The minimum vertex cover problem (MinVertexCover) is an important optimization problem in graph theory, with applications in numerous fields outside of mathematics. As MinVertexCover is an NP-hard problem, there currently exists no efficient algorithm to find an optimal solution on arbitrary graphs. We consider quantum optimization algorithms, such as the Quantum Alternating Operator Ansatz (QAOA+), to find good minimum vertex covers.

This thesis presents new 'second degree' mixing Hamiltonians for MinVertexCover in the QAOA+ framework, which allow mixing between solutions Hamming distance 2 apart. The performance of these new Hamiltonians is evaluated on a small graph. Methods to extend this idea by constructing Hamiltonians which allow mixing between solutions at Hamming distance n are also presented, along with a generalization of second degree mixing Hamiltonians to other optimization problems.

Finite element analysis and multibody dynamics are popular mathematical modeling techniques for sev- eral mechanical applications. The finite element method is a costly method from a runtime standpoint. Moreover, both multibody dynamics and finite element methods require approximating the geometry of the application either by meshing or by analytical surfaces. Therefore, IsoGeometric Analysis (IGA) has been proposed as an alternate method for parameterizing geometric surfaces. IGA enables a uni- fied mathematical representation of both geometries and solution fields using B-Splines or NURBS. NURBS are the de-facto standard in modern Computer-Aided Design (CAD) workflows, but their use for analysis in finite elements is rare. By directly using the geometric information for the analysis, IGA reduces the cost of mesh generation, therefore enabling a more efficient design process.
The IGA-based rigid body contact algorithm offers several advantages over the traditional FEM- based approach. First, it provides an accurate representation of the geometry. This improves the efficiency of the design process and reduces the need for mesh refinement. Second, by looking for the local orthogonal projection in the parameter space rather than the coordinate space, the algorithm simplifies the definition of boundary conditions. This makes the process of defining boundary conditions easier and more intuitive. Third, the algorithm approximates the contact area by mapping it from the parameter space to the physical space. This allows for the specification of the level of accuracy of the contact area, which can permit more precise solutions.
The paper compares the IGA-based rigid body contact algorithm with the existing rigid FE mesh- based contact method in Simcenter Madymo. The results show that for the same level of accuracy, the IGA-based method can largely reduce execution time. This suggests that IGA could be a valuable alternative for use in the field of contact simulation.

In an attempt to find alternatives for solving partial differential equations (PDEs)
with traditional numerical methods, a new field has emerged which incorporates
the residual of a PDE into the loss function of an Artificial Neural Network. This
method is called Physics-Informed Neural Network (PINN). In this thesis, we study dense neural networks (DNNs), including codes developed in the context of this bachelor project. We derive the backpropagation equations necessary for training and use different configurations in a DNN to test its interpolating accuracy. We distinguish between a-PINNs which use automatic differentiation to evaluate a PDE, and n-PINNs which approximate differential operators in a PDE with numerical differentiation. We compare both PINNs on the harmonic oscillator, the 1D heat equation and the 1-soliton and 2-soliton solutions of the Korteweg-De Vries (KdV) equation. Both PINNs could accurately converge to the solution, except to the 2-soliton solution, where the a-PINN outperformed the n-PINN. Furthermore, we tested a highly nonlinear problem of the KdV equation, which can be described by a train of solitons. We observed that PINNs are inaccurate if insufficient training samples are used for training. Adding training samples on the interior from a numerical solution leads to a good qualitative agreement, though more effort is required to find a better network configuration to obtain more accurate predictions.
Additionally, PINNs were used for inverse problems to derive an unknown coefficient in a PDE and proved to be highly accurate for noiseless data. When we
generated training samples with 10% noise from a uniform distribution, the PINN
results’ relative error stayed within a margin of under 2%. However, inverse PINNs are much more inefficient compared to nonlinear least squares methods like the Levenberg–Marquardt algorithm.
As of now, PINNs are still very early in development and stand no match against
traditional numerical methods to a known PDE. They may, however, provide a
useful alternative in the future as they are constantly being improved.

