the results. For example, the assumption of compressibility can significantly impact the distribution of the effective stress in the soil and thus the results.
Biot’s model is more sensitive for changing the compressibility parameter than the new model. Both models give similar solutions to the water pressure. However, they give different solutions to the other variables like the volumetric strain and displacements which appear in both models. Furthermore, the new model in one dimension is in line with the momentum balance equations and satisfies the volume balance equation. On the other hand, the standard is to solve Biot’s model by imposing Terzaghi’s principle at the boundary. For the new model we found promising results for the water pressure, when validating with the data of two experiments. At the end, which model predict the best solutions for volumetric
strain, water pressure and displacements depends on what kind of problem the model is used for and the corresponding physics. The used code can be found at https://github.com/fpmklein/Compressiblevs.-incompressible-pore-water-in-fully-saturated-poroelastic-soil.

This thesis addresses the challenge of segmenting ultra-high-resolution images. Limitations of current approaches to segment these are that either detailed spatial contextual information is lost or many redundant computations are necessary. To overcome these issues, we propose a novel approach combining the U-Net architecture with domain decomposition strategies to balance incorporating spatial context and maintaining computational efficiency. Our proposed method partitions input images into non-overlapping patches, each processed independently on a separate device. A communication network facilitates the exchange of information between patches, enhancing the model's understanding of the spatial context.

Through theoretical analysis and practical experimentation on device memory usage during training, we demonstrate that our approach incurs minimal additional memory overhead for inter-device communication. Evaluation on synthetic and realistic datasets, including the Inria Aerial Image and DeepGlobe Satellite Segmentation datasets, demonstrates the effectiveness of our approach. Our model achieves competitive performance compared to the baseline U-Net model, with consistent class predictions around boundaries. Visualization of feature maps highlights the role of the communication network in transferring contextual information. Furthermore, it is shown that our approach remains scalable even when trained on limited subdomains.

In conclusion, our proposed model offers an intuitive solution for segmenting ultra-high-resolution images by effectively incorporating spatial context. Future research could explore variations of our model, such as overlapping subdomains or communication on different levels of the U-Net, to further enhance boundary consistency and information transfer.

The rise of quantitative investment strategies has been driven by increased data availability and advancements in financial modeling. This thesis introduces Causal Factor Investing (CFI), a novel approach that integrates causality and machine learning to enhance the performance and explainability of factor investing strategies. Traditional factor investing often suffers from specification errors due to its reliance on correlations rather than causal relationships, and machine learning methods are frequently criticized for their `black box' nature. CFI addresses these issues by using causal discovery methods, which are based on the mathematical properties of graph theory, to identify factors that have a cause-effect relationship with asset returns. These causal factors are then utilized as features in machine learning models to predict future returns, serving as investment signals for portfolio construction.

Our empirical analysis in the European corporate bond market utilized causal discovery algorithms including Fast Causal Inference (FCI) and Greedy Equivalence Search (GES). The use of GES in CFI improves portfolio performance compared to traditional factor investing, while FCI led to insufficient causal graphs. For the portfolios constructed with neural networks in CFI, the use of causal factors resulted in the best-performing portfolio.

Altogether, CFI contributes to the field of quantitative finance by offering an explainable and profitable approach to factor investing. For further research, we suggest exploring alternative causal discovery algorithms, including time-series causal discovery methods and other algorithms that account for hidden confounders to increase the accuracy of the causal graphs. Additionally, a practical improvement would be including transaction costs and adjust the model for risk constraints through optimization approaches.

