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.

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 emission of reactive nitrogen species increased rapidly in the twentieth century causing a significant perturbation of the nitrogen cycle. This causes many detrimental effects for both humans and ecosystems such as eutrophication, acidification and biodiversity loss. Ammonia, which is a reactive nitrogen species is difficult to quantify because the compound is hard to observe with measurement equipment such as ground-based measurement stations and remote sensing instruments such as CrIS. Therefore, this thesis aims to improve the monitoring of surface ammonia concentrations with a machine learning technique called random forests (RF). These type of models can detect complex and non-linear relationships between variables and are frequently used in other air pollution studies. In this study, a RF model has been built to estimate the ammonia surface concentration with vertical column density (VCD) datasets, meteorological variables and land-use variables. Different combinations of VCD datasets (either modelled or CrIS VCD data) have been used to train the model and to predict the surface ammonia concentration. The result of the study is that RF models are statistically more accurate at estimating the surface ammonia concentration than the LOTOS-EUROS model when validated by ground-based measurements stations from the MAN and LML network. Especially trained RF models that have been supplied with CrIS VCD data during the prediction phase show strong performance. Moreover, when comparing the RF model that has been trained without VCD data to the ‘complete’ RF models, the complete RF models show better performance, proving the added benefit of incorporating satellite data in RF modelling. Recommendations for further research include performing similar experiments with other input variables and other machine learning algorithms, validating the performance of the RF model in different years and considering using other datasets as the ground-truth variable for RF models – such as ground-based measurement data from MAN and LML.

Isogeometric Analysis (IgA) is an extension of the more well known Finite Element Method (FEM). It allows for more accurate descriptions of boundary value problems on irregular domains. However, many of the traditional iterative solution strategies that are known to work well in FEM do not show the same behavior in IgA, especially for increasing order of basis functions p.

A method shown to have fast convergence in this situation is a p-multigrid method with a smoother based on a Block ILUT factorization. Most of the blocks of this factorization are efficiently calculated. The same holds for the smoothing steps.

It is therefore our objective to make changes to the Block ILUT smoother. Inspiration is taken from methods where ILU factorizations are constructed using a fixed-point iteration. We combine these with the existing Block ILUT smoother.
This ultimately leads to two new proposed methods we will call Block Fixed-point ILU and Block ParILUT.

The existing methods as well as the newly suggested methods are tested and compared on computational costs of the factorization, the number of nonzero entries in this factorization and the number of multigrid iterations needed
to reach convergence, if these factorizations are to be used as smoother. The benchmark used for these tests is a convection diffusion reaction (CDR) equation on a multipatch geometry with 4, 16 or 64 patches.

Large textures that can provide realistic details are widely used in modeling, gaming, art design, etc. Texture synthesis is a way to create large textures based on a small sample pattern, which can be obtained by image examples or hand­drawn work by an artist. Different methods that aim to achieve better visual effects on reducing or avoiding seams and distortions during synthesis are proposed. It has been a popular topic both in computer graphics and computer vision for many years. However, most of the synthesis methods are automatic and take images as the input. There are few methods designed for artists who want to control and create patterns manually. The goal of our Diffusion Mosaics is to introduce a graphics tool that allows the artist to create tiles that can be seam­ lessly concatenated. In this way, non­periodic textures of arbitrary size can be produced at very low memory costs. We rely on a special kind of tiles called Wang tiles, which are squares with colored edges. Neighboring tiles should share the same edge color at the common border. The texture tiles are designed in such a way that if this constraint is fulfilled, the transition from one tile to the next will be seamless. It becomes possible to create a large non­periodic final texture. In our system, each tile can be filled with hand­drawn patterns using diffusion curves, which is a vector graphics primitive created by diffusing the given colors of defined Bezier curves, which has been proven to be a useful alternative to purely pixel­based design that artists typically rely on. They also match our application well, as, by adding border­color constraints to the tile, solving the diffusion process results in tiles that match the neighboring tiles naturally, hence avoiding any seams. The tool gives creative freedom to the artist that they can even draw the pattern outside the tile area, resulting in an automatic update of all other tiles when needed to ensure consistency. Finally, for more complex illustrations and back­ wards compatibility to software such as Photoshop, pixel­based images are also supported and can be integrated.

