This thesis explores Markov chain Monte Carlo (MCMC) methods for a discrete linear Bayesian inverse problem with non-Gaussian priors. The non-Gaussian priors are total variation and Besov space priors, which are called edge-preserving due to their ability to model sparse features and discontinuities. Three sampling algorithms are compared: random walk Metropolis-Hastings, preconditioned Crank–Nicholson (pCN), and randomise-then-optimise (RTO). Prior transformations are developed to adapt RW, pCN and RTO for edge-preserving priors. Results show a trade-off between computational efficiency and accuracy. RTO with prior transformations yields more accurate reconstructions and credible intervals, but at significant computational cost and with sensitivity to prior choice. pCN is faster, more robust to discretisation, and provides more control over the sampling process but produces highly correlated samples and less accurate estimates.

This thesis investigates efficient Monte Carlo methods for estimating the 99.9% Value-at-Risk of
concentrated credit portfolios modelled through a normal copula framework. Crude Monte Carlo
simulation is inefficient when estimating extreme loss levels. To address this inefficiency, variance
reduction techniques are applied, including importance sampling and its adaptive variant. This thesis
provides empirical evidence that these techniques make extreme quantile estimation computationally
feasible in a realistic portfolio credit risk setting.
Multiple methods from the literature are reviewed to approach an appropriate importance sampling
proposal. Bernoulli tilting conditional on the common factors is an effective method when obligors
are weakly correlated. However, when large losses are mainly driven by common factors in the credit
portfolio, shifting their means turns out to be more effective. This has to be done with care, since an
improper shift may increase the variance. To this end, a deterministic method that approaches the mode
of the zero-variance distribution is discussed. Furthermore, adaptive importance sampling is studied,
which aims to minimize the variance directly by iteratively updating the shift based on past samples.
This thesis introduces a method that uses the generalized Poisson–Binomial distribution, together with
an inverse Fourier transform, to approximate the zero-variance distribution nearly exactly for simple
portfolio cases.
A deterministic mean-shifted importance sampling (M-IS) and an adaptive importance sampling
(AIS) approach are applied on stylized portfolios. These portfolios are constructed such that they mimic
real-world conditions. The M-IS method achieves an estimated variance reduction factor of 175 on a
large portfolio with 20 000 obligors. Performance on the actual Triodos Bank portfolio delivered similar
performance, but detailed results are withheld for reasons of confidentiality. For a moderately sized
portfolio consisting of 1 000 obligors, M-IS is no more effective than crude Monte Carlo because of the
higher impact of idiosyncratic concentrations. In this case, AIS finds a better balance between the
various kinds of concentrations. With conservative parameters, AIS achieves an estimated variance
reduction factor of 5 on the moderately sized portfolio. Bernoulli tilting could further increase the
variance reduction in these portfolio scenarios. Moreover, as the number of samples increases, AIS further
reduces variance. Finally, this thesis investigates how to construct asymptotic confidence intervals for
the extreme quantile estimates of these stylized portfolios and demonstrates via experiments that they
exhibit asymptotic normal behaviour despite the discreteness of the loss distribution.

Stemmatology is the study and reconstruction of textual genealogy and has several similarities to phylogenetics, the study of evolutionary histories of species. Current methods in computational stemmatology often borrow tools from phylogenetics, yet classical phylogenetic models are not well suited to the structural and labelling requirements of manuscript traditions. In particular, phylogenetics typically assumes leaf-labelled trees or networks and lacks the means to accommodate internal labels—a feature which is crucial in stemmatology. Nonetheless, these models are frequently used as there are few formal alternatives.

This thesis addresses that gap by studying the encoding and reconstruction of internally labelled trees and networks from rooted triplets. Specifically, we consider three classes of graphs: multifurcating rooted trees, general rooted trees (allowing nodes with out-degree one), and level-1 networks. Each graph is assumed to have labels on a subset of its vertices, including all leaves and some internal nodes. We prove that, under limited assumptions, a complete set of rooted triplets uniquely determines the structure and labelling of each of these graph classes up to isomorphism. This generalises earlier results for leaf-labelled binary trees and extends triplet-based encoding to structures with internal labelling and limited reticulation.