The topic of this thesis project is to use the particle-tracking framework to study salt intrusion.
This project was proposed by Deltares and is a small part of the SALTISolutions project, that attempts to build a digital twin model of the Rhine-Meuse Delta. From a mathematical perspective, the particle-tracking framework is inspired by the theoretical connection between partial differential equations that describes substance transport and stochastic differential equations (SDEs). The idea is to use replace the continuous substance distribution in the fluid by many particles. After tracking the particles, the consequent substance distribution can be approximated by the particles' distribution at certain time instant. There are two major achievement of this project. Firstly, different implementation of non-penetrating and flux boundary conditions for the particle-tracking framework are tested and compared. The recursion algorithm is preferred for the previous and the ``Neumann-wise'' is recommended for the latter. Secondly, the convergence order of SDE numerical schemes and their influences on the simulation of the vertical transport physics are examined by studying the one-dimensional pure diffusion test case. We see that most simple Euler scheme contaminates the stratification completely in a theoretical setting even using a small timestep, indicating that it might be worthwhile to switch to high-order schemes whenever possible. However, even if one can implement other higher order numerical scheme correctly and efficiently, which is not easy, it might be equivalently important to have enough particles and good interpolation of velocity and diffusion field in the computation domain, because they may introduces error comparable to the diffusion error. Since salt transport is already resolved by the Delft3D-FM model, it is of nonsense to use the particle-tracking framework to study transport. The one-dimensional pure diffusion test case is mainly used for verification of the method. Actually, the particle-tracking framework should be used in combination of the diagnostics timescales and tracer methods. This often requires calculation of the statistics of the particle ensemble. It will be interesting to conduct such analysis on a realistic three-dimensional test case, and this could be the field of future research.

Financial markets continue to see an increase in the share of trades executed by algorithmic trading systems. A key component of an efficient algorithmic trading system is its ability to accurately estimate the probability an order will be executed: the fill probability. This thesis aims to determine whether the dynamics of the order book can be captured accurately by a stochastic model, which subsequently leads to the following research question: Can we accurately compute fill probabilities of limit orders, conditional on the state of the order book? Our research builds upon the stochastic model proposed by Cont, Stoikov and Talreja, which assumes orders and cancellations have unit size and arrive according to independent Poisson processes [12]. There are two main advantages of the model proposed by Cont et al., which include: • Straightforward parameter estimation based on observable order book data, • The possibility to compute various probabilities of interest semi-analytically. These probabilities include the fill probability of limit orders and the direction of the next mid-price move. In this model, the number of orders at each price level in the order book is modelled as a birth-death process, in which births represent an increase in the number of orders and conversely, deaths represent a decrease. The first-passage time is the time it takes for such a process to reach a certain state for the first time. Since the order arrivals follow Poisson processes, we can find an expression for the Laplace transforms of the density functions of the first-passage times. By numerically inverting these Laplace transforms we can calculate the probabilities of interest. The parameters of the Poisson processes that determine the arrivals of orders are calibrated using observable order book data of the Euro-US Dollar currency pair of the foreign exchange (FX) spot market. We extend the model of Cont et al. by integrating spread-dependent arrival rates of orders and cancellations, meaning that the arrival rates change for different sizes of the spread. This extension is based on the order book data, which clearly shows a difference in arrival rates for various spread sizes. Additionally, we provide an expression to compute fill probabilities of orders posted not only at the best bid price, but also for one price level below. This expression can be generalised to obtain an expression for the fill probabilities of orders posted even deeper in the order book. Finally, by employing this stochastic model, we also show that this model has the potential to be applied to asset classes other than equities, which in our case is the FX market. To evaluate this model, we compare the computed probabilities with the empirical probabilities based on the FX data. Our findings indicate that the model has the ability to effectively capture the dynamics at each price level and the short-term movements of the mid-price. We are able to show that the fill probability for orders posted at the best quote can be estimated with reasonable accuracy when the order has the highest priority of execution. For orders with lower priority, the model is not able to capture all the dynamics, resulting in most cases in an overestimation of the fill probability. Additionally, we evaluate the method for computing the fill probability at one price level lower than the best quote. We also compare the results of this method to empirical probabilities. However, it is challenging to evaluate the performance of the model due to limited data availability and the small magnitude of the computed probabilities, which are typically less than 1.0%, resulting in large relative errors. Overall, this research contributes to better understanding order book dynamics and provides insights on the precision of the computation of fill probabilities via an extension of [12]. Further improvements and refinement of the model, such as a more realistic order flow or allowing multiple order sizes, could lead to more accurate estimations and a better comprehension of the market dynamics and fill probabilities.

Humans are efficient at moving due to their exceptional mastery of bipedal locomotion. Several models have been made that attempt to model the motion of the centre of mass with a spring-mass system with various degree of success. For example, a two dimensional model tracks the height of the centre of mass well and a three dimensional model explains the normal force exerted on the ground.

In this report a three dimensional model is constructed to describe the motion of the centre of mass, with the purpose to determine the spring constant and the rest length in a simple prosthetic leg. The research question is: can a three dimensional spring-mass model accurately track the centre of mass and can this be used to determine the optimal properties of the spring of a simple prosthetic leg?