The development of optical metamaterials in recent years has enabled the design of novel optical devices with exciting properties and applications ranging across many fields, including in scientific instrumentation for space missions. This in
turn has led to demand for computational methods which can produce efficient device designs. Traditional optical devices admit a closed-form solution for this inverse design problem. However, in the presence of strong multiple scattering, which is often the case when considering optical metamaterials, the inverse problem becomes ill-posed. As a result, many optimization and machine learning techniques have been applied towards discovering good solutions.
In this MSc thesis project, several of the most promising of these techniques are applied to a specific problem, the discovery of silicon metamaterial lens designs for the CoPILOT high-altitude balloon project. Ultimately, a software tool capable of producing effective and admissible designs is produced and demonstrated.
First, an overview of the CoPILOT design problem is presented. Next, relevant background material topics, including properties of metamaterials and computational methods for simulating them, are covered in some detail. After this, methods used to solve optical design problems in past literature are described and contrasted. Then, a comprehensive explanation of the method developed and used for this project, including important design considerations, is given. The best solutions found using this lens optimization method are shown and compared. Finally, fruitful areas of future work on this topic are listed.

For scaling up the qubits in silicon quantum computers, it is vital to determine crosstalk effects that can lower the fidelity of the computer.
In this computational project, we examine single-qubit gate-fidelities in the presence of crosstalk for uncoupled spin qubits that are driven with X-gates via electron dipole spin resonance (EDSR). We introduce two models: the first model introduces the AC Stark shift and the novel second model expands on this by adding a resonance frequency shift on top. We assume the latter resonance frequency shift to be due to heating effects. We optimize the gate-fidelity for a qubit coupled to up to six drives as a function of the overall driving time and -frequency of a single drive for both models using the Nelder-Mead algorithm.

Using the AC Stark shift model, we still obtain 0.99999 fidelity if we do not account for the crosstalk. However, when using the second model, the fidelity drops to 0.69 in the presence of two drives when we do not correct for the heating-induced resonance frequency shift and the AC Stark shift. Furthermore, the fidelity decreases linearly with the number of drives coupled to the qubit, implicating that the resonance frequency shift will become a significant problem for the scalability of silicon quantum computers. We find that we can correct for the resonance frequency shift entirely by using optimized driving time and -frequency, where most gain comes from optimizing the driving frequency. Moreover, we discover that there is a linearly increasing dependence of the resonance frequency shift at the theoretical driving time as a function of the total drives. Up to a translation factor of 0.5 MHz, we discover the same linear relationship for the correction needed on the theoretical driving frequency to hit maximum fidelity as a function of the total drives.

The main objective of this thesis was to explore the capabilities of neural networks in terms of representing governing differential equations, primarily in the purview of fluid/aero dynamic flows. The governing differential equations were accommodated within the loss functions for training the neural networks, thereby making them 'physics-informed'. Subsequently, this idea of physics-informed neural networks (PINNs) was extended to parameterized geometries generated with the help of commercial auto-encoders developed by the UK based company Monolith AI pvt. ltd. because neural networks have the capability to learn the desired PDEs over variable/parameterized geometries without the need to recompute the solution for every minor change in the input geometry, which proves out to be a huge advantage over classical numerical techniques. The advantages, limitations and scope for further research in the field of physics-informed deep learning have been discussed in the contents of the underlying thesis report.

Convection-dominated flow problems are well-known to have non-physical oscillations near steep gradients or discontinuities in the solution when solved with standard numerical methods, such as finite elements or finite difference methods. To overcome this limitation, algebraic flux correction (AFC) can be used, which is a stabilization method. However, AFC contains time-consuming computations, therefore, alternative approaches are explored. The rapidly rising field of machine learning in the mathematical world, so called scientific machine learning, has successful applications in solving partial differential equations. In this work, the focus is on convection-dominated flow problems, in particular the steady state convection-diffusion equation in one-dimension. To solve this, two alternative approaches based on neural network-learning have been developed that are able to mimic the AFC limiter with a certain accuracy and performance. In some cases, the neural network-based limiter is outperforming the AFC limiter.

Differentiable ray-tracing is an exciting new development in computer graphics to approach all sorts of 3D scene design problems by obtaining gradients of renders produced by ray-tracing with respect to parameters that define the scene. These gradients can then be incorporated in a gradient-descent type optimization pipeline.

