The standard bin packing problem is about distributing a set of items with positive sizes over bins of certain capacity in such a way that the number of bins used is minimized. In this thesis a relatively undocumented version of bin packing with negative items is discussed. Negative items added to the standard problem can reduce the load in a bin. In context, this means energy is regenerated or recharged. This forces us to take a look at the problem in a different way. The goal of this thesis is to find methods that give solutions with a guaranteed desired quality. For the standard problem, we already have well-working heuristic algorithms such as the First Fit (FF), Best Fit (BF), and their decreasing-order variants (FFD and BFD). These algorithms have nice bounded approximation ratios. However, introducing negative items sabotages these nice approximation ratios. We know negative items can reduce the total number of bins, so naively applying classical algorithms does not perform well, as these methods do not take the potential of negative items to offset large positives and reduce bin usage into account. To try and solve these problems, we introduce the adjusted heuristic algorithms Negative First Fit (NFF), Negative Best Fit (NBF), and their decreasing-order variants, which work by placing all negative items into the first bin before applying the standard algorithm. By placing all negative items in the first bin, we can get a packing that uses 1 bin or more. If the packing uses only 1 bin, the problem is trivial. If it uses more than 1 bin, we know that all negative items in the first bin have been canceled out, so we can retain the logic of the standard problem and thus the approximation ratio. Although the negative adjusted algorithm has a nice approximation ratio, putting all negative items in the first bin does not make sense in a real-life context. That is why we also explore the heuristic method of dividing negative items over large positive ones before applying standard packing algorithms. Here we focus on the notion that fewer big items, where we define big items as items larger than half the capacity, lead to a smaller number of bins for the solution because all big items need to be in a separate bin. First we have Decreasing Division (DD), which pairs the largest big item with the smallest negative items. Unfortunately, this does not guarantees a reduction of big items. To try and improve on this, we also introduce Increasing Division (ID). ID pairs the smallest big item to the smallest negative item. This way, ID can give a bigger guarantee to reduce the number of big items. DD and ID both have the same approximation ratio and both give more logical packing results. However, we proved that for DD there is no better approximation ratio. For ID however, we saw that a better approximation ratio may still be found. This thesis is mostly based on literature research, mathematical analysis and explanatory examples. We show the potential performance and limitations of different heuristic algorithms for bin packing
with negative items. The results achieved give a different perspective on the bin packing problem and leave open future research in optimizing bin packing with negative items. But most importantly, the goal of finding methods that work on the bin packing problem with negative items was achieved.

In this thesis, we study fractional differential equations with Hilfer derivative operators. Solutions are approximated using a newly developed Bernstein-splines approach and subsequently applied to the Van der Pol oscillator with fractional damping. Fractional derivatives generalize differentiation to the order α ∈ (0,∞), resulting in order α differential operators. The Hilfer-derivative of order α ∈ (0,1) and type β ∈ [0,1] is one of such operators and combines two of the most commonly used operators through parametrization: the Riemmann-Liouville-derivative (β = 0) and Caputo-derivative (β=1). The choice of β results in different kernel behavior of the operator, commonly yielding singular behavior of solutions of initial value problems (IVP's) and boundary value problems (BVP's) for β ∈ [0,1). Based on existing results for IVP's, we develop a new proof for existence and uniqueness of solutions to BVP's for Hilfer-fractional derivatives. To obtain solution approximations numerically for IVP's and BVP's, a Bernstein splines solution approach is developed and implemented, providing accurate convergence results for nonlinear IVP's and BVP's in an efficient vectorized approach. Implementation difficulties for the singular behavior of solutions for β ∈ [0,1) are successfully resolved through time-domain transformation approximation techniques. Finally, the methods are applied to numerically study the behavior of the fractionally damped Van der Pol oscillator, a nonlinear equation of interest in electrical engineering and control theory. We study the approximate limit cycle, of which behavior corresponds with existing analytical results for the Caputo-derivative (β=1). Convergence of solutions can also be obtained for Hilfer type values of β ∈ [0,1), appearing to be of little influence on the long-term limit cycle. Furthermore, when forcing is applied, we observe periodic, quasiperiodic and chaotic behavior.

