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.

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.

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.

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.

Groundwater is an essential ingredient in farming, knowledge about how this is expected to change over time can help farmers improve yields and save water. Predictions about the groundwater level can be made using a mathematical model. This model takes into account the precipitation and evaporation, the flux towards a deeper aquifer and flux to lateral open structures, with parameters for the different resistances and the storage. This model needs input data which is acquired from among others the KNMI (Koningklijk Nederlands Meteorologisch Instituut), with data about the groundwater level at the points of interest acquired from the measurements from the Water board Drenths Overijsselse Delta. The input from the model can be used to generate predictions, whilst the measurement data can be compare with the predictions to obtain more accurate predictions and calibrate the parameters in the model. Combining these factors can be done with the Kalman Filter, a mathematical tool from Data Assimilation which can be used to combine a Mathematical Model and Data into an optimal estimate. Furthermore, it is the wish to improve the parameters in the mathematical model, this causes the state to expand thereby enlarging the problem, and the parameters included in the state causes the problem to become non-linear. The Kalman Filter is only suitable for small linear models but an extension of the regular Kalman Filter can account for these problems, the Ensemble Kalman Filter. To apply all this information for groundwater modeling in a single location, first the numerical errors associated with the different model solutions are tested. It was found that a semi-analytical solution which assumed the input to stay constant over a day to produce the best result. After this the Kalman Filter is compared with the Ensemble Kalman Filter to see whether the Ensemble Kalman Filter can perform comparably in state estimation, it was found that en ensemble size of $n = 200$ this was the case. To test if the parameter calibration scheme works as desired and to test the quality of the parameters found, artificial data is created using the model with all parameters set to one. Then, the parameters are shifted to a different value and the parameter Ensemble Kalman Filter is used to try to find the original parameters back. It is found that in almost all cases the parameter calibration cause the parameter to change in the direction of one, within the first 2/3 months most change within the parameter takes place and the standard deviation of the parameters decreases. After this initial period the parameter remains more stable and the standard deviation changes little. Since the model uses multiple parameters for the resistances, the possibility of calibrating these separately was investigated. It was discovered that in the data usually one resistance parameter is lot smaller than the other, it was enough to calibrate only this parameter to account for the resistance. After this, the parameters in the model are calibrated with actual data. The model always becomes a lot closer to the measurements, and interestingly all parameter become larger than 1. Finally the quality of the predictions is tested. Here we saw that the quality improved fast, already 1/3th year was enough to find parameter far better than originally.

A one-dimensional model of respiratory deposition is developed based on an Eulerian approach. The model simulates aerosol deposition in all generations of the respiratory tract by numerically solving the aerosol general dynamics equation (GDE) for a range of aerosol diameters. The lung geometry is described by Weibel’s morphometric model, with a time varying alveolar geometry to accommodate inhalation dynamics. The model is first valuated by comparing it with numerical and experimental results. Afterwards a series of parameter studies is performed by changing breathing conditions, particle parameters and lung geometries. An increase in Tidal volume and decrease of breathing period resulted in an increase of the total deposition fraction for coarse particles and a decrease of the total deposition fraction for ultrafine particles. An increase in the particle density resulted in an increase in the total deposition fraction. A decrease in the airway diameter generally resulted in an increase of the total deposition fraction. This difference was most noticeable in the tracheobronchial region. Decreasing the airway diameter in the tracheobronchial region mostly effects coarse particles while decreasing the airway diameter in the alveolar region mostly effects ultrafine particles.

Coupled mathematical models are used successfully to represent systemic blood flow with different degrees of precision: for example, 3D representations in zones of particular interest (e.g., the aortic bifurcation) are embedded into 1D models who account for systemic blood circulation. When it comes to the lymphatic system, various 1D models for systemic lymph flow have been developed, however it is not clear whether they can be readily used in the context of a coupled mathematical model. In this work, we give an overview of the lymphatic system and its components in order to justify the interest for a 1D model of systemic lymph flow which can be coupled with 3D representations of specific zones. Then, we develop a one-dimensional model for lymph flow with exactly this property. In particular, we propose a formulation for the compliance of the walls of lymphatic vessels and of the sinuses of lymph nodes, as well as a novel formulation to mimic the phase contractions of lymphatic endothelial cells. Moreover, we give details and insight on the eigenvalues and characteristics of our proposed system for one-dimensional lymph flow, in order to highlight why and how coupling with 3D representations is possible. Finally, we perform a set of numerical experiments to study the applicability of our model.

