In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.

We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.

We consider the Cauchy problem for the Helmholtz equation with a domain in with N cylindrical outlets to infinity with bounded inclusions in . Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Mazya proposed an alternating iterative method for solving Cauchy problems associated with elliptic, selfadjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R-2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters mu(0) and mu(1), the Robin-Dirichlet alternating iterative procedure is convergent.

We propose a variance reduced algorithm for solving monotone variational inequalities. Without assuming strong monotonicity, cocoercivity, or boundedness of the domain, we prove almost sure convergence of the iterates generated by the algorithm to a solution. In the monotone case, the ergodic average converges with the optimal O(1/k) rate of convergence. When strong monotonicity is assumed, the algorithm converges linearly, without requiring the knowledge of strong monotonicity constant. We finalize with extensions and applications of our results to monotone inclusions, a class of non-monotone variational inequalities and Bregman projections.

We present an implementation of a very recent approximation algorithm for scheduling jobs on a single machine with precedence constraints, minimising the total weighted completion time. We also evaluate the performance of this implementation. The algorithm was published by Shi Li in 2021 and is a (6+ε)-approximation algorithm for the multiprocessor problem P|prec|∑_j w_jC_j. We have implemented a version which is a (2+ε)-approximation algorithm for the single processor problem 1|prec|∑_j w_jC_j. This special case can easily be generalised to the multiprocessor case, as the two algorithms are based on the same LP relaxation of the problem. Unlike other approximation algorithms for this and similar problems, for example, those published by Hall, Schulz, Shmoys and Wein in 1997, and by Li in 2020, this algorithm has been developed with a focus on obtaining a good asymptotic run time guarantee, rather than obtaining the best possible guarantee on the quality of solutions. Li’s algorithm has run time O((n+κ) · polylog(n+κ) · log³ p_max · 1/ε²), where n is the number of jobs, κ is the number of precedence constraints and p_max is the largest of the processing times of the jobs. We also present a detailed explanation of the algorithm aimed at readers who do not necessarily have a background in scheduling and/or approximation algorithms, based on the paper by Li. Finally, we empirically evaluate how well (our implementation of) this algorithm performs in practice. The performance was measured on a set of 96 randomly generated instances, with the largest instance having 1024 jobs and 32 768 precedence constraints. We can find a solution for an instance with 512 jobs and 11 585 precedence constraints in 25 minutes.

Studies of life expectancy and death probabilities are crucial for life insurance. Payments for life insurance are completely dependent on whether an individual is alive or not, or is in various health conditions. In order to be able to price premiums correctly and set aside reserves, it is therefore of great importance to model life expectancy in the most accurate way possible. The insurance industry today uses historically proven well-functioning models that go as far back in time as 200 years. There are models even further back in time, but the models used today are mainly Gompertz (1826), Makeham (1860) and Lee-Carter (1992). Although these models perform well, it is always necessary to investigate whether there may be alternative models that model mortality better.

In this thesis, affine short-term interest rate models are applied for modeling the force of mortality that forms the basis for most interesting actuarial variables. As these models introduce stochastic mortality, the uncertainty and dependence over time can thus be described. The three short-term interest rate models examined in this project, which are common in financial theory; are Ornstein-Uhlenbeck, Feller and Hull-White. These models are then compared against each other in terms of the modeled force of mortality as well as the expected remaining life expectancy and the one-year probability of death. One aspect of stochastic mortality modeling that is not found in the existing literature but which is examined in this thesis is the modeling of mortality over time as this is one of the most important aspects in the life insurance mathematical industry. Finally, for validation purposes, all short-term interest rate models are evaluated using back-testing. The second main part of the work consists of generating results for the same quantities as above based on the DUS method in order to compare a commercial method with more theoretical and less approved ones.

The results show a great potential in several of the short-term interest rate models versus DUS both in terms of modeling over ages and calendar years. However, the results are not completely impeccable for individual calendar years where large spikes occur due to inaccurate parameter calibration. The satisfactory modeling of the short-term interest rate models over time was above the expectations as the models are not designed to capture decreasing trends. This is something that can be considered a great flexibility of the short-term interest rate models as they are more or less as accurate as the Lee-Carter model used in DUS, both in terms of age and time modeling of mortality.

This thesis focuses on statistical and machine learning methods designed for sequential and repeated measurement data. We start off by considering the classic general linear model (MANOVA) followed by its generalization, the growth curve model (GMANOVA), designed for analysis of repeated measurement data. By considering a binary classification problem of normal data together with the corresponding maximum likelihood estimators for the growth curve model, we demonstrate how a classification rule based on linear discriminant analysis can be derived which can be used for repeated measurement data in a meaningful way.