To diagnose diseases using low-field MRI devices, automatic image recognition techniques are required to detect anomalies in a given image. A critical component of this process is image segmentation, which involves dividing the image into coherent regions with similar features to detect the anomalies. Low quality images make segmentation difficult by the existence of noise in the images. The Chan-Vese model is a segmentation technique to segment images into two regions. This report aims to segment the low quality images obtained by the low-field MRI devices with use of the Chan-Vese model. This model is found to be particularly well-suited for handling the challenges posed by low-quality images which the low-field MRI devices produce. Future research should investigate if it is possible to further segment the region with the objects, such that each object has a separate region.

Local search approaches are often used to find solutions for optimisation problems. However, these approaches easily get stuck in local optima, who are not yet globally optimal. More complex variants of those approaches cannot always escape those local optima easily. Thus, the need for another effective technique arises. This thesis proposes such a technique: potential KPIs. A potential KPI is an addition to the objective function such that the local search algorithm accepts solutions with `potential', i.e., solutions that are closer to the global optimum in some sense.
This thesis focuses on two optimisation problems, the bucketised planning problem (BPP) and the travelling salesman problem, to investigate the performance of potential KPIs. Various potential KPI functions and employment methods are tested to determine the efficacy of potential KPIs and to study their behaviour.
The findings of this thesis indicate that adding a potential KPI to the objective function can significantly improve a basic local search algorithm. The performance of the method with potential KPI is highly dependent on the chosen potential KPI function. Moreover, defining such a function is challenging since some knowledge about the local and global optima is needed. While a random function can already improve the algorithm a lot, using a well-constructed function performs even better as potential KPI.
Furthermore, the performance of the method is dependent on how the chosen function is employed. The method should always end with the local search with the original objective function, but whether to start with or without potential KPI is dependent on the function. Nevertheless, starting without potential KPI seems to be slightly better due to the lower runtime and ability to be always better than the basic local search. The weight of the potential KPI in comparison with the original objective should be high and instance-dependent. Additionally, the initial solution also influences the found solution of the method, as a better initial solution, such as a greedy solution, leaves less room for improvement for the basic local search. Thus, the technique of a potential KPI can significantly improve the basic local search, but its effectiveness is dependent on the method of employment and constructed function. Moreover, a potential KPI can only be used when enough information about the optimisation problem is known.

The computational demands of finite element simulations, particularly in predicting the time-dependent response of high-dimensional non-linear dynamical systems, pose significant challenges. To overcome these challenges, researchers have developed model order reduction (MOR) methods, which aim to reduce computational complexity by utilizing lower-dimensional models. This thesis proposes a MOR technique that simultaneously learns both the projection to, and the reduced dynamics on, a lower-dimensional manifold using autoencoders, a type of neural network. During training, the known linear part of the reduced dynamics is used to aid the optimization process, leading to an effective method of simultaneous projection and linear informed training (SPLIT). SPLIT demonstrates outstanding performance on the test case of a 2D cantilever beam, and is capable of making non-linear forced response predictions, even though being trained on unforced decaying trajectories. Even in scenarios involving highly non-linear behaviour, such as when the beam folds over itself, SPLIT continues to make accurate predictions, while other MOR techniques fail. This work highlights the potential of autoencoders to advance the field of MOR and improve the efficiency and reliability of simulations for complex dynamical systems.

Phylogenetic trees are commonly utilised in evolutionary biology. These trees represent the evolution of a set of species. In case of gene transfer however, a phylogenetic tree is insufficient. In such cases, tree-based phylogenetic networks are more suitable as they can depict reticulate evolution.
Nonetheless, tree-based phylogenetic networks have their limitations as they are unable to represent reticulate events occurring between different environments, families, or lineages of species. To address this, forest-based networks offer a solution.
A forest-based network is a collection of leaf-disjoint phylogenetic trees with arcs added between different trees in case of reticulate evolution.
In this thesis, an integer linear programme (ILP) is created to determine whether a binary phylogenetic network is forest-based. It is important to note that this thesis exclusively focuses on binary phylogenetic networks.
This ILP model is expected to be valuable in the development of new models and algorithms for both binary and non-binary phylogenetic networks, thereby enhancing the understanding of reticulate evolution. Additionally, the ILP may lead to time-related challenges when processing bigger phylogenetic networks.