Existence of commuting THB-spline projectors is of importance in the field of numerical mathematics. These projectors are required to show that numerical solutions to the abstract Hodge Laplace problem are stable and consistent. We have introduced a local THB-spline projector based on Bezier projection, that commutes in the one-dimensional case. Additionally, to show that the projector is well posed in the univariate and multivariate case, we have derived so called projection elements, that are grouped mesh elements, on which the THB-splines are linearly independent.

Wave phenomena play an important role in many different applications such as MRI scans, seismology and acoustics [41, 49, 47]. At the core of such applications lies the Helmholtz equation, which represents the time-independent version of the wave equation. Simulating a Helmholtz problem numerically with accurate numerical solutions for large wave numbers is challenging. Numerical solvers for the Helmholtz problem have to balance having accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity with respect to the grid nodes. Currently, there is no numerical Helmholtz solver that can satisfy these requirements at once.
We developed Schwarz domain decomposition preconditioners which leads to wave number independent convergence for wave numbers in 2D and 3D, while remaining to have accurate numerical solutions. The preconditioners use two-level Schwarz preconditioners, with the coarse problem being constructed using higher-order interpolation with quadratic rational Bézier curves. The developed domain decomposition preconditioners are designed to leverage parallel computing in the future in an attempt for the preconditioners to acquire the ability to solve with linear time complexity.

In this research, the preconditioner resulting in wave number independent convergence and the lowest iteration count is the two-level scaled hybrid Schwarz preconditioner with a coarse problem constructed using higher-order Bézier interpolation. This preconditioner uses a deflation method to remove unwanted eigenvalues. Removing these unwanted eigenvalues results in a clustering of the eigenvalues which is more favourable for GMRES. Currently, all the developed preconditioners suffer from high computational cost for large wave numbers, due to the coarse problem becoming large. Decreasing the coarse problem size of the preconditioners, while remaining to have wave number independent convergence, has shown to been unsuccessful. To better understand the required conditions for wave number independent convergence of the preconditioners, we investigated the relationship between the number of coarse grid nodes and the wave length, too see if there is anything generalizable about this relationship and wave number independent convergence of the solvers.
In conclusion, the balancing for a Helmholtz solver to have accurate numerical solutions, requiring a number of iterations to reach convergence that is independent of the wave number and solving with linear time complexity
is again shown to be difficult. This work provides the initial development and testing of promising wave number independent Helmholtz solvers, from which more research should follow that tackle its biggest computational problems.

A wide range of practical problems involve computing multi-dimensional integrations. However, in most cases, it is hard to find analytical solutions to these multi-dimensional integrations. Their numerical solutions always suffer from the `curse of dimension', which means the computational complexity grows exponentially with respect to the dimension.

There is an existing approach that approximates multivariate functions by a tensor of truncated multi-dimensional Fourier series coefficients, and uses the Stochastic Gradient Descent method to solve the lower-rank CPD model, which is used to reduce the computational complexity of the coefficient tensor. In contrast to this work, this thesis project extends its application to solve multi-dimensional integrations, utilizing the Fourier-cosine series expansion to represent the integrand. This project also replaces the SGD method with the Conjugate Gradient method, which improves the function matching accuracy significantly and also has great integration accuracy. The computational cost is reduced regardingly as well.

This thesis also tests an expectation operator related to the COS method, which can be used to compute the expectation of functions of several random variables with much less computational complexity. This method filters out insignificant Fourier-cosine basis functions of the marginal distribution functions, and uses the selected `principal' basis functions to compute Fourier-cosine coefficients of the joint density function by the high-dimensional COS method. The test results show that more than half of the Fourier-cosine series terms can be dropped per dimension while the expectation accuracy is kept, and the correlation does not influence the expectation accuracy significantly for target functions of normally distributed random variables.

