This dissertation is a summary of the graduate study in the past few years. In first part, we develop efficient spectral methods for the spectral fractional Laplacian equation and parabolic PDEs with spectral fractional Laplacian on rectangular domains. The key idea is to construct eigenfunctions of discrete Laplacian (also referred to Fourier-like basis) by using the Fourierization method. Under this basis, the nonlocal fractional Laplacian operator can be trivially evaluated, leading to very efficient algorithms for PDEs involving spectral fractional Laplacian. We provide a rigorous error analysis for the proposed methods, as well as ample numerical results to show their effectiveness.<div><br>In second part, we propose a method suitable for the computation of quasiperiodic interface, and apply it to simulate the interface between ordered phases in Lifschitz-Petrich model, which can be quasiperiodic. The function space, initial and boundary conditions are carefully chosen such that it fix the relative orientation and displacement, and we follow a gradient flow to let the interface and its optimal structure. The gradient flow is discretized by the scalar auxiliary variable (SAV) approach in time, and spectral method in space using quasiperiodic Fourier series and generalized Jacobi<br>polynomials. We use the method to study interface between striped, hexagonal and dodecagonal phases, especially when the interface is quasiperiodic. The numerical examples show that our method is efficient and accurate to successfully capture the interfacial structure.</div>

The planning process of public transit systems have for long been a subject treated in operations research. In recent years, research within the public transit area focus on adapting this planning process for electric vehicles. This thesis evaluates a heuristic approach to the vehicle scheduling problem for electrical buses without the use of any licensed software. Among the previous research is an article that presents a mathematical model for scheduling of electrical buses using AMPL and CPLEX. However, due to not finding optimal solutions for all instances of the problem, the question of a heuristic approach is raised. Literature studies indicate that large neighbourhood search (LNS)-based heuristics have shown previous success for scheduling of vehicles. Results from the implementation of a constructive heuristic combined with an LNS-heuristic are compared with the results from the mathematical model and CPLEX. We see some success using the heuristic approach. However, the method we compare against still provide better solutions for almost all instances. More specifically, the difference between the results (number of buses needed) increases with an increasing complexity of the problem. Finally, due to the lacking results, some recommendations are given for improving the performance of the heuristic.

The study examines the Projection method and the simultaneousapproximation-term (SAT) method as boundary treatment for the dynamic beam equation using summation-by-parts (SBP) operators for handling the inner domain. The methods are examined for both the homogeneous constant coefficient case, and the inhomogeneous piecewise constant coefficient case with a coupled interface. The outer boundaries are handled by SAT or Projection, the coupled interfaced is handled by Projection or a mix between Projection and SAT. Solutions are integrated in time with finite central difference schemes and compared to analytical solutions. A convergence study is conducted with respect to the spatial discretization to measure the accuracy, and the stability is examined by numerical simulations of the CFL-condition. The study shows that Projection has the same accuracy as SAT for most boundary conditions while allowing for a larger timestep. A discontinuity in the medium is found to be handled equally accurate by Projection and the Projection and SAT mixture for all but one case studied, where the mixture was slightly more accurate.

A common method to design trajectories is by cubic splines to interpolate a set of via-points for a robot. The trajectories are optimized with respect to an objective function including the total time with a set of kinematic constraints. The execution time for this problem is longer then the sampling time of an embedded computer and restricts the use of this method in on-line applications. To modify this procedure to be usable on-line a point-to-point motion planner is developed using the output from the optimization programming at each iteration. This point-to-point motion planner runs on a separate process and contains a virtual vehicle to restrict the end-effector speed in its operational space. This results in an adaptive motion planner capable of handling kinematic actuator constraints and end-effector speed constraints in the operational space. The thesis ends with results from simple simulations demonstrating the effects. / En vanlig metod för attutforma banor är genom kubiska splines för att interpolera en uppsättning avvia-punkter för en robot. Banorna är optimerade med avseende på en målfunktionav den totala tiden med en uppsättning av kinematiska begränsningar.Exekveringstiden för detta problem är ofta längre än samplingstiden för eninbyggd dator och begränsar användningen av denna metod i applikationeron-line. För att metoden skall kunna användas on-line skapas enpunkt-till-punkt planerare med hjälp av utdata från den iterativa lösaren avoptimeringsproblemet vid varje iteration. Punkt-till-punkt banorna planeras ochkörs på en separat process och innehåller en virtual vehicle för att begränsaend-effectorns fart i den operativa rymden. Detta resulterar i en adaptiv rörelseplaneraresom hanterar kinematiska manövrerings begränsningar och fart begränsningar avend-effectorn i den operativa rymden. Arbetet avslutas med resultat från enklasimuleringar som visar effekterna.