This thesis treats the thin-film equation which models the film height $h$ for a viscous film in the complete wetting regime. We show existence and uniqueness to the thin-film equation with mobility m(h) = hⁿ and mobility exponent n∈ (1,3/2)∪ (3/2,3). The thin-film equation is rewritten as an abstract Cauchy problem and usage of semi-group theory yields maximal L^p-regularity for the linearized problem. With a fixed point argument, analogous to the one used by Giacomelli, Gnann, Knüpfer and Otto in their 2014 paper, the nonlinear problem is treated. Under a smallness condition on the initial value to a suitably transformed version of the thin-film equation, we obtain a solution in L^p(0,∞;H_{k-2,α-1/2})∩ \dot{W}^1,p(0,∞;H_{k+2,α+1\2}), where the H-spaces denote weighted Sobolev spaces. The novelty of this work lies in the usage of L^p-spaces in time, where the existing literature only deals with L²-spaces. It is found that the L^p setting allows for treatment of all n∈ (1,3/2)∪ (3/2,3).

Central configurations provide the only closed-form analytical solutions of the n-body problem. All possible central configurations of three bodies have been extensively studied along with the stability of the associated periodic orbits. Stable cases have been found for the Lagrangian triangle configuration, which we see occurring with the Trojan asteroids. However, the knowledge about four-body central configurations remains limited. An explicit parameterization of a family of kite shaped four-body central configurations has recently been published. The present research investigates the stability of periodic solutions provided by these central configurations. An analytical treatment of linear stability is carried out and the eigenvalues for circular periodic orbits are calculated. This is complemented with a numerical estimation of Floquet multipliers to determine the linear stability of eccentric periodic orbits. While most of the kite configurations are found to be unstable, regions of linearly stable cases are discovered for both circular and eccentric orbits. Further, numerical simulations of the non-linear system are performed as an independent approach to validate the linear stability results. Perfect agreement with the linear analysis is found, suggesting that stable kites may be observed in the universe.

Modelling natural phenomena via Dynamical System Theory has become commonplace amongst mathematicians, physicists, engineers, and the like. As such, research in this field is currently underway, with a notable focus on chaos. The Competitive Modes Conjecture is a relatively new approach in the field of chaotic Dynamical Systems, aiming to understand why a strange attractor is chaotic or not. Up till now, the Conjecture has only been used to study multipolynomial systems because of their simplicity. As such, the study of non-multipolynomial systems is sparse, filled with ambiguity, and lacks mathematical structure. This paper strives to rectify this dilemma, providing the mathematical background needed to rigorously apply a large set of non-multipolynomial systems to the Competitive Modes Conjecture. Examples of this new theory include application of Lorenz System, the Chua System, and the Wimol-Banlue System. As far as the authors are aware, any previous application of the latter two systems to the Conjecture has not been attempted. Therefore, this paper presents the first applications of a whole new set of dynamical systems to the Conjecture.

Cholera is an infectious disease caused by the bacterium Vibrio cholerae. Infections can occur directly via contact with infected people, but also indirectly via infected water. Because of these direct, but especially indirect infections, it is useful to investigate the spread of the disease using mathematical epidemic models. In this report a compartmental epidemic model for cholera is discussed: the SIRB model. This model consists of four compartments: susceptibles, infected, recovered and the bacterial concentration in water. The interaction between the groups is given by a system of differential equations. The equilibrium solutions and their stabilityare mathematically analysed. The main focus of this report, however, is on two different spatial models. The patches model consists of multiple patches connected by migration of people. Different measures against cholera are studied using this model. The second spatial model that has been numerically implemented is a diffusion SIRB model. The influence of the diffusion coefficients on the solution patterns will be researched for different initial conditions. The results presented in this report are building blocks for further research into compartmental models that simulate the spatial dynamics of cholera.