The spring mass model consists of two springs connected to the ground and the centre of mass. The springs do not exert a force on the centre of mass if they are extended. The model incorporates steps by moving the ground connection points instantaneously over fixed points on a rail. The model can be used in two distinct ways. The first generates a periodic trajectory by finding the optimal initial positions y₀ and z₀ (PPFA) and the second finds the optimal spring parameters k and u to fit a data set (DFA). The optimal conditions are found by discretizing the parameter space and using an iterative process. The space around the best parameters becomes the parameter space for the next iteration.

The PPFA shows that the initial lateral displacement is nearly constant with respect to the spring constant when the rest length is chosen such that the centre of mass is at constant height if the model is stationary. The PPFA also shows that there are two distinct y-trajectories possible for walking. Running, however, has only one trajectory. Several versions of the model are fit to the data: only discretised step widths, discretised step widths and a x₀-offset, and distinct legs. There is not much difference between these models. The x₀-offset is only Δs = -0.02m confirming that the position of the feet does align with the extreme values of the y- and z-position. The model fits the general characteristics well but fine details in the motion are lost. Future research should investigate how cumbersome these deviations of the data are perceived by people using a simple prosthetic.

In many fields we are interested in inference for a complex stochastic process given limited observations regarding its state over time. This thesis therefore introduces an expectation propagation approach to backward filtering forward guiding for high-dimensional finite-state space models. The backward filtering forward guiding method is first derived for such models with a specific emphasis on the temporal dynamics, after which factorised guiding terms which exploit the inherent structure of the latent state space are introduced.

Performance of the method is assessed by comparing numerical results for statistical inference of a Susceptible-Infected-Recovered example problem. The expectation propagation approach performs comparably with existing methods in a particularly simple setting where state variables are observed individually, and performs very well in a more difficult setting which the familiar methods can not deal with. We conclude that a more advanced treatment of the approximate likelihood filtering phase may be warranted in such complex settings.

The research summarised in this work provides a first effort towards the development of more general expectation propagation based backward filtering procedures for other types of high-dimensional sequential data models. The work additionally elucidates connections between backward filtering with backward marginalisation and alternative approximate likelihood filtering procedures, suggesting multiple avenues for future research.

The salinity profile in a coastal sea-river system is the result of balancing of salt transport due to mixing of the water by tides and the freshwater discharge from the river. When for example during times of drought freshwater discharge suddenly declines, the salinity in the river may increase significantly and threaten drinking water supplies and agriculture. The time it takes for the salinity to reach the new equilibrium may be considerable. Thorough understanding of the adaptation process is essential for taking appropriate prevention measures. However, much about this adjustment process is still unknown, in particular regarding the interaction between the river and the adjacent sea. In this study, a simplified linear model is developed to describe a well-mixed river and the adjacent sea where the water flows radially away from the mouth of the river. The unique assumptions, most notably a time-independent intrusion length, allow for a unique analytical approach using the method of eigenfunction expansion. The smallest eigenvalues are found to define a time scale of the adjustment process. The eigenfunctions provide insight about how the salinity adjustment time varies within the system. The adjustment time does not only depend on the parameters that describe the new equilibrium, but also on the initial salinity. The eigenvalue time scales corresponding to the sea and to the river can then be used to determine the time scale of the coupled system. It is found that when the sea adjusts much faster than the river, the time scale of the river is leading the adjustment. When the eigenvalue time scale of the sea is similar to or larger than the eigenvalue time scale of the river, the adjustment time is significantly increased by the sea. When salinity is increasing, the sea restricts the inflow of salt in the river, while for decreasing salinity, the sea keeps transporting salt to the river by mixing which delays the adjustment. The model is applied to analyse a sudden decrease in freshwater discharge in the Rotterdam Waterway, which is part of the Dutch Rhine-Meuse delta. Results show that the salinity response is delayed compared to freshwater discharge, indicating that the adjustment time of the river indeed plays an important role. Further research should consider a more general dispersion relation or an extra vertical dimension to better describe estuaries that are not well-mixed. Using numerical methods there are many other possibilities to extend the model. The effect of tides on the presence and dynamics of the fresh water bulge in the sea is of specific interest.