Due to their attractive characteristics, convertible and callable bonds became a more important class of fixed-income products within the financial market. Therefore, the need for fair and accurate pricing of convertible and callable bonds increases. Where the convertible option can be considered as a right for the bondholder, the callable option is a right assigned to the bond issuer. Moreover, due to the hybrid nature of a convertible bond, it both contains characteristics of fixed income and equity products. As a consequence, both the risk of default and the possible equity profits need to be taken into account when valuing convertible bonds. Furthermore, as many convertible bonds also include a call option, an extra early exercise feature due to this call option needs to be taken into account. This early exercise option further increases the complexity of the valuation problem, as a possible buyback of the bond before maturity needs to be considered in the value of the bond. Computing a fair and accurate price for convertible and callable bonds, therefore gives rise to a complex valuation problem which leads to the need for further research. 
In this thesis, a structural default model is used to value callable convertible bonds. Contrary to reduced-form models, structural default models are characterized by the rationale behind the default event. Due to the better rationale and new insights into possible better market fits, structural default models have gained new interest in academic research. Recent research is conducted on finding new numerical techniques used for approximating the prices of complex financial products. This thesis discussed different valuation methods as proposed by Longstaff and Schwartz, Lord et al. and Oosterlee and Grzelak applied to the valuation of callable convertible bonds. In particular, a convolution-based method (CONV) and a cosine-based (COS) method are discussed. Whereas the Monte Carlo methods and the CONV method are already used in different articles, the COS method was not yet applied to the callable convertible bond valuation. As the COS method has been proven to be an efficient algorithm for approximating the values of financial derivatives, this thesis uses the COS method to compare its convergence to seek more insights into the convergence of the CONV method. Monte Carlo methods are used to verify the obtained approximations. 
For the problem considered under constant interest rates, this thesis shows that the COS method can be derived and applied. For small values of the grid size, the COS method showed to converge much faster than the CONV method. For larger values of the grid size, the CONV method showed to catch up with the COS method to become almost equally accurate. The results also showed that, contrary to the COS method, the CONV method was robust under the choice of the hyper-parameter concerning the integration grid. For the COS method, it was shown that a bad choice of the hyper-parameter could lead to a bad approximation. Although the two-dimensional CONV method shows to converge to the value obtained under Monte Carlo simulation, the results are still off for grid sizes of intermediate size. Results seem to indicate that the amount of grid points is not sufficient for the proposed integration interval and that therefore more grid points are needed. A greater amount of grid points, however, will also indicate a requirement for a greater amount of resources which may not always be available. 
From the results of the zero-coupon case, the COS method showed a more rapid convergence towards the approximations obtained by the Monte Carlo methods than the approximations obtained using the CONV method. Furthermore, when taking the hyper-parameters into account, the CONV method showed less robust features than the COS method in the two-dimensional case. Only for a small range of the hyper-parameters convergence is obtained for the CONV method. On the other hand, the COS method clearly shows convergence for a much wider range of hyper-parameters.

Chemical transport models (CTMs) are used to improve our understanding of the complex processes influencing atmospheric composition, as well as provide operational air quality forecasts and model potential future air quality scenarios. Numerical tracers in CTMs track the concentration of chemical species, while operators simulate various physical processes such as advection. One such CTM, LOTOS-EUROS, uses a volatility basis set (VBS) approach to represent the formation of organic aerosol (OA) in the atmosphere, which contributes to the concentration of total particulate matter. The added dimensionality of the VBS tracers in LOTOS-EUROS slowed down computation of the advection operator by a factor of two, limiting their representation in operational forecasts.
To keep the detailed process representation of OA formation, while reducing the computational costs, we develop an unsupervised machine learning method to compress the VBS tracers to a set of superspecies for use in advection, and subsequently decompress superspecies back to the tracer space for OA-relevant calculations. The focus of this machine learning method is physical interpretability, allowing for operators to resolve equations using the superspecies. This method conserves mass to machine precision and retains important information like phase (gas or aerosol) on compression. This data-driven approach reduces the dimensionality of the system more than a second proposed approach based on partitioning theory. The ML superspecies approach was integrated into LOTOS-EUROS for online calculations, showing numerical stability over a model simulation time of two weeks under various conditions. With the superspecies, the computation time for advection is reduced by 56% to 66% of the time for advection of the VBS tracers. The results of this approach show potential for use in accelerating air quality operational forecasts, as well as pathways forward for integration of ML box models of atmospheric chemistry into CTMs.