We proceed to the topics of neural networks which serve as our second method of classification. The reader is introduced to classic neural networks and relevant subtopics are discussed. We present a generalization of the classic neural network model to the recurrent neural network model and the LSTM model which are designed for sequential data.

Lastly, we present three types of data sets with an total of eight cases where the discussed classification methods are tested.

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

Visualization and image filtering are important parts of cryo-electron tomography analysis. ArtiaX, a plugin developed for UCSF ChimeraX, has been extended to improve the functionality of these two parts. For the visualization, a method of moving 3D surfaces to remove overlap between them has been developed and implemented. To accommodate this, a Monte Carlo approach using Poisson disc sampling for approximating volume of overlap between 3D surfaces is used, and a novel method for measuring overlap has been invented, called the Normal Projection Method, useful for measuring the depth of overlap between surfaces. For the image filtering, tomogram averaging and frequency filters have been added to the ArtiaX toolbox.

In this paper, an optimal actuator placement problem with a linear wave equation as the constraint is considered. In particular, this work presents the frameworks for finding the best location of actuators depending upon the given initial conditions, and where the dependence on the initial conditions is averaged out. The problem is motivated by the need to control vibrations induced by pedestrian-bridge interactions. An approach based on shape optimization techniques is used to solve the problem. Specifically, the shape sensitivities involving a cost functional are determined using the averaged adjoint approach. A numerical algorithm based on these sensitivities is used as a solution strategy. Numerical results are consistent with the theoretical results, in the two examples considered.

Congestion-aware scheduling in case of downlink cellular communication has ignored the distribution of diverse content to different clients with heterogeneous secrecy requirements. Other possible application areas that encounter the preceding issue are secure offloading in mobile-edge computing, and vehicular communication. In this paper, we extend the work in Arvanitaki et al. (SN Comput Sci 1(1):53, 2019) by taking into consideration congestion and random access. Specifically, we study a two-user congestion-aware broadcast channel with heterogeneous traffic and different security requirements. We consider two randomized policies for selecting which packets to transmit, one is congestion-aware by taking into consideration the queue size, whereas the other one is congestion-agnostic. We analyse the throughput and the delay performance under two decoding schemes at the receivers, and provide insights into their relative security performance and into how congestion control at the queue holding confidential information can help decrease the average delay per packet. We show that the congestion-aware policy provides better delay, throughput, and secrecy performance for large arrival packet probabilities at the queue holding the confidential information. The derived results also take account of the self-interference caused at the receiver for whom confidential data is intended due to its full-duplex operation while jamming the communication at the other user. Finally, for two decoding schemes, we formulate our problems in terms of multi-objective optimization, which allows for finding a trade-off between the average packet delay for packets intended for the legitimate user and the throughput for the other user under congestion-aware policy.

Explicit bounds are given for the Kolmogorov andWasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some sigma-algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the sigma-algebra. The proof is based on Steins method. As an application, we consider the Yule-Ornstein-Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

This thesis aims to produce a control strategy that minimizes the energy usage of the pumps in the sewer system of Växjö City. The pumps are located in a small number of pumping stations. These are connected by a network of pipes in a way that requires the control of the pumps to be coordinated, both within and between the pumping stations. A model of the system based on fluid mechanics is proposed and from this an optimization problem is constructed for the control problem. When the optimization problem is reformulated as a hierarchy of smaller problems these problems can be solved sequentially and the computational requirements are low. The resulting control strategy gives an optimal control at each flow rate to the waste water treatment plant, and the results show that an optimal control can yield significant energy savings. The control strategy is also shown to be stable with regards to errors in the model.

The Cauchy problem for Helmholtz equation, for moderate wave number k(2), is considered. In the previous paper of Achieng et al. (2020, Analysis of Dirichlet-Robin iterations for solving the Cauchy problem for elliptic equations. Bull. Iran. Math. Soc.), a proof of convergence for the Dirichlet-Robin alternating algorithm was given for general elliptic operators of second order, provided that appropriate Robin parameters were used. Also, it has been noted that the rate of convergence for the alternating iterative algorithm is quite slow. Thus, we reformulate the Cauchy problem as an operator equation and implement iterative methods based on Krylov subspaces. The aim is to achieve faster convergence. In particular, we consider the Landweber method, the conjugate gradient method and the generalized minimal residual method. The numerical results show that all the methods work well. In this work, we discuss also how one can approach non-symmetric differential operators by similar operator equations and model problems which are used for symmetric differential operators.