This thesis consists of two papers considering different aspects of stochastic process modelling and the minimisation of computational cost. In the first paper, we analyse statistical signal properties and develop a Gaussian pro- cess model for scenarios with a moving receiver in a scattering environment, as in Clarke’s model, with the generalisation that noise is introduced through scatterers randomly flip- ping on and off as a function of time. The Gaussian process model is developed by extracting mean and covariance properties from the Multipath Fading Channel model (MFC) through coarse graining. That is, we verify that under certain assumptions, signal realisations of the MFC model converge to a Gaussian process and thereafter compute the Gaussian process’ covariance matrix, which is needed to construct Gaussian process signal realisations. The obtained Gaussian process model is under certain assumptions less computationally costly, containing more channel information and having very similar signal properties to its corresponding MFC model. We also study the problem of fitting our model’s flip rate and scatterer density to measured signal data. The second paper generalises a multilevel Forward Euler Monte Carlo method intro- duced by Giles [1] for the approximation of expected values depending on the solution to an Ito stochastic differential equation. Giles work [1] proposed and analysed a Forward Euler Multilevel Monte Carlo method based on realsiations on a hierarchy of uniform time discretisations and a coarse graining based control variates idea to reduce the computa- tional effort required by a standard single level Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretisations generated by adaptive algorithms developed by Moon et al. [3, 2]. These adaptive algorithms apply either deterministic time steps or stochastic time steps and are based on a posteriori error expansions first developed by Szepessy et al. [4]. Under sufficient regularity conditions, our numerical results, which include one case with singular drift and one with stopped dif- fusion, exhibit savings in the computational cost to achieve an accuracy of O(T ol), from O(T ol−3 ) to O (log (T ol) /T ol)2 . We also include an analysis of a simplified version of the adaptive algorithm for which we prove similar accuracy and computational cost results.

The goal of this project is to implement and verify an optimization model for finding a min-cost selection of switches and train paths at railway stations. The selected train paths must satisfy traffic requirements that commonly apply to regular railway traffic. The requirements include different combinations of simultaneous and overtaking train movements. The model does not rely on timetables but does instead utilize different path sets that are produced via algorithms based on a network representation of the station layout. The model has been verified on a small test station and also on the real station layout at Katrineholm. These tests show that the model can solve the problem for mid size stations with through traffic. In addition, we have performed a literature study regarding maintenance problems for switches and crossings. We have also looked at articles regarding the scheduling and routing of trains through railway stations. Finally we present some possible ways to further improve the model for more realistic experiments.

Fixed income markets are vast markets, involving a large number of actors including financial institutions, state actors, asset managers and corporations. An import part of these markets are contracts written on the xIBOR rates. This report is concerned with the trying to provide prices for options written on these rates, in particular for swaptions that are not at-the-money (atm) utilizing prices in the cap market. Different methods have been suggested in the literature for solving this problem. In particular we study the method suggested by Hagan et al where one calibrates a SABR model to the caplet surface with the same expiry as the swaption. One then assumes that the swaption contract with the same expiry follows the same SABR dynamics as the caplet, but with a recalibrated initial volatility to fit the atm point. We also study the approach suggested by Rebonato and Jäckel. They derive a model for swaption prices based on the individual volatilities of the forward rates that the underlying interest rate swap consists of, as well as the correlation between the forward rates.  Both of these approaches are studied empirically for the STIBOR market. The data set span between 2016 and 2021 and consists of the yield curve, flat cap volatilities and swaption volatilities. We use the 1Y1Y and 5Y5Y swaption surfaces, where the prices are not only quoted atm, to verify our model. We conclude that despite the SABR model being able to fit the caplet prices well, the method suggested by Hagan does not capture the swaption smile. The Rebonato and Jäckel approach also falls short of capturing the smile and produces similar results as the Hagan et al method. This is suggested to be due to the Hagan method capturing the caplet smile well, and the constant correlation assumption made in this thesis.