One such type of design problem is caustic design with free-form lenses. This is the process of obtaining the geometry of lenses in an optical system such that the light that passes through this system forms a desired illumination distribution on a target screen. A typical application of this is distributing the light from streetlights or car headlights in a pleasing and efficient way over the street.

The non-imaging optics literature offers many methods to design free-form lenses for caustic design, but differentiable ray-tracing is largely unknown in this field. Therefore this thesis proposes a method of optimizing free-form lenses for caustic design with differentiable ray-tracing.

The optimization pipeline starts with a multi-layer perceptron neural network which outputs parameters that define one free-form lens side in the form of a B-spline surface. A specially implemented ray-tracer then produces a caustic render by tracing through this lens from either a plane wave or a (grid of) point source(s). Back-propagation and optimization takes place using automatic differentiation in PyTorch and the Adam optimizer.

The results, which are verified with LightTools, show great promise for this technique, especially for the plane wave optimizations. The point source grid optimizations proved to be more challenging, but also here the optimization was able to achieve improvement. This shows that the proposed technique also has potential in positive etendue optimizations.

In this work, we consider how to optimize an optical system, specifically one with diffractive optical elements (DOE). We start by describing optical theory called Fourier optics also known as wave optics. This type of optics is found by making assumptions from the Maxwell equations for magnetic and electrical fields. This leads us to the Rayleigh-Sommerfeld diffraction integral, which we need to propagate light. To optimize an optical system, we introduce the standard optimization methods used when gradients are available and also dive into data-driven methods. Two wellknown algorithms in each category: the Adam optimization method, which is an extension of normal gradient descent methods, and the UNet convolutional neural network. To make the optimization methods work with our physics simulation, we use an automatically differentiable implementation which gives the gradients for the optimization. Combining the two optimization methods with our optics engine, we optimize optical designs such that the resulting intensity on the sample plane resembles some target intensity. We are able to optimize systems with single and multiple DOE and for high and low resolution DOE designs. We find that more lenses makes the optimization better and increases the variability in the created projection. We also find that increasing the resolution severely slows down the optimization with the Adam method. Although, the optimization method Adam is well suited for this optimization task. It becomes computationally very expensive on high resolution due to the physics simulation at every optimization step. Some physical simulations require high resolution to make sure the simulation does not contain to much artefacts. We show that the data-driven approach has potential to solve this issue. We train a network that takes as input a target intensity and outputs the lens that produces that intensity. Combining these results, we conclude that modern optimization methods are well suited for optical system optimization and we find that there is a large untapped potential for data-driven methods in optics.

The Material Point Method (MPM) is a numerical method primarily used in the simulation of large deforming or multi-phase materials. An example of such a problem is a landslide or snow simulation. The MPM uses Lagrangian particles (material points) to store the interested physical quantities. These particles can move freely in the spatial domain. Each time step, these physical quantities are projected on a background grid, which is necessary to evaluate these quantities at a future time step. Once all necessary properties are projected, the acceleration on this grid can be updated using the conservation of linear momentum. At last, this updated acceleration can be projected to the particles to compute their updated velocity, position and other quantities.

The main problem addressed in this thesis arises in the projection of the particles to the background grid. The use of linear basis functions in the classical MPM causes instabilities, which can be resolved by using higher order B-splines. However, another problem arises when particles move between the background cells. Since these particles move freely through the spatial domain, it is possible for some cells to by \textit{nearly-empty}. In this thesis, the use of Extended B-splines is explored to increase the stability in these areas by deactivating unstable B-splines and \textit{extending} stable ones.

Furthermore, higher dimensional B-splines are constructed by a tensor product of one dimensional B-splines. The scalability can be a problem, as these B-splines cannot be refined locally. Therefore, Truncated Hierarchical B-splines are introduced to locally refine a geometry. The effectiveness of this technique in combination with the EB-splines is investigated.

The thesis shows that the use of EB-splines in the context of MPM increases the stability in the case of nearly-empty cells, and can also improve the quality of the solution near the boundary. Furthermore, in the neighborhood of high stress concentrations, the THB-splines have a similar accuracy as the regular B-splines, while drastically reducing the computational costs. In combination with EB-splines, THB-splines can be used to accurately refine a geometry in the presence of nearly-empty cells.