Algorithmic optimization is a viable tool for solving complex materials engineering issues. In this study, a data-scarce Bayesian optimization model was developed to research the composition of bio-based composites. The proof-of-concept program adjusts the natural materials' weight ratios to optimize towards user-defined mechanical properties. Preliminary results show that the bio-composites proposed by the program had improved properties compared to existing bulk-moulding compounds. However, the algorithm choice is often arbitrary or based on anecdotal evidence. In parallel, this thesis proposed a data-driven framework for general data-scarce optimization problems to adapt the meta-heuristic during optimization. Guided by the 'No Free Lunch' theorem, we verified that the effectiveness over a selection of algorithms is dependent on problem-specific features and convergence. This effectiveness was captured in a unique identifier metric by optimizing a generated training set of optimization problems. The average solution quality was improved by combining several meta-heuristics in series, based on these problem-specifics. During the optimization of problems in the testing set, the same unique identifier was constructed at predefined stages in the optimization process. Subsequently, the problem was classified, and the meta-heuristic was adapted to the best-performing algorithm based on similar training samples. Experiments with various classifiers and a different number of predefined assessment stages were performed. Results show that the data-driven heuristic decision strategy outperformed the individual optimizers on the testing set. Despite the use of binarization techniques, the classification accuracy was heavily influenced by the imbalanced training set. In terms of computational resources, the various adaptions of the data-driven heuristic strategy are 2.5 times faster in runtime compared to the best-performing meta-heuristic Bayesian Optimization. Lastly, the framework was benchmarked against the 'learning to optimize' study and shows excellent performance on the logistic regression problem compared to the autonomous optimizer. In conclusion, it has been shown that even with the limited information of black-box optimization problems, data-driven optimization effectively improves the current standard of materials engineering processes.

When creating electronic devices, it is essential to model what happens when an electromagnetic field hits the device and it scatters. Conventionally, this can be modelled using the Marching-on-in-Time algorithm. This can become computationally expensive for complex systems. To speed up the algorithm, the Plane-Wave Time-Domain algorithm is combined with the MOT algorithm. To accelerate the process even more, part of the algorithm is implemented using a Graphics Processing Unit, or GPU.

To test if using GPUs for this type of problem is actually beneficial, three experiments are set up. The first one tests the basic operations of addition and multiplication on matrices and vectors of various sizes, to determine if and when the computation time of the GPU is lower than that of a CPU. The second experiment tests the use of Fast Fourier Transform planner functionality and compares the CPU computation time with that of the GPU for the FFT of matrices of various sizes. The third experiment compares an example of the PWTD algorithm on the CPU and the GPU. These experiments are performed on three different devices.

The results from experiment 1 and 2 show that, after a certain point, the GPU is almost always faster, no matter the operation. Experiment 3 shows that the current GPU implementation is currently not as fast as the regular PWTD algorithm, though one of the devices is only 0.003\% slower.

In conclusion, theoretically a decrease in computation time is expected. From experiment 3 it follows that it is not the case yet, though with more optimisation the GPU implementation would almost certainly become faster.

In the ocean resides a large amount of plastic. This has severe environmental consequences for the oceans. To be able to clean up all this plastic the location of the plastic is needed. A lot of times plastic is cleaned up at the source, but to clean up here this location needs to be known. However, what if the start location is not known and only the end location is known. This thesis looks at the following research question: Can the method of particle filtering be used to find the start location of a plastic particle of which the end location is known?. The method of particle filtering compares the end locations of the released particles and the end location of the end particle. (The end particle is the particle of which the end location is known and we want to know the start location.) The method calculates weights on the basis of this comparison for each of the released particles. Then the weights are used to calculate a probability density function that gives the likelihood of possible start locations being the right start location of the end particle. The method is applied to the MSC Zoe case, which is a container accident, and a simplification of this case. The MSC Zoe lost in this accident hundreds of containers above the Wadden Islands in the Netherlands, in these containers were bags of plastic particles. A lot of these particles washed ashore on the Wadden Islands. However, it is not known exactly where all the containers fell overboard. So the start location of the plastic particles that washed ashore is not known. The method of particle filtering applied to the simplification showed that it can indeed be used to find the likelihood of possible start locations. However, it also showed that for some values of the variables it was possible to get multiple peaks with weights that are zero in between. This is not a desired outcome for a probability density function (pdf). So a kernel estimation is used to smooth out the pdf's. It uses the weights calculated in the method of particle filtering to estimate the pdf. This gives results that also in this way the likelihood of possible start locations can be found and now without multiple peaks. The kernel estimation was also applied to the MSC Zoe case and the results were calculated for multiple end particles. The locations that came out of this were very close to a few of the possible main locations were containers fell overboard. These possible main locations came from the international investigation into the MSC Zoe accident. A simplification of this research is that the start location is found for one specific plastic particle. For future research it could be interesting to look if the method can be adapted to find the start location of concentrations of plastic particles.