Graphene is expected to open up a world of new possibilities in the field of micro and nano sensors, due to its outstanding properties. However, graphene membranes suffer from high damping which limit their performance. The physical mechanism behind this damping is still unknown. Therefore, in this work, it is investigated how coupling between eigenmodes due to nonlinear stiffness causes damping. Besides, a method to construct this nonlinear stiffness for reduced order models is improved. Using this, numerical simulations are compared to experimental results for validation.

Ballistic capture is a transfer method which was first applied in 1990. It allows a spacecraft to approach a target celestial body and enter a (temporary) orbit around it without requiring manoeuvres in between. Ballistic capture is a promising concept, as it is expected to be safer, cheaper, and more flexible in terms of launch windows than a traditional Hohmann transfer. Currently, a computationally efficient method which simulateneously allows for an inisghtful description of the dynamics of the ballistic capture problem remains to be found. A potential solution lies within the field of Lagrangian Coherent Structures (LCS). LCS is defined as a separatrix of regions in a flow with distinct dynamics. It may be possible that LCS around a planet have some correspondence to results found using stable set manipulation, a classic technique for obtaining capture trajectories. In this research three new areas within the field relating LCS to ballistic capture are explored. Firstly, it has not yet been shown what LCS can be found in an area around a planet, without making use of a priori stable set information. Furthermore, it is unclear what the effect is of changing the integration time in the procedure of extracting LCS. Finally, there has not yet been an analysis to show how the LCS relate to stable sets with different number of revolutions n. In this work two algorithms for extracting LCS have been developed. One is based on the simple but efficient computation of the Finite Time Lyapunov Exponent (FTLE). Another is based on the more involved Variational Theory. Both algorithms are validated on a toy problem used frequently in LCS extraction studies, and are then applied to the Elliptic Restricted Three Body Problem (ERTBP). It is shown that LCS around a planet yield resemblance with stable set results. The FTLE-based algorithm is able to quickly and efficiently identify the shape of the stable set. The Weak Stability Boundary, however, can not be extracted distinctly. The Variational Theory-based algorithm yields more distinguishable results for the Weak Stability Boundary. It is shown that large and constant integration times are beneficial. It is shown that extracted LCS form an approximation of the average resulting WSB for all stable sets.

Using an idealized width-averaged model, the influence of spring-neap cycles on the transport and trapping of suspended fine sediment in tide-dominated estuaries is investigated. To this end we introduce a multiple time scale expansion. This provides a mathematically sound argument for treating the fast ebb-flood cycle and the slow spring-neap variations as independent time scales. With this expansion, semi-analytical approximations of the water motion and suspended sediment concentrations can be found as functions of the spring-neap time scale. The advantage of this is that the transport of sediment is then investigated in a tidally-averaged sense using analytically obtained temporal dependencies and model simulations based on the conditions in 2005 of the Ems-Dollard estuary. We found that sediment import is strongly enhanced during spring tide and that a combination of vanishing import processes and sediment export due to river discharge resulted in a net export of sediment during neap tide. Furthermore the spring-neap varying deposition of fine sediment in a bottom pool on the river bed is investigated. We found that the time needed for the system to adjust to new equilibrium conditions can not be neglected within a spring-neap cycle as temporal lag effects are clearly visible in the dynamic behaviour of the bottom pool and suspended sediment concentrations. These temporal lag effects are sensitive to the choice of the parameter governing erosion. Assuming different values of the erosion parameter we have shown that the long-term characteristics of the bottom pool are sensitive to these temporal lag effects. Choosing a low value of the erosion parameter results in the presence of a bottom pool throughout the spring-neap cycle. With a high value of the erosion no bottom pool is formed at any time in the spring-neap cycle.