Physics Informed Neural Networks are a relatively new subject of study in the area of numerical mathematics. In this thesis, we take a look at part of the work that has been done in this area up until now, with the ultimate goal to develop a new type of PINN that improves upon the old concept. We introduce the concept of parameterized PINNs, which allow a single trained network to solve multiple partial differential equations for multiple boundary conditions and geometries by parametrizing these variables as an input for the network. Two methods are tested: one using global basis functions, and one using B-splines. The proposed methods are tested for Laplace’s equation and Poisson’s equation in multiple dimensions, most of which show that these methods are viable alternatives for the current style of collocation-based PINNs.

This thesis poses a new geometric formulation for compressible Euler flows. A partial decomposition of this model into Roe variables is applied; this turns mass density, momentum and kinetic energy into product quantities of the Roe variables. Lie derivative advection operators of weak forms constructed with this decomposed model naturally follow to be self-adjoint, which results in skew-symmetric discrete advection operators in any number of dimensions. Under certain conditions these conserve products of the Roe variables, leading to a discrete model formulation with advection operators that simultaneously conserve mass, momentum, kinetic energy, internal energy and total energy in compressible Euler flows. While this idea is not new the novelty of this work lies in its extension to mimetic finite element methods and its application to discontinuous compressible Euler flows.The regular geometric Euler model and its Roe variable decomposition have been discretized through mimetic isogeometric analysis. At the core of mimetic discretization methods lies the idea of retaining the De Rham sequence of differential form spaces when projecting these to finite-dimensional approximations and when constructing discrete operators. B-spline differential form spaces have been defined such that the exterior derivative maps in a topologically exact and metric-free way, while the interior product has been discretized in a weak way in order to retain its map between appropriate spaces in the De Rham sequence. Only primal grids are used; the Hodge star operator is discretized through the definition of an L2 inner product to resolve weak forms. Cartan’s homotopy formula allows for a consistent discretization of the Lie derivative through compositions of the interior product and exterior derivative.Several tests were carried out to determine the efficacy and behavior of this regular geometric Euler model, its Roe variable decomposition and the resulting discrete advection operators. Testing one-dimensional linear advection and Burgers’ equation on periodic domains shows that discretizations of the self-adjoint advection operators are consistent with conservative formulations. Sod’s shock tube is used as one-dimensional discontinuous compressible flow test. Both the regular Euler model and its Roe variable decomposition display strong oscillatory tendencies, yet solution convergence is obtained without any issues. While application of a simple moving average-filter removes the worst oscillations more sophisticated methods are necessary for obtaining solutions that are free of unphysical oscillations. The Roe variable decomposition negatively affects the accuracy of shock speed predictions. The presence of strong oscillations likely affects the numerical conservation errors of both methods. Compared to finite volume package Clawpack and the nodal Discontinuous Galerkin (DG) method of Hesthaven & Warburton both models have less numerical diffusion on coarse meshes with the regular model outperforming the Roe variable decomposition. Convergence of momentum conservation error is slow and the errors are large compared to the reference methods. For two-dimensional periodic vortices both the regular geometric Euler model and its Roe variable decomposition outperform both reference methods for stationary and moving vortices. Clawpack displays diffusive behavior, resulting in large L2 errors and high amounts of numerical diffusion. While the L2 errors of the DG method are comparable to those of the two models developed in this work for all basis function orders considered, the resulting DG discretization requires significantly more degrees of freedom to attain this. To obtain similar levels of numerical diffusion the DG method needs up to ten times as many degrees of freedom as the methods presented in the current research.

The aerospace engineering industry is continuously striving for faster methods to solve and optimize engineering and research problems with a higher degree of accuracy. It is therefore relevant to investigate possibilities that are expected to accelerate computational speed, such as quantum computing. For this reason, the objective of this thesis is to investigate the feasibility of optimizing 2D determinate truss structures with a quantum algorithm run on a Gate-Based Quantum Computer (GBQC). The Quantum Approximate Optimization Algorithm (QAOA) is currently expected to be the most suitable quantum algorithm candidate for optimization problems, because of its simplicity and robustness. The most important take-away from this thesis is that it is possible to map a truss structure to QAOA format to optimize it on a GBQC. The program proves to be working on quantum virtual machines. However, it is currently not possible to obtain correct results when running on real quantum hardware due to noise.