Building on these encoding theorems, we develop polynomial-time algorithms to reconstruct each graph class from its full triplet set. For trees, our methods generalise previously proposed algorithms by allowing multifurcations, nodes with out-degree one, and labels at internal vertices. For level-1 networks, we design a reconstruction algorithm that correctly identifies cycle structures and label placement, adapting earlier techniques for dense triplet sets. All reconstruction algorithms are proven correct and theoretically efficient under the given assumptions. The algorithms have been tested on many instances and run quickly.

To compare internally labelled graphs, we introduce an extension of the classical triplet distance. This adapted metric counts differences in triplet sets between two graphs on the same label set. We evaluate the metric and reconstruction algorithms on synthetic and real-world data, demonstrating their ability to capture meaningful structural differences and to recover known graphs from complete or near-complete triplet information. These results show that rooted triplets form a robust foundation for reasoning about internally labelled structures in both tree-like and mildly reticulate settings. The theory and algorithms presented in this thesis provide new tools for computational phylogenetics and stemmatology, enabling the reconstruction and comparison of complex transmission histories from local relational constraints.

Link to the GitHub page containing all the code - https://github.com/TMALevert/LAOML

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

Market making, the act of providing liquidity to the market by simultaneously buying and selling, is a difficult problem to solve. The use of reinforcement learning to solve for market making is increasing, as academics and practitioners alike look for novel ways to approximate for optimal policies in increasingly complex markets. This thesis examines the use of reinforcement learning to solve the market making problem in limit order books. To this end, we provide the theoretical background on market making, modelling limit order books, and reinforcement learning. Furthermore, we implement and compare ’classical’ algorithms, such as Q-learning and value and policy iteration, and compare their policies with newer techniques involving deep learning, namely deep Q-networks and double deep Q-networks. To train and compare these models, we use two models to simulate the dynamics of an order book. Experiment results and the ultimately learned policies are presented and discussed. We propose several ideas that could be worked on in the future.

Insurers are required to have buffers to be able to meet financial obligations that result from their portfolios, which are determined using a cash flow model. The input of such a cash flow model consists among of things, of two mortality tables and the portfolio of an insurer. Mortality rates are simulated using the Lee-Carter model. These simulated rates are in turn used to simulate the cash flow corresponding to a portfolio. This results in one possibility of incoming and outgoing money over a period of time. Lots of simulations are required to get a reliable estimate for the future cash flow which is (depending on the number of simulations) computationally heavy and therefore time consuming. The calculation time is decreased by applying an extreme gradient boosting (XGBoost) machine learning method in which cash flows are considered target variables and the mortality tables are considered features of the model. The trained XGBoost model can predict the cash flows based on the mortality tables. The standard XGBoost model is extended to a multi-target regression model which is able to predict multiple target variables at once. This XGBoost model reduces the computation time and ensures that 99.5% of the predictions deviates within either 1% or 0.5% of the observed values. XGBoost gives a good method of determining a reliable estimate of the future cash flow.

The stochastic FitzHugh-Nagumo equations are a system of stochastic partial differential equations that describes the propagation of action potentials along nerve axons. In the present work we obtain well-posedness and regularisation results for the FitzHugh-Nagumo equations with domain R^d. We begin by considering the weak critical variational setting, where we prove global well-posedness for the case d=1. We subsequently consider the strong variational setting, which allows us to extend our well-posedness results to d <= 4. To prove well-posedness and regularisation for arbitrary d, we consider the FitzHugh-Nagumo equations in the L^p(L^q)-setting. Building on earlier results for reaction-diffusion equations, we first prove well-posedness on the d-dimensional flat torus and use bootstrapping techniques to prove instantaneous regularisation of the solution. We subsequently extend the theory for reaction-diffusion equations to the unbounded domain R^d to finally prove well-posedness and regularisation for the FitzHugh-Nagumo equations on R^d.

