1 

Estimating the extreme value index for imprecise data
In extreme value theory the focus is on the tails of the distribution. The main focus is to estimate the tail distribution for a rounded data set. To estimate this tail distribution the extreme value index should be estimated, but due to the rounded data this extreme value index oscillates heavily. Therefore a correct estimate can not be obtained. By adding a small uniform stochast the rounded data can be smoothend out and in this way the oscillation can be cancelled.

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2 

Optimal boundary point control for linear elliptic equations
This master thesis is concerned with optimal boundary point and function control problems for linear elliptic equations subject to control constraints. The elliptic partial differential equation with Robin boundary condition is considered. The control is chosen as a linear combination of the Dirac delta functions in the point control problem. The weak formulation and optimality conditions are obtained for the function control problem. The main goal is to examine the existence of the weak solution and to derive optimality conditions of boundary point control. Introducing sufficient discretization methods such as a finite volume and a finite element methods, we obtain finite dimensional problem.
We apply efficient numerical methods including primaldual active set strategy, projected gradient and conjugate gradient methods. The test examples are presented clarifying the performance of numerical methods mentioned earlier. The conjugate gradient method is compared to the unconstrained matlab function QUADPROG and primaldual active set strategy is compared to the constrained QUADPROG. The projected conjugate gradient method is applied to improve the projected gradient method. Due to its importance, the sparse point and function control problems are studied. We apply primaldual active set strategy where the sparse parameter f is chosen differently. All of the results are presented in both point and function control problems.

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3 

Optimal distributed point control of linear elliptic equations
In this master thesis, we consider the optimal function and point control problems governed by linear elliptic partial differential equations together with bilateral control constraints. The aim of this work is to choose a control function by linear combination of the Dirac delta and solve the optimal distributed point control problem. In this work, optimality systems of point and function control problems are derived by Lagrangian principle and reduced functional respectively. The optimality system is discretized by the finite element method (FEM) and finite volume method (FVM). We apply a semismooth Newton (SSN) method which is equivalent to a Primal dual active set strategy (PDASS) to solve the discretized optimality system. As a second solution method, we propose Projected gradient (PG) method for the same problem. For each method, we compare the result of FEM and FVM and give a preference. In order to have the best solution method, we compare the results of PDASS and PG methods to the results of matlab command QUADPROG. We have to solve two partial differential equations in every iteration, namely the state and the adjoint equations. Therefore, we develop a Multigrid Preconditioned Conjugate Gradient (MGPCG) method for solving the discretized optimality system as fast as possible. Finally, we consider the linear elliptic optimal control problems with L1 norms in the cost functional which results sparse control problem. Due to L1 norm, objective functional becomes nondfferentiable and the optimal controls are identically zero on large parts of the control domain. Using an appropriate smoothing of the nondifferentiable terms for the cost functional, we solve the optimal sparse control problems theoretically and numerically.

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4 

Investigation of Different Solvers for Radiotherapy Treatment Planning Problems
Radiotherapy treatment planning involves solving inequality constrained minimization problems. The currently used interior point solver performs well, but is considered relatively slow. In this thesis we investigate two different solvers based on the logarithmic barrier method and Sequential Quadratic Programming (SQP) respectively. We argue that the behaviour of the logarithmic barrier solver is uncertain, thereby making it generally unreliable in this context. In addition we substantiate that the performance of the SQP solver is solid, but lacks efficiency in computing the minimizers of its related quadratic subproblems.
We conclude that without serious improvements, none of the solvers investigated are faster than the currently used interior point optimizer.

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5 

Investigation of Different Solvers for Radiotherapy Treatment Planning Problems
Radiotherapy treatment planning involves solving inequality constrained minimization problems. The currently used interior point solver performs well, but is considered relatively slow. In this thesis we investigate two different solvers based on the logarithmic barrier method and Sequential Quadratic Programming (SQP) respectively. We argue that the behaviour of the logarithmic barrier solver is uncertain, thereby making it generally unreliable in this context. In addition we substantiate that the performance of the SQP solver is solid, but lacks efficiency in computing the minimizers of its related quadratic subproblems. We conclude that without serious improvements, none of the solvers investigated are faster than the currently used interior point optimizer.