Structure Optimization has been an important subject with many applications for centuries. In the last sixty years, numerical optimization has facilitated large advancements in this field. One of the areas in Structure Optimization is Topology Optimization, which is used for Additive Manufacturing purposes. In this thesis we explore Static and Dynamic Topology Optimization. In the optimization problems matrix equations of the type Ax = b, with A sparse and badly conditioned, are accelerated using deflation techniques in addition to preconditioning. We have applied several iterative methods, preconditioners, and deflation types to the topology optimization problems. The static problem concerns compliance optimization of a two-dimensional MBB-beam. The deflation type that reduced the number of iterations the most without introducing large overhead costs was rigid body modes deflation, divided over element squares. It was found that dividing the rigid body modes vectors over density based regions did not reduce the number of iterations. The dynamic problem concerns eigenvalue optimization of a three-dimensional moving wafer stage that is used for laser-printing computer chips. The optimization formulation contains a shifted eigenvalue problem that is solved using model order reduction. In the computations of bases for the reduction matrix equations appear, to which deflation techniques were applied. All deflation types reduced the number of iterations and needed time to solve the matrix equations. The best deflation type was using rigid body modes (RBM) divided over element cubes combined with eigenvectors from the previous iteration. The next best was the same combination without the division over cubes. Using eigenvectors or RBM separately were the least effective deflation types. There is an optimal amount of element cubes to use when dividing the RBM. The tests on a few grid sizes suggested a quadrupling of the amount when the grid size doubles, but more research is needed to really identify a relation between the grid size and optimal amount. When increasing the grid size to a level where parallel computing on a cluster was required, the deflation type using element cubes could not be used due to complications with parallel implementation. Using the deflation type RBM and eigenvectors, reductions by a factor of 1.60 and 1.75 in the total needed time for 150 optimization iterations were achieved for grid sizes 120x120x20 and 180x180x30, respectively. In the first case the objective function of the optimized design converged further in the same amount of iterations when using deflation. future research could include using the promising deflation type RBM in cubes + eigenvectors in parallel computing to obtain an even larger time reduction in the optimization of the wafer stage with large grid sizes.

In the past decades, several EOR (Enhanced Oil Recovery) techniques have been developed to increase the amount of oil recoverable from a reservoir. Among these techniques, polymer flooding consists in the injection of a solution of water and polymer into the reservoir, and it is performed in order to lower the mobility of injected water. A peculiarity of a water-polymer flow through a porous medium is the velocity enhancement, or hydrodynamic acceleration, that affects the polymer molecules. Experimentally, polymers are observed to travel faster than inert chemical species. Therefore, when simulating numerically a polymer flooding, a velocity enhancement effect has to be incorporated into the governing equations to accurately predict oil recovery. The simple model that introduces a constant enhancement factor, usually implemented in commercial simulators, leads to an ill-posed problem, causing stability issues in the simulations, which further leads to a monotonicity loss in the solution: the polymer is predicted to pile-up at the water-oil interface. While accumulation of polymer at the water front is not necessarily unphysical, it should not be the result of a mathematical ill-posed problem. Hence, an alternative model, employing a saturation dependent factor, has been proposed in the work of Bartelds et al. (G.A. Bartelds, J. Bruining, and J. Molenaar. The modeling of velocity enhancement in polymer flooding. Transport in Porous Media, 26(1):75–88, 1997), and has been extended by Hilden et al. (S.T. Hilden, H.M. Nilsen, and X. Raynaud. Study of the well-posedness of models for the
inaccessible pore volume in polymer flooding. Transport in Porous Media, 114(1):65–86, 2016) to overcome some physical restrictions. In this thesis, these models are re-examined through a thorough analytical study in the one-dimensional case. An analytical solution, resulting in an acceleration of the polymer front, is computed for the well-posed model proposed by Bartelds, while the extended enhancement factor derived by Hilden is shown to lead again to an ill-posed problem. A study of the monotonicity of the numerical schemes reveals that accumulation of polymer at the water front is strictly related to ill-posedness.
Obtaining accurate numerical solutions is not an easy task. Commercial simulators adopt fully implicit schemes to solve the multi-phase flow, and transport of chemical agents is solved through the development and coupling of separate modules. In this case, the use of high-resolution methods for the transport of polymer is not compatible with the first order solver for the underlying flow.
To investigate the interaction of the enhancement model proposed by Bartelds with other physical phenomena, adsorption of polymer onto the reservoir rock is added to the governing equations. The problem maintains the well-posedness property, but because of a raise in equations complexity, only numerical results are presented. The model is then extended to the two-dimensional case and, relying on numerical solutions, similar conclusions as in the one-dimensional case are derived.