We analyze three different methods that can approximate the expected shortfall of a financial portfolio in a nested simulation. In this simulation process, the outer simulation generates risk scenarios, and the inner simulation approximates the value of the financial portfolio under each risk scenario. The first method is the most standard one, and therefore we call it ’the standard Monte Carlo method’. This method uses the same amount of computational cost for each inner simulation. The second method adapts the computational cost of the inner simulation to the output of the outer simulation. Therefore, we call it ’the adaptive sampling method’. This technique has already been proven to work more efficiently than the standard Monte Carlo method. The third method is called ’the multilevel Monte Carlo method’ (MLMC) and is based on the technique that has been introduced by Giles. This method approximates the expected shortfall multiple times and with different levels of accuracy. These estimators are used to give an accurate overall approximation of the expected shortfall. To the best of our knowledge, it has never been examined how efficiently the expected shortfall can be approximated in this manner by the MLMC method. This thesis will thoroughly explain how each method can be applied in a nested simulation to approximate the expected shortfall. In addition, an analysis is given on how to ensure that each method is used as efficiently as possible in the nested simulation. We also perform a numerical experiment in which we simulate a simplified version of a financial portfolio using stochastic processes. With the help of the numerical experiment, we examine which method is most efficient to approximate the expected shortfall.

In various scientific disciplines, measurement data is collected across space and over time with the aim of inferring information regarding an underlying stochastic spatiotemporal phenomenon. The high computational costs of current methods render this task intractable for the large datasets typically encountered in spatiotemporal statistics.

We propose a model based on Gaussian random fields, defined as the solutions to a class of space-time fractional stochastic partial differential equations (SPDEs) driven by Gaussian noise. The tunable parameters characterizing this class are expected to admit intuitive interpretations, in which case an efficient numerical approximation scheme for solutions to these SPDEs yields a powerful spatiotemporal model which is usable in practice.

The study of this class of SPDEs is the focus of this work. We define weak and mild solution concepts and show their equivalence under lenient conditions on the differential operators involved. Subsequently we establish results linking the well-posedness, spatiotemporal regularity and asymptotic covariance structure of the SPDEs to conditions on the model parameters, confirming their interpretability. Lastly, we specialize the SPDEs to a class of fractional ordinary differential equations in time, and describe a method to compute numerical approximations of their solutions. Future work is needed to analyze this scheme and generalize it to the original class of SPDEs.

Designing engineering structures relies on accurate numerical simulations to predict the behaviour of a structure before its realization. In the design process, many variables influence the final structure. There is an incentive for optimizing the design based on the desire for cheaper structures and less material usage. Although a wide variety of optimization techniques exist, in practice many structures are not fully optimized and could fulfill their purpose with less material usage. This partly results from optimization techniques generally requiring many simulations for problems with a high number of variables. In creating increasingly complex models, simulating its performance can take up to days or weeks to compute, incentivizing the development of optimization methods that use as little simulations as possible. The goal of this thesis is to provide a method of optimization which significantly reduces the required number of simulations to be performed, aiming to overcome the issue of required computational effort. The optimization problem used in this thesis is the geometric optimization of a fiber reinforced polymer (FRP) microstructure. Each evaluation requires a Finite Element analysis, and many parameters are required to describe the geometry. The goal is to find the maximum stress at perfect plasticity during uniaxial tension, interpreted as a measure of strength. While generally only fibers with a circular cross-section are used, here a morphing parameter is introduced that changes this shape between being circular and square. Furthermore, fibers are allowed to overlap, effectively creating a single fiber with a complex shape. This geometry optimization provides a challenging problem for which different optimization techniques can be compared. The approach taken in attempting to minimize the number of function evaluations is a combination of several machine learning methods. Based on a limited number of initial samples, Bayesian optimization (BO) is applied. In BO a prediction model in the form of a Bayesian Neural Network (BNN) is created and used to inform further sampling. This prediction model provides a mean and a standard deviation in its prediction, both of which are used to find the best point to sample next. The number of initial samples required to create an accurate prediction model increases exponentially with the number of parameters. It is therefore opted to first use a variational autoencoder (VAE) to reduce the number of parameters in which BO is performed, by encoding the parameters in a lower dimensional representation. The variational autoencoder does this without requiring any function evaluation. Results show that when encoding the original parameters using the VAE, the encoded representation is unable to recreate all possible configurations of the original parameters. Furthermore, by transforming to a lower dimensional representation, within this representation the complexity of a function increases compared to the original function, generally leading to many local optima. This complexity proves difficult for the BNN to accurately predict based on a limited number of samples. As a result, BO does optimize the result, but does not reliably find the global optimum of the reduced parameter space. No clear conclusion can be made on the overall performance of the method compared to alternative optimization methods. Recommendations are made for what additional studies could be performed. Further recommendations are given for solving issues limiting the current performance as well as possible additions to the framework. The fiber geometry optimization serves as a good case to compare different optimization methods. It is shown that the geometry has a significant influence on the mechanical performance of a FRP microstructure, and optimization is therefore beneficial. The load case considered is however too specific for the optimized result to have direct practical use. Still, it does demonstrate that the results of an optimization study can be generalized. As the optimization framework is data driven, the optimization objective could easily be extended to study more realistic problems.