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6 

Solving Stochastic PDEs with Approximate Gaussian Markov Random Fields using Different Programming Environments
This thesis is a study on the implementation of the Gaussian Markov Random Field (GMRF) for random sample generation and also the Multilevel Monte Carlo (MLMC) method to reduce the computational costs involved with doing uncertainty quantification studies. The GMRF method is implemented in different programming environments in order to evaluate the potential performance enhancements given varying levels of language abstraction. It is seen that the GMRF method can be used to generate Gaussian Fields with a Mat{\'e}rn type covariance function and reduces the computational requirements for large scale problems. Speedups of as much as 1000 can be observed when compared to the standard Cholesky Decomposition sample generation method, even for a relatively small problem size. The MLMC method was shown to be at least 6 times faster than the standard Monte Carlo method and the speedup increases with grid size. It is also seen that in any Monte Carlo type methods, a Krylov subspace type solver is almost always recommended together with a suitable preconditioner for robust sampling.
This thesis also studies the ease of implementation of these methods in varying levels of programming abstraction. The methods are implemented in different languages ranging from the most common language used by mathematicians (MATLAB), to the more performance oriented language (C++PETSc/MPI), and ends with one of the newest programming concept (ExaStencils). The GMRF method featured in this thesis also is one of the earliest application to be implemented in ExaStencils.

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7 

Simulating sprouting angiogenesis: using a new 3D substrate dependent cellbased model
Angiogenesis1 is the biological mechanism by which new blood vessels sprout from existing ones. It differs from vasculogenesis, which is the de novo growth of the primary vascular network from initially dispersed endothelial cells (ECs). Vasculogenesis is predominant in embryonic tissue whilst new vasculature in the adult body arises mostly from angiogenesis. ECs, lining the inside of blood vessels, react to different angiogenic stimuli and inhibitors. Among the stimuli is the vascular endothelial growth factor (VEGF) which is upregulated in tissue where the vascular structure is damaged or insufficiently developed to meet oxygen demand.
The identification of the processes involved in angiogenesis is quite recent and has stirred increased interest in therapeutic and clinical applications according to Carmeliet et al. [1]. One can think of tissue repair in wound beds, inhibition of growth of tumorous tissue or vascular reform during the female reproductive cycle. Rossiter et al. [2] showed that VEGF induced angiogenesis is crucial for wound healing in an experiment where wounds were inflicted upon normal and VEGFdeficient mice. New vasculature ensures supply of oxygen and lymphocytes and disposal of carbon dioxide and lactates, accelerating wound healing and tissue reconstruction. The increased creation of new vasculature around tumorous tissue is believed to follow the same process and inhibiting angiogenesis is therefore an important topic in clinical studies on cancer treatment.
Biochemical laboratory experiments can be hard, time consuming, expensive or unethical. Computational models can be used to provide an easy, quick and cheap way to get insights that would otherwise require laboratory experiments. The understanding of biological processes needs quantification and in this sense mathematical formulation of the relations involved becomes useful. Their mathematical interpretation and experimental verification is an iterative process resulting in better understanding of the process itself. Computer simulation will never make laboratory experiments obsolete, but it can provide guidance in targeting viable hypotheses before conducting in vitro or in vivo experiments.
Mathematical modeling of biological cellular processes dates back to the simulation by Glazier and Graner in 1992. They describe natural sorting behavior of different cell types [3] and different rearrangement patterns driven by the differential adhesion hypothesis [4]. This hypothesis states that cells of different types have specific potential energies upon adhesion, driving sorting behavior. In these simulations, the cellular Potts model2 (CPM) is used. A CPM for vasculogenesis based on this work was made byMerks et al. [5, 6] in which a layer of partial differential equations (PDEs) models the chemoattractants. Later, Merks added Vascular Endothelial cadherin (VEcadherin) caused contactinhibited chemotaxis to simulate angiogeniclike sprout formation [7]. From an initial clump of ECs in the model sprouting behavior appears. Merks postulates that both vasculogenesis and angiogenesis must be driven by the same principles. To produce these results, a generic library called the Tissue Simulation Toolkit (TST) was written in C++ starting from 2004 modeling the CPM described by Glazier et al. [4] in a generic way. Merks [7] extensively describes the advantages of a cell based approach over a continuum approach that is widely used in mathematical biology. Although his CPM is a nice method that increases insight in the angiogenic process, it is computationally heavy, limiting the scalability of the tractable problem domain.
Vermolen and Gefen [8] described tissue behavior using a semistochastic cellbased formalism to model the migration of cells in colonies in the context of wound healing, tumor growth, bone ingrowth and contraction formation. Movement of cells is assumed to be the result of a strain energy density working as a mechanical stimulus. Like the CPM, the model tracks displacement and viability of individual cells.
The aim of this study is to adapt this semistochastic cellbased formalism to describe angiogenesis, hence connecting this modeling approach to the subject ofMerks’ work. The need for such a model is clearly stated in the discussion of Vermolen’s work [9]. Thanks to the computational less heavy character in comparison with the CPM, we hope to be able to simulate larger areas to get a better glance at large scale behavior whilst still being able to benefit from the cellbased character of the model. We also improve the biochemical model for the degrading of the substrate by the cells and formulate all relevant parameters based on local properties.
The challenge is to translate the advantages of Merks’ CPM, like cell shape specific behavior, tracking of elongation patterns and cellcell contact behavior, to this new formalism without compromising the computational simplicity.
To verify our simulation results with biochemical experiments, this study is performed in collaboration with the Dermatology Department of the VU Medical Center. This department does in vitro laboratory research on many processes that occur in the skin, for example the role of endothelial cells during skin wound healing.
The first aim of this research is tomimic their in vitro angiogenesis sprouting assay using our computational model, simulating the response to different chemical stimuli like VEGF. Formulating a way to visually and numerically compare the laboratory work to the simulated results is key to making the model applicable in practice.