Temperature control is important for the steel making process. Knowledge of the amount of thermal energy stored in the ladle allows for better predictions of the steel temperature during the process. This has a potential to improve the quality of the steel. In this work, we present a mathematical model of the heat transfer within a ladle during the production process. The model can be used to compute the current, and also the future, thermal status of the ladle. The model is simple and can be solved efficiently. We also present results from numerical simulations intended to illustrate the model.

We present a MATLAB implementation of the symmetric rank-one (SC-SR1) method that solves trust-region subproblems when a limited-memory symmetric rank-one (L-SR1) matrix is used in place of the true Hessian matrix, which can be used for large-scale optimization. The method takes advantage of two shape-changing norms [Burdakov et al. 2017; Burdakov and Yuan 2002] to decompose the trust-region subproblem into two separate problems. Using one of the proposed norms, the resulting subproblems have closed-form solutions. Meanwhile, using the other proposed norm, one of the resulting subproblems has a closed-form solutionwhile the other is easily solvable using techniques that exploit the structure of L-SR1 matrices. Numerical results suggest that the SC-SR1 method is able to solve trust-region subproblems to high accuracy even in the so-called "hard case." When integrated into a trust-region algorithm, extensive numerical experiments suggest that the proposed algorithms perform well, when compared with widely used solvers, such as truncated conjugate-gradients.

This thesis focuses on the establishment and analysis of residuals in the so called GMANOVA-MANOVA model. The model is a special case of the Extended Growth Curve Model. It has two terms where one term models the profiles (growth curves) and the other the covariables of interest. This model is useful in studying growth curves in short time series in fields such as economics, biology, medicine, and epidemiology. Furthermore, in the literature, residuals have been extensively studied and used to check model adequacy in univariate linear models. This thesis contributes to the extension of the study of residuals in the GMANOVA-MANOVA model. 

In this thesis, a new pair of residuals is established via the maximum likelihood estimators of the parameters in the model. One residual indicates whether an individual is far away from the group means and a second residual is used to check assumptions about the mean structure. Different properties of these residuals are verified and their interpretation is discussed. Moreover, using parametric bootstrap, the empirical distributions of the extreme elements in the residuals are derived. 

Finally, testing bilinear restriction in the MANOVA model is considered. One can show that the MANOVA model with bilinear restrictions is nothing more than a GMANOVA-MANOVA model. Furthermore, the likelihood ratio test can be shown to be given as a function of the residuals to the GMANOVA-MANOVA model, which can be used to understand the appropriateness of the model and test the bilinear hypothesis. 

The main goal of this paper is to study residuals in a special case of the extended growth curve model, called the GMANOVA-MANOVA model. With the help of an example, emphasis is put on the model formulation, interpretation of the model and residuals that vanish, with a discussion about the reasons behind this fact and the consequence of it.

In this article, the GMANOVA-MANOVA model is considered. Two different matrix residuals are established. The interpretation of the residuals is discussed and several properties are verified. A data set illustrates how the residuals can be used.

Two different matrix residuals in a special GMANOVA-MANOVA model have previously been established (see Byukusenge et al., 2021, “On residual analysis in the GMANOVA-MANOVA model”). The residual that is studied in this article is constructed via the difference of the observed group means and the estimated mean structure. The residual provides information about the appropriateness of the model assumptions concerning the mean structure. The aim of this paper is to study the distribution of the largest elements (by absolute value) of the residual via two data sets. Parametric bootstrap is used to identify thresholds so that extreme elements of the residuals can be identified.

In this thesis, we set out to model the market risk exposure for 251 stocks in the S&P 500 index, during a ten-year period between 2011-04-30 and 2021-03-31. The study brings to light a model not often mentioned in the scientific literature focused on market risk estimation, the linear mixed model. The linear mixed model makes it possible to model a time-varying market risk, as well as adding structure to the idiosyncratic risk, which is often assumed to be a stationary process. The results show that the mixed model is able to produce more accurate estimates for the market risk, compared to the baseline, which is here defined as a CAPM model. The success of the mixed model, which we in the study will refer to as the ADAPT model (adaptive APT), most certainly lies in its ability to create a hierarchical regression model. This makes it possible to not just view the set of observations as a single population, but let us group the observations into different clusters and in such a way makes it possible to construct a time-varying exposure. In the last part of the thesis, we highlight possible improvements for future works, which could make the estimation even more accurate and also more efficient.

The goal of this thesis is to create an optimization model to optimize the routing of trains within railway stations. This problem is known as the train platforming problem, and the model we present is an integer programming model. By this model we aim to optimize factors such as walking distance, switch usage or platform usage.

We validate the model by implementing the model for Linköping station, which is a typical mid size station in the Swedish railway network. This implementation is done for different time horizons, ranging from 2 hours to one day, which corresponds to train sets ranging from 27 to 265 trains.