This report reviews two quantum key distribution (QKD) protocols: the BB84 protocol and the measurement device independent (MDI) QKD protocol. The goal of this report is to recreate the security proof of the BB84 protocol, to generate the secret key rate for a practical application of the BB84 protocol, both with and without decoy states, and to review the MDI-QKD protocol and look at the advantages it has for practical QKD. The proof security of the BB84 protocol is done by designing an equivalent theoretical protocol and proving its security by bounding the information of the eavesdropper to an exponentially small number. The best distance over which secret key rate can be generated with the practical model of the BB84 protocol that is presented in this report, is 165 km. This is achieved by employing decoy states and is more than three times as far as the model can achieve without decoy states, which gave a distance of 52 km at best. The zero distance key rate of the model with decoy states is R = 1.21 · 10^-2 bits per pulse (bpp), at best. This is an order of magnitude larger than R =1.02 · 10^-3 bpp, which is what the model without decoy states could produce. The decoy state method is therefore an improvement for practical QKD. By reviewing the MDI-QKD protocol it can be concluded that because it eliminates the measurement device side channels, it is very useful for practical QKD, since it makes security analysis simpler and more precise. It also necessarily employs decoy states, which improves the secret key rate. MDI-QKD therefore is a promising protocol to use for future quantum communications.

Quantum computing is an emerging technology that combines the principles of both computer science and quantum mechanics to solve computationally challenging problems significantly faster than the current classical computers. In this thesis, a proof of concept to generate hardware-executable quantum circuits for Noisy Intermediate-Scale Quantum (NISQ) devices that follow the paradigm of approximate computing is presented.

We adopt a multi-level optimization approach and consider the placement of native quantum gates that can be physically implemented on one of the existing quantum systems (IBM, Rigetti) as a circuit topology optimization problem and the adjustment of parameters of these native quantum gates as a continuous design optimization problem. The outer topology problem is solved with the aid of a genetic algorithm, whereas the gradient-descent method is used for the inner design optimization.

Our approach starts from the reference circuit and transforms it into an executable circuit with tunable parameters by replacing the high-level quantum operations with a set of approximate decompositions, e.g., single qubit rotation quantum gates or a combination of single qubit rotation gates (matrix multiplication). Later, the inner design optimization ensures that the circuit created is tuned to be executed to achieve an expectation value close to that of the reference circuit. This is complemented by compiler-based optimizations to further reduce or aggregate the quantum gates of the optimized circuit. This three-step pipeline is embedded into an outer genetic algorithm framework that inspects many different circuit designs (the approximate-decomposition replacements are by no means unique) and returns a set of unique approximate quantum circuits that can be executed on hardware.

We have tested our circuit generation approach for superconducting quantum systems such as IBM and Rigetti hardware for many different benchmark circuits such as Quantum Fourier Transform, Toffoli, quantum adder, multi-controlled operations and three oracle-based algorithms, namely, Grover's search algorithm, Deutsch and Bernstein-Vazirani algorithm. In nearly all cases, our approach outperforms the quantum compiler frameworks by IBM and Rigetti even on their highest optimization level and produces significantly smaller circuits, up to 2x reduction in the number of gates. In addition, the circuits generated using the proposed approach provide better (a) performance (on average, improved by 24.25%) and (b) reliability (on average, improved by 23.50%) compared to the IBM generated circuits when executed on the ibmq_5_yorktown physical quantum device.

This project aims to recreate intensity patterns using Fraunhofer diffraction as a means of simulation. These intensity patterns are created by phase shifting specific parts of an incoming field of light. These phase shifts are determined by a B-spline surface, which is in turn controlled by so-called control points. Only a handful of control points can describe a whole surface. The position of these control points is then determined using machine learning and specifically a technique inspired by ‘physics-informed neural networks’, which were introduced last year by Raissi et al. [1]. With this method, simple experiments which sought to recreate intensity patterns known to be in the solution space were carried out. These experiments showed some success, but suffered from the fact that they used too sensitive parameters in the input of the machine learning model, reducing the sophisticated method to a Monte Carlo search, or they used no input at all, which degraded the machine learning model to simple parameter optimization. Nevertheless, these experiments showed that this method has the potential to be used in more flexible optical setups, where multiple configurations can yield the same intensity pattern or where changing the parameters defining the setup do not induce enormous changes in the resulting intensity pattern. In addition, the proposed method relies upon Fraunhofer diffraction, which, when discretized for numerical computation, introduces aliasing issues when the incoming field changes too rapidly. This phenomenon was especially apparent when using point sources that create spherical wave fronts. A possible solution for this issue is to consider ray tracing techniques in future research.