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8 

Unit Root Testing for AR(1) Processes
The purpose of this study is to investigate the asymptotics of a first order auto regressive unit root process, AR(1). The goal is to determine which tests could be used to test for the presence of a unit root in a first order auto regressive process. A unit root is present when the root of the characteristic equation of this process equals unity. In order to test for the presence of a unit root, we developed an understanding of the characteristics of the AR(1) process, such that the difference between a trend stationary process and a unit root process is clear.
The first test that will be examined is the DickeyFuller test. The estimator of this test is based on Ordinary Least Square Regression and a ttest statistic, which is why we have computed an ordinary least square estimator and the test statistic to test for the presence of unit root in the first order auto regressive process. Furthermore we examined the consistency of this estimator and its asymptotic properties. The limiting distribution of the test statistic is known as the DickeyFuller distribution. With a Monte Carlo approach, we implemented the DickeyFuller test statistic in Matlab and computed the (asymptotic) power of this test. Under the assumption of Gaussian innovations (or shocks) the limiting distribution of the unit root process is the same as without the normality assumption been made. When there is a reason to assume Gaussianity of the innovations, the Likelihood Ratio test can be used to test for a unit root.
The asymptotic power envelope is obtained with help of the Likelihood Ratio test, since the NeymanPearson lemma states that the Likelihood Ratio test is the point optimal test for simple hypotheses. By calculating the likelihood functions the test statistic was obtained, such that an explicit formula for the power envelope was found. Since each fixed alternative results in a different critical value and thus in a different unit root test, there is no uniform most powerful test available. Instead we are interested in asymptotically point optimal tests and we will analyze which of these point optimal tests is the overall best performing test. By comparing the asymptotic powercurve to the asymptotic power envelope for each fixed alternative we could draw a conclusion on which fixed alternative results in the overall best performing test.
On the basis of the results of this research, it can be concluded that there does not exist a uniform most powerful test, nonetheless we can define an overall best performing test.

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9 

Anomaly detection for internet banking using supervised learning on high dimensional data
Nowadays, a high number of transactions are performed via internet banking. Rabobank processes more than 10 million transactions per day. Most of these transactions are (part of) normal behaviour. On the other hand, some transactions are considered to be out of the ordinary. These anomalous events occur relatively infrequently (less than 10 per day). Employees, that try to find these anomalous events, combine the transactions data, historical knowledge of the anomalous events and their expertise to detect and quantify them. Several types of anomalies are considered to be interesting and so they are labelled. These anomalies need to be detected, so they can be prevented in the future. The employees try to find events similar to known anomalies. Characteristics of anomalies change over time and employees also need to detect this slightly changed, but similar, behaviour. It is not our goal to detect completely new types of anomalies. In this thesis, the focus lies on finding events similar to the known anomalies. In order to assist these employees, a model that uses the transaction data and incorporates known anomalous events is built. Our model is able to score new incoming transactions and use these to update the model parameters. The scores can be returned to the employees to assist them in finding transactions that are similar to a particular type of anomaly. The AdaGrad algorithm with diagonal matrices is used. Also, l1regularization is used on the parameter to create a more sparse solution.

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10 

On the assessment of scientific research
At universities there is a need to measure the quality of academic achievements in order to distribute available funds in a fair and objective manner.
This happens based on the scientific journals in which the several departments publish. These, at their turn, are being judged on the number of citations an average article receives. However, there exist sharp differences in the amount of citations among different research fields, and between different types of articles. An adjusted version is used at TU Delft where these journal ratings are adjusted in some way to correct for these differences.
We will investigate the distributions of these statistics to determine their consistency.