In the conclusion we see that the model is efficient for optimizing the train platforming problem for the implemented station and timetables, and that the model has a possibility to optimize the four objectives tested. Furthermore we see that optimizing certain objectives gives solutions that are also good with regards to other objective functions.

The emerging concept of Over-the-Air (OtA) computation has shown great potential for achieving resource-efficient data aggregation across large wireless networks. However, current research in this area has been limited to the standard many-to-one topology, where multiple nodes transmit data to a single receiver. In this letter, we address the problem of applying OtA computation to scenarios with multiple receivers, and propose a novel communication design that exploits joint precoding and decoding over multiple time slots. To determine the optimal precoding and decoding vectors, we formulate an optimization problem that aims to minimize the mean squared error of the desired computations while satisfying the unbiasedness condition and power constraints. Our proposed multi-slot design is shown to be effective in saving communication resources (e.g., time slots) and achieving smaller estimation errors compared to the baseline approach of separating different receivers over time.

In this paper the authors take an approach over the likelihood ratio test for double complete symmetry, which enables a particularly simple implementation of the test as well as a particularly adequate way towards obtaining very sharp approximations for the exact distribution of the test statistic. Numerical studies show the very good performance of the near-exact distributions derived, namely their very good performance for very small sample sizes and their asymptotic behavior for increasing numbers of variables involved.

Purpose: High dose-rate prostate brachytherapy has been implemented in Sweden in the late 1980s and early 1990s in six clinics using the same schedule: 20 Gy in two fractions combined with 50 Gy in 25 fractions with external beam radiation therapy. Thirty years have passed and during these years, various aspects of the treatment process have developed, such as ultrasound-guided imaging and treatment planning system. An audit was conducted, including a questionnaire and treatment planning, which aimed to gather knowledge about treatment planning methods in Swedish clinics. Material and methods: A questionnaire and a treatment planning case (non-anatomical images) were sent to six Swedish clinics, in which high-dose-rate prostate brachytherapy is performed. Treatment plans were compared using dosimetric indices and equivalent 2 Gy doses (EQD(2)). Treatment planning system report was used to compare dwell positions and dwell times. Results: For all the clinics, the planning aim for the target was 10.0 Gy, but the volume to receive the dose differed from 95% to 100%. Dose constraints for organs at risk varied with up to 2 Gy. The dose to 90% of target volume ranged from 10.0 Gy to 11.1 Gy, equivalent to 26.0 Gy EQD(2) and 31.3 Gy EQD(2), respectively. Dose non-homogeneity ratio differed from 0.18 to 0.32 for clinical target volume (CTV) in treatment plans and conformity index ranged from 0.52 to 0.59 for CTV. Conclusions: Dose constraints for the organs at risk are showing a larger variation than that reflected in compared treatments plans. In all treatment plans in our audit, at least 10 Gy was administered giving a total treatment of 102 Gy EQD(2), which is in the upper part of the prescription doses published in the GEC/ESTRO recommendations.

PURPOSE: The aim was to evaluate a postprocessing optimization algorithms ability to improve the spatial properties of a clinical treatment plan while preserving the target coverage and the dose to the organs at risk. The goal was to obtain a more homogenous treatment plan, minimizing the need for manual adjustments after inverse treatment planning. MATERIALS AND METHODS: The study included 25 previously treated prostate cancer pa-tients. The treatment plans were evaluated on dose-volume histogram parameters established clin-ical and quantitative measures of the high dose volumes. The volumes of the four largest hot spots were compared and complemented with a human observer study with visual grading by eight oncologists. Statistical analysis was done using ordinal logistic regression. Weighted kappa and Fleiss kappa were used to evaluate intra-and interobserver reliability. RESULTS: The quantitative analysis showed that there was no change in planning target volume (PTV) coverage and dose to the rectum. There were significant improvements for the adjusted treatment plan in: V150% and V200% for PTV, dose to urethra, conformal index, and dose nonhomogeneity ratio. The three largest hot spots for the adjusted treatment plan were significantly smaller compared to the clinical treatment plan. The observers preferred the adjusted treatment plan in 132 cases and the clinical in 83 cases. The observers preferred the adjusted treatment plan on homogeneity and organs at risk but preferred the clinical plan on PTV coverage. CONCLUSIONS: Quantitative analysis showed that the postadjustment optimization tool could improve the spatial properties of the treatment plans while maintaining the target coverage. (c) 2022 The Authors. Published by Elsevier Inc. on behalf of American Brachytherapy Society. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

This thesis aims to study how flat dielectric lenses can be designed. The usage of flat lenses is steadily increasing as they are smaller and less bulky than traditional convex lenses. Instead of a lens with a curved surface the