In this project, we develop a mathematical framework that automates AC power flow simulations by solving both converging and diverging power flow problems. On the basis of the mathematical framework, we develop an algorithm that can be used to perform year-round AC power flow simulations with little manual intervention. We demonstrate the applicability of the algorithm by implementing it on artificial test networks, and two real-world transmission networks - the Dutch transmission network and a sub-European transmission network.

The single-step single nucleotide polymorphism best linear unbiased prediction (ssSNPBLUP) model can potentially be used in animal breeding for genetic evaluations. It has been reported that this model has convergence issues when the Preconditioned Conjugate Gradient (PCG) method is applied. This is due to the linear system being ill-conditioned. Moreover, in a recent research a subspace decomposition deflation method has been proposed. Unfortunately, the method requires too many deflation vectors, which results in an implementation of the Deflated PCG (DPCG) algorithm that requires a high demand of RAM. In this thesis a different subdomain decomposition strategy is proposed. This subdomain decomposition method utilises the $k$-means algorithm applied on the matrix of correlation among the single nucleotide polymorphisms (SNPs). This method has been applied on a simulated data set of dairy cattle. This method results in halving the number of deflation vectors required for the same rate of convergence compared with the previous subdomain decomposition deflation method.

In practice for genetic evaluations the data sets increases over time by incorporating new information. In this thesis a specific initial solution has been investigated. This initial solution closely resembles the solution of the updated system to solve the ssSNPBLUP model efficiently. Also a deflation method that utilises the previous solutions of genetic evaluations can be applied. The Proper Orthogonal Decomposition (POD) based deflation method is proposed in this thesis to full fill this requirement. Moreover, only a few deflation vectors are needed to improve the rate of convergence.

Furthermore, the subdomain decomposition deflation method deflates the largest eigenvalues of the preconditioned coefficient matrix and the POD deflation method deflates the smallest eigenvalues of the preconditioned matrix. Combining the deflation vectors of both methods results in an improvement in the rate of convergence compared with the methods on their own.

Based on the results the best performing method is a combination of selecting the initial solution as the previous genetic evaluation, the POD deflation method and the k-means++ clustering applied on the subdomain decomposition deflation method. In this research a reduction of 81% in iteration time and 19% in total computation time has been observed. Note that the initial solution as the previous genetic evaluation approach can only be employed if that solution is available. Likewise, the POD method can only be applied if multiple solutions of genetic evaluations are available.

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.

It is possible to estimate derivatives of real functions with complex methods. These complex methods can increase speed or accuracy. The functions needs to be known to determine the derivative with a complex method.

In dit onderzoek hebben we een nieuwe dienstregeling voor tramlijn 19 opgesteld. Deze tramlijn rijdt nu tussen Delft en Leidschenhage. In 2020 zal het traject worden uitgebreid met haltes op de campus van de TU Delft. In dit onderzoek hebben we twee dienstregelingen opgesteld: één voor in de spitsuren, en één voor in de daluren. Er zijn een aantal voorwaarden meegenomen in het onderzoek. Zo moet de dienstregeling aansluiten op de collegetijden van de studenten die gebruik maken van tram 19. Ook moeten we de voorwaarden die we in de huidige dienstregeling vinden, meenemen in het nieuwe traject. Denk bijvoorbeeld aan de frequentie waarin trams rijden, of wanneer de dienstregeling start op een dag. Daarnaast hebben we onderzocht hoe het beste kan worden overgegaan van een spitsuur dienstregeling naar een met daluren. Dit alles is gedaan met behulp van max-plus algebra, waarbij het nemen van een maximum en optellen, de conventionele operaties van optellen en vermenigvuldigen vervangen. Het power algoritme helpt ons bij het kiezen van de juiste vertrektijden, zodat we een constant vertrekschema krijgen. Ook maken we gebruik van Petrinetten omhet gehele netwerk temodelleren. In dit onderzoek wordt steeds eerst gekeken naar een kleiner of versimpeld tramnetwerk. Hierna passen we de theorie toe op het toekomstige traject. Op het einde hebben we een dienstregeling kunnen opstellen voor tramlijn 19. In deze dienstregeling vindt afwisseling plaats tussen spits-en daluren, met elk een andere dienstregeling en hoeveelheid trams. Hierbij is rekening gehouden met de voorwaarden die wij voor ons model hebben opgesteld.

The finite element method (FEM) is an ubiquitous method for the analysis of boundary value problems. Specifically, it can be used to find approximations to solutions of boundary value problems on a specific domain...