The computation of multivariate expectations is a common task in various fields related to probability theory. This thesis aims to develop a generic and efficient solver for multivariate expectation problems, with a focus on its application in the field of quantitative finance, specifically for the quantification of Counterparty Credit Risk (CCR).

The proposed COS-CPD method utilizes the COS method to recover the exposure distribution by its Fourier-cosine series expansion, from which measures such as the PFE and EE can be obtained. The key insight is that the corresponding Fourier coefficients are readily available from the characteristic function, which can be solved using numerical integration methods. However, the efficiency of standard quadrature rules is limited to only a few risk factors, as the dimension of integration is determined by the number of risk factors involved.

To address this limitation, the COS-CPD method reduces the dimension of integration of the characteristic function through two steps. Firstly, the joint density function of the risk factors in the characteristic function is replaced by a dimension-reduced Fourier-cosine series expansion, which is obtained through CPD. With CPD, the computational complexity of computing the Fourier coefficient tensor is reduced to a linear growth with respect to the number of dimensions. Secondly, the portfolio is divided into segments that share the same risk factors. These two steps reduces the evaluation of the characteristic function to the calculation of only one- and two-dimensional integrals, which are solved by the Clenshaw-Curtis quadrature rule. As a result, the COS-CPD method is suitable for portfolios with more than three risk factors.

Numerical comparisons of the COS-CPD method and Monte Carlo (MC) method are made for netting-set PFE and EE profiles of multiple derivative portfolios up to five risk factors. For similar accuracy levels, the COS-CPD method greatly outperforms the Monte Carlo method in computation time. This difference increases for larger portfolios, which makes the COS-CPD method a much more efficient alternative for the MC method, especially for large portfolios.

Furthermore, the COS-CPD method is applied in the context of multi-asset option pricing. A six-dimensional basket option is considered, and the results are compared to a recently developed sparse grid method. The comparison shows that the COS-CPD method outperforms the sparse grid method in both accuracy and computation time. Moreover, the COS-CPD method allows the computation of the option value for multiple strike prices simultaneously, with no significant additional computational cost.

In biology, phylogenetics is the study of the evolutionary history of and relations between e.g. species. Such data are often represented in trees. Remarkably, trees lack the representation of reticulation events, such as hybridization, while such events are believed to be important. One of the reasons why trees are widely used, is the enormous complexity of inferring a network directly from DNA data.
Therefore, indirect approaches are studied. One option is to construct a network from trees, which have been constructed from DNA data. Another option, which is studied in this work, is to create a network which represents all clusters from the trees. We are interested in finding a network having the lowest level possible, i.e. the lowest number of reticulations per biconnected component. The Cass algorithm is one of the best algorithms in this respect. However, it does not always give optimal results. In this thesis, an optimal algorithm is presented called STCass, of which many ideas are inspired from Cass. We lay
theoretical foundations on how to build an optimal network. Furthermore, (greedy) steps are proposed to solve the problem, which seem to work very well in practice. Based on several optimizations that are proposed, an improved algorithm called MSTCass is presented, which runs faster than STCass, in general. In order to speed up any Cass-based algorithm or derivatives of it, several lower bounds are presented on the reticulation number r(N) per biconnected component N of an optimal network. Besides, a new upper bound on r(N) is conjectured which can be evaluated in seconds, although e.g. an upper bound obtained by the HybridInterleave or PIRN algorithm is mostly lower and which can be evaluated often in twenty minutes for the tested instances.