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11 

Replication and risks of the ATM Forward Percentage Call Spread

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12 

Estimation of Relative Permeability Parameters in Reservoir Engineering Applications
In a reservoir application, there are a large number of parameters that can be considered for estimation during history matching of the simulated model to the production data. If traditionally, parameters like porosity and absolute permeability have been most often included in such applications, this study focuses on researching the possibility of estimating the relative permeability curves in an assisted procedure by means of the Ensemble Kalman filter (EnKF).
Stand alone estimation of the relative permeability parameters, as given by the Corey parametrization, as well as combined absolute permeability  relative permeability estimation experiments were performed on a synthetic study case and the ability of the EnKF to recover the true values of the parameters, given bottomhole injector pressure and oil/water producer rates measurements, was tested.
The influence of the initial distribution of the relative permeability parameters on the value of the estimates and the reduction of uncertainty, as well as that of the number of the measurements and the length of the assimilation period, in terms of covering or not the water breakthrough moment, were also investigated.
Results show that some of the relative permeability parameters (the Corey oil coe±cient) can be recovered from the measurements, while others (the relative permeability end points) are not very sensitive to data assimilation and that estimating relative permeability has a positive e®ect on the estimation of absolute permeability, at the loss of accuracy of the relative permeability parameters estimations.

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13 

Distancebased Parameterisation and History Matching
The aim of this work is to use a distancebased ensemble representation of the uncertainty in a reservoir model in order to map the problem to a feature space, and use this mapping in combination with some historymatching technique, in order to better preserve and estimate the realism required in the different geological features.

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14 

Wind Speed Modeling

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15 

Multivariable feedback control of a Dividing Wall Column

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16 

A Solution Method for Vehicle Routing Problems with TimeDependent Travel Times
The goal of this thesis is developing an efficient method that produces good quality solutions to reallife Vehicle Routing Problems with timedependent travel times. The TimeDependent Vehicle Routing Problem (TDVRP) is one of the most challenging combinatorial optimisation problems and belongs to the category of NPhard problems. Therefore, several heuristic algorithms are explored and the most promising that handles the TDVRP is implemented.

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17 

Meervoudig hypothesen toetsen, toegepast op microarrays.
Mulitple testing, applied to microarrays.
Hoe gaat het in zijn werk als we niet één maar n>1 hypothesen toetsen. Hoe moeten type I en type II fouten worden gegeneraliseerd. Welke beslisregels worden er gebruikt om meervoudig te toetsen. Welke methoden hebben het meeste onderscheidende vermogen? Meervoudig hypothesen toetsen wordt veel gebruikt in microarray toepassingen. Dit onderwerp wordt ook besproken in deze scriptie.
What kind of multiple testing procedures are applied in multiple testing problems. We have to make generalizations of the type I and type II error so we can use them in multiple testing problems. Which multiple testing procedure is the most powerfull? A large range of application are the microarrays, we also discuss the multiple testing in microarrays.

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18 

The Heston model with term structure
The purpose of this project is to extend the Heston model in order to incorporate the term structure (TS) of the implied volatility surface. This includes implementing a TS within the Heston model and its calibration to a set of market instruments. The TS Heston model with piecewise constant parameters is implemented to match the TS and the COS pricing method is used for fast option pricing. We calibrate the model to the EUR/USD and USD/JPY market data and historic data is also used to test the robustness of the model. Then the model with calibrated parameters are used to price exotic options by means of Monte Carlo simulation with a new control variate we propose. Finally we also propose the COS method for pricing discrete barrier options as future research directions.

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19 

Market liquidity risk and market risk measurement
The main aim of the thesis is to formulate a concept of liquidity risk and to incorporate liquidity risk in market risk measurement. We first review two types of liquidity risk and the relation between liquidity risk and market risk. To achieve our aim, we use a new
framework of portfolio theory introduced by Acerbi. A novelty of Acerbi’s framework is that portfolio valuation includes a consideration of liquidity risk in portfolio valuation. Under the new framework, the valuation of a portfolio becomes a convex optimization problem. We give some examples of calculation schemes for the convex optimization problem. Equipped with the new portfolio theory, we can quantify market liquidity risk
and introduce a new market risk measure which includes the impact of liquidity risk. We end the thesis by giving some possible questions for further study.

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20 

Periodicity and a problem of powers a padic perspective
We investigate an old numbertheoretical problem by Mahler. Using betaexpansions and padic valuations we obtain some new results. An important extension on a theorem on periodicity concerning expansions with algebraic base is proven.

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