Backward stochastic differential equations (BSDE) are known to be a powerful tool in mathematical modeling due to their inherent connection with second-order parabolic partial differential equations (PDE) established by the non-linear Feynman-Kac relations. The fundamental power of BSDEs lies in the fact that with them one does not merely obtain the solution of the corresponding PDE but also its spatial gradient through the control process Z. Classical numerical methods tackling the system face the so-called curse of dimensionality and cannot be used to solve high-dimensional problems. In recent years, multiple approaches have been developed to overcome this computational burden, building on deep learning and showing remarkable empirical success even beyond 10 dimensions. However, such Deep BSDE methods struggle with giving accurate approximations for the Z-process throughout the whole time horizon. In this thesis we propose a novel approach aimed to give better estimations for the control problem, exploiting the natural dynamics of the Z-process given by Malliavin calculus. The proposed methods use deep learning parametrizations taking advantage of the universal approximation capability of neural networks. The Malliavin derivatives are estimated through the Malliavin chain rule. Two discrete numerical methods are developed which are called One-Step Malliavin (OSM) and Multi-Step Malliavin (MSM) schemes respectively. An error analysis is carried out proving the consistency of the algorithms and showing first-order convergence under certain assumptions. Numerical experiments are presented to demonstrate the efficiency of the Malliavin formulation compared to other Deep BSDE solvers.

Safe hypothesis tests are tests that are robust under accumulation bias, namely when there are dependencies between the results of previous studies and the decision whether to conduct further studies. We construct two types of safe test for the 2 × 2 contingency table, the conditional and unconditional safe tests. In general safe tests are given by an information projection that may be difficult to compute. The conditional tests we construct however are given either in explicit form or implicitly via a defining equation. The same can be said of many of the unconditional tests we construct, for which we prove a number of theoretical results enabling their quick calculation when not given explicitly. The method we develop to accomplish this may perhaps be used to identify optimal safe tests in many other scenarios.

In recent years, the offshore wind industry has grown significantly. The wind turbines are constructed in wind farms, and are serviced with relatively small crew transfer vessels. These vessels transport repair crews from the mainland to the farms and back within a day. One of the big challenges is that these transits can cause motion sickness. If one crew member gets sick, the vessel is legally required to return to the harbour, without performing the repair. It is of interest to predict motion sickness, since both a failed repair and a broken wind turbines that could have been repaired are costly. This can be done by predicting the motion sickness incidence (MSI), which is an objective measure for motion sickness.
The goal of this thesis is to predict MSI using the wave conditions measured in or near the wind farms, by means of an analytical and numerical approach using machine learning. Testing is done on two sites: the Greater Gabbard wind farm and the Westermost Rough wind farm. The analytical approach relates the recorded wave spectrum to the motion spectrum of the vessel, but proves infeasible as a stand-alone method. Gaussian regression, tree ensemble regression and neural networks with Bayesian regularization and backpropagation are the best performing machine learning techniques for MSI prediction. We conclude that the methods from this thesis have sufficient accuracy, by comparing our results with those from Orsted, whose model is already successfully deployed.