Computational fluid dynamics is nowadays one of the pillars of modern aircraft design, just as impor­tant as experimental wind tunnel testing. Very ambitious goals in regards to performance, efficiency and sustainability are being asked of the aviation industry, the kind that warrant a virtual exploration of the edges of the flight envelope. High­ order has come to be regarded as a necessary ingredient to achieve breakthrough advances in this direction. So much so, in fact, that the major aircraft manufac­turers, governmental aerospace research agencies and top universities worldwide have been acting in coordination to increase the technology readiness level of these approaches, for the last ten years. In this work, I study in significant detail the one aspect that makes high-­order methods ideal candi­dates for enabling the use of more advanced turbulence models in industry: the cost­-efficiency of their discretization—they offer minimal amounts of dispersion and dissipation errors, for a given number of degrees of freedom. I consider three research objects: discontinuous Galerkin spectral method (DGSEM), flux reconstruction (FR) and a novel B-­spline-based discontinuous Galerkin formulation stabilized via algebraic flux correc­tion (DGIGA­-AFC), and characterize their order of accuracy, linear and nonlinear stability characteristics, as well as dis­persion and dissipation errors as a function of wavenumber. Afterwards, I experimentally investigate their relative suitability towards scale­-resolving simulation of compressible and turbulent flows, by solv­ing a number of simple test cases of increasing difficulty (linear advection, inviscid Burgers and Euler equations; all in 1D) using a purpose­-made MATLAB implementation. The proposed isogeometric method (DGIGA) has been found to be at least as viable as the other two, a priori, for the resolution of high­-speed turbulent flows. Moreover, I have found that low dispersion and dissipation need not always be associated to high order, but to a high number of degrees of freedom per patch instead; these two coincide in the more conventional schemes, yet not necessarily in DGIGA. As a nonlinear stabilization mechanism, however, the proposed combination of DGIGA with AFC has turned out to be inferior to existing limiters.

Large fault­tolerant universal gate quantum computers will provide a major speed­up to a variety of common computational problems. While such computers are years away, we currently have noisy intermediate­scale quantum (NISQ) computers at our disposal. In this project we present two quantum machine learning approaches that can be used to find quantum circuits suitable for specific NISQ devices. We present one gradient­based and one non­gradient based machine learning approach to optimize the created quantum circuits, to best mimic the behaviour of a given function up to measurement. We make sure that the created quantum circuits obey the restrictions of the chosen hardware, therefore the approaches can be used to find circuits perfectly suited for specific NISQ devices. This enables the user to make the best use of quantum technology in the near future. In doing this we created our own quantum simulator which can be used to simulate small quantum circuits that obey hardware restrictions. We also present the method used to implement this simulator. Finally we present the results of applying both machine learning approaches to different problem types and compare their performance.

Recent developments in quantum annealing have shown promising results in logistics, life sciences, machine learning and more. However, in the field of geophysical sciences the applications have been limited. A quantum annealing application was developed for residual statics estimation. Residual statics estimation is a highly non-linear problem in geophysical subsurface imaging. The problem is run on a quantum annealer as well as on a hybrid solver. Quantum annealers solve binary quadratic models, which are a specific type of NP-Hard problem. The hybrid solver uses quantum annealers together with classical computers to solve optimisation problems. Two binary quadratic models were developed for the purpose of residual statics estimation, one based on the one-hot encoding and the other on standard binary encoding. These two models were theoretically analysed and subsequently implemented and tested. Tests with synthetic data were done using the quantum annealer. The solutions found with quantum annealing were often only a few bit-flips away from the global optimum. Therefore, the results were further improved by post-processing of the data with the steepest descent method. It was found that the one-hot encoding with steepest descent performed superior to the other methods. As a final test the hybrid solver was used to solve a business-sized and realistically difficult problem. The hybrid solver outperformed a current industry standard cross-correlation solver.