Reducing cost and improving computability of reservoir simulation is an important goal in the process of enabling CCS (Carbon Capture \& Storage) as a large-scale technology for mitigating CO2 emissions. In terms of computation time data-driven approaches have potential to outweigh the performance of numerical reservoir simulators, learning instead of solving an approximation of the subsurface physics. In this research a Graph Neural Network (GNN) is trained on 2D reservoir model samples to construct a ML autoregressive reservoir simulator. As such the aim of the model is to integrate a spatiotemporal learning task in a graph representation learning problem. Where GNNs have shown their potential in various graph representation and node classification tasks, their use in a reservoir simulation setting is practically unexplored. Graph methods are studied in this setting because of their potential to handle data in non-Euclidean space, which is desired in reservoir simulation to efficiently handle unstructured grids. A simple GNN autoencoder is introduced, implementing the common GCN (Graph Convolutional Network) layer as a locally-operating message-passing operator. The model does not work with spatial graph information, accepting arbitrary graph sizes and structures in 2D or 3D. The model is trained, validated and tested against 2D simulated datasets, generated with the INTERSECT numerical reservoir simulator. The trained model remains numerically stable and makes correct predictions for >140 autoregressive steps. It also shows capability to translate point-source information (the injection rate at the injection well) correctly into a CO2 saturation propagation pattern and corresponding pressure field across the reservoir grid. The model exhibits the correct behaviour around geological layers with variable permeability/porosity, indicating the model successfully adapts to geological variability in the subsurface. Finally the model shows it can infer on totally new reservoir structures. The latter showcases the local method's potential to train a ML reservoir simulator for arbitrary reservoir grids by training it with small generalized subsurface samples, dismissing the need to fully (re)train on usual reservoir grid sizes. Predicting using the GNN methods gives a 60x speed-up compared to the numerical simulator on the small training grids, which likely improves for larger grids given the complexity scaling of numerical simulators. As expected, capturing the elliptic behaviour of the PDE governing pressure is challenging for the local GNN method. While the resulting expressiveness of pressure is passable, ideas for improvement are suggested which can be incorporated using the same proposed GNN architecture.

Rendering accurate water reflections is crucial for achieving realism in computer graphics. Their integration with VR and AR technologies can further elevate experiences in sectors like education, engineering, and medicine. Even though various monoscopic reflection techniques exist they either demand substantial computational resources or cannot be straightforwardly adapted to stereoscopic media, leaving a big hole in research concerning stereo-aware reflections and their applications. This thesis researches and analyses the different techniques for water reflections. Further, it makes significant strides in the development of stereo-consistent real-time water reflections for XR platforms. This is achieved by combining implicit functions and bounding volume hierarchies (BVH) into a unified system called Adaptive Hierarchical Signed Distance Fields Reflections (AH-SDFR). It was found that AH-SDF outperforms leading methods such as Hierarchical Screen Space Reflections (HSSR) and Reflection Probes in both visual quality and performance, and as a result, it offers a promising solution for XR applications. However, challenges persist, mainly due to the large memory demands of intricate 3D scenes and perceptual errors from inaccuracies in volume textures.

This thesis presents a comprehensive exploration of the rough Heston model as a means to enhance financial derivative pricing and calibration in the context of the complex behavior of market volatility. Recognizing the limitations of classical models, such as the Black-Scholes and the standard Heston model, which assume constant or mean reverting volatility, this research delves into the application of rough volatility models that account for the empirical ’memory’ effect observed in financial markets. These models, inspired by the fractional Brownian motion with a Hurst parameter less than 0.5, offer a more accurate representation of the volatility surface.
A significant portion of the thesis is dedicated to the development of a novel cosine tensor network that expedites the supervised learning of the characteristic function of the lifted Heston model. This advancement is pivotal for the rapid pricing and calibration of European options under the rough Heston framework. The cosine tensor network, leveraging the characteristic function’s availability, enables the efficient application of Fourier-based methods, such as the COS method, for option pricing. This approach is further extended to the pricing of path-dependent options like barrier and Bermudan options through the 2-dimensional COS method.
The thesis is methodically structured, beginning with a foundational overview of option pricing and volatility, followed by a literature review that situates rough volatility within the broader context of quantitative finance. Subsequent chapters detail the mathematical framework of the rough Heston model, its derivation from market microstructures, and the introduction of the lifted Heston model as a multi-factor approximation.
Empirical analysis is provided to validate the lifted Heston model against traditional methods, demonstrating its superior performance and accuracy. The thesis culminates in the presentation of a new calibration method, supported by real market data, and a novel benchmark for pricing complex derivatives under the rough Heston model.
The research encapsulated in this thesis not only sets a new standard for computational speed and precision in the pricing and calibration of financial derivatives under the rough Heston model but also opens avenues for future research, particularly in the application of rough volatility models to other areas of financial mathematics.

Barrier options, although highly liquid financial derivatives, present notable pricing challenges. In this thesis, we present a novel pricing approach for valuing continuously-monitored knock-out barrier options within the framework of stochastic volatility models.