Time-of-Flight (ToF) cameras are becoming an increasingly common sensor in smartphones. These sensors are able to produce depth measurements in a grid at a relatively high rate. Using these depth measurements, point-clouds representing the captured scene can be produced. Previous research has been conducted in using ToF or LIDAR images to localize the camera. In this paper, we investigate several methods to localize the camera using point-clouds and surface meshes. Small alterations were made to some of the algorithms but the concepts remain the same. The main algorithms consisted of ICP variants as well as a relatively recent method called Corrective Gradient Refinement (CGR). The results obtained from generated data indicate that some of the methods are applicable for real-time applications, and the position estimates are comparable to those found in previous results. / Time-of-flight (ToF)-kameror blir en allt vanligare sensor i mobiltelefoner. Dessa sensorer kan producera djupmätningar i ett rutnät med relativt hög frekvens. Med hjälp av dessa djupmätningar kan ett punktmoln som representerar den fångade scenen produceras. Tidigare forskning har gjorts med hjälp av ToF- eller LIDAR-bilder för att lokalisera kameran. Här undersöks flera metoder för att lokalisera kameran med hjälp av ett punktmoln och en triangulering av en modell. Algoritmerna bestod till största delen av ICP-varianter samt en relativt ny metod som heter Corrective Gradient Refinement (CGR). Resultaten som erhållits från genererade data indikerar att vissa av metoderna är lämplig för realtidsapplikationer och felet på positioneringen är jämförbart med dem som hittades i tidigare resultat.

Increasing global warming with droughts, forest fires and melting polar ice has forced the world to speed up the transition from fossil fuel to fossil free energy. A big part of it is the use of solar energy. To obtain solar thermal energy, different sorts of solar collectors have been developed. One of these are the parabolic solar collector, a trough which concentrates the sun rays onto a receiver tube, which in turn absorbs the thermal heat. Absolicon Solar Collector AB is a Swedish company developing a parabolic solar collector named T160. These collectors are optically verified in the end of the Absolicon production line to decide if they meet the expected criteria. The verification of the parabolic shape is of utmost importance for the performance of the trough while a shape error can cause the beam to hit the receiver tube in a sub-optimal angle or miss it completely. If this were to happen, all of the energy can not be extracted. Earlier research have developed different methods for finding slope errors, deviations in the normal angles, in the trough but does not investigate the connection between slope errors and the trough shape errors that might have caused the deviations. This report aims to develop an algorithm based on slope errors in a parabolic trough collector which identifies four predetermined common shape errors in the trough. Identifying shape errors help to quickly identify and correct systematic deviations. In addition, this work aims to implement a new acceptance criteria based on slope errors for the solar collectors to make sure they hold up to their standard. The algorithm should also be compatible with a new camera system being implemented in the optical verification at Absolicon. This is done by deriving mathematical expressions for the normal angles in the trough with respect to the shape errors. By using the Pinhole Camera Model, the Law of Reflection and geometric properties of the solar collector, it is possible to convert pixel coordinates of receiver tube edges in images to normal angles. The resulting deviation in normal angle compared to the ideal ones are analysed and fitted to the mathematically derived expression for the normal angles by a build in minimization method in the tool lmfit in Python which uses non-linear least squares to detect type shape errors. The acceptance criteria and compatibility with the new system is implemented and taken into account. The results show that the calculation of slope errors from the data is valid with an uncertainty of 0.82 mrad and expected differences in the acceptance criteria quality value is seen when dealing with solar collectors with different type shape errors. The type shape error algorithm finds the correct shape errors for noisy self-created data which shows that the method works. The results when testing on real collectors with forced shape errors show potential but is in need of further adjustments and more clean precise data to produce certain accurate results. The algorithm is a good start to create a tool for finding typical shape errors in parabolic solar collectors.

Dredged rivers remain from when lumber was floated downriver for processing and shipping purposes. This has left parts of some rivers in Sweden with shallow rocky beaches and deep central furrows. Restoration efforts of dredged unregulated rivers has shown positive results for fish populations. This report compares, by use of CFD two regulated parts of Luleälven, one dredged and one natural. The report looks at habitats where the Riverine Grayling prefers to lay its eggs and evaluates whether velocity and depth conditions could give an explanation for the lower population of the Riverine Grayling in the dredged river. For the water levels tested, usable areas exist for all cases, but the dredged river has substantially smaller areas suitable as a habitat. The river geometry seems to be the large issue for Mattisudden as its shallow wide riverbanks are not conducive to habitats for the Riverine Grayling. Surprisingly the "lower" test case for the dredged river showed a larger usable area than for the "higher" water level, for the same river and higher usable area than the natural river. For this case the water level is now so low that the shallow wide riverbanks are above the water level and only the central furrow is filled with water. This shows two avenues that can be used to improve the habitats for the Riverine Grayling, lower the water level, or raise the riverbanks.