The underlying process is firstly modelled under geometric Brownian motion and, subsequently, under Heston's stochastic volatility model. A key insight is that the value of a barrier option can be expressed as a single-dimensional integral, whereby the integrand involves the so-called survival density function, which captures the barrier-brea-ching information. Therefore, the option can be valued using the one-dimensional COS method for European options, once the Fourier series coefficients of the survival density are obtained.

The coefficients of the sine series expansion of the survival density function are, in fact, a continuous function, which is closely related to the characteristic function of the density. This motivates us to directly recover that function, which we refer to as the target function herewith, by selecting an appropriate series expansion for it. Thereafter, we insert this series expansion into the partial differential equation (PDE) that the target function should satisfy, which can be derived from the pricing PDE. This results in a linear system, solving which we obtain the coefficients needed to reconstruct the target function. Notably, this approach is particularly advantageous when the reference values of option prices are limited or unavailable, as it relies solely on the PDE to calibrate the series coefficients.

Our choice of series expansion is driven by the need for precise global and local approximations. Our research shows that a proper expansion for the target function can be built up from integrating a two-dimensional Fourier series of its first derivative with respect to time, marking our second pivotal insight. That results in a trigonometric expansion that significantly enhances the accuracy compared to a direct Fourier series expansion on the target function. Finally, our third pivotal insight: applying a change of variables further improves error convergence.

Extensive testing results suggest that, for similar accuracy levels, our approach greatly outperforms Monte Carlo simulations in terms of computation time. It also demonstrates superior computational efficiency and accuracy compared to other advanced numerical methods in the existing literature.
The benefits of this method become even more prominent when pricing a large number of options simultaneously.

Numerical tests reveal algebraic convergence for the series expansion reconstruction. For the option price, theoretical error analysis aligns with our findings, also predicting algebraic convergence.

The right to use a certain amount of capacity in an electrical cable between two countries for the purpose of trading energy is an asset that can be bought. Each hour of capacity can be seen as a real spread option with the energy prices of each country being the underlying processes. In this thesis we build a model to find the fair value of a month of cross-border capacity and use it to predict this value for the Germany-France andNetherlands-Germany border for the months of July-October 2021. Our model is implemented as an IOHMM with 4 states following a multinomial distribution and a non-stationary Normal Inverse Gaussian distribution for each output distribution. We use SPOT prices of each bordering country to create a ’spread process’, which are then used as target values. As inputs various market drivers for each respective country were used. The model produces a mixture probability distribution for the spread for each hour of the month. Taking the discounted expected value then gives the hourly capacity value; certain Greek calculation is also possible. The results show that the capacities follow a comparable trend to the market. The predicted values for September and October fall short of the realized prices; we expect this to be due to unprecedented volatility in the energy market. Finally, we tested whether certain key drivers were mapped to the spreads as one would expect; this was the case for the Dutch and French drivers, but not for the German drivers.

Optimal control is a paradigm for solving optimization problems involving dynamical systems, which are to be controlled. It is able to solve fish harvesting problems, in which we want to optimize harvesting out-take by considering fishing as a control function that acts on the state of the dynamical system, which represents the growth of fish species in the environment. Other modelling aspects of optimal control are defining terminal costs and running costs, e.g. maximizing profit. We keep the terminal condition comparable for a different number of species. It is based on the initial population.
By using the optimal control Hamiltonian and Pontryagin's Maximum Principle we can calculate the optimal state trajectories corresponding to suitable optimal controls. The Hamiltonian is dependent on the state equation and the running costs. We present two approaches of modelling the running costs. An approach that is not directly translatable to the fish harvesting problem, but it leads to a smooth Hamiltonian, which greatly simplifies derivation and computation. The other, which is equivalent to maximizing profit, leads to a non-smooth Hamiltonian. This leads to jump-discontinuous derivatives needed for computation. We propose to regularize the derivatives of the Hamiltonian using suitable smooth functions, such that it is equivalent to regularizing the Hamiltonian directly. We give details for implementing both approaches up to systems of n competing species. After which we go into detail on algorithms and programming structure implemented. Finally, in modest numerical experiments, for one and two species, we show the relation between the optimal control and the terminal costs. But more interestingly, that the smooth Hamiltonian models are inadequate and regularized Hamiltonian models are the preferred choice. Intriguingly, the latter approach results in steady state solution, where the control acts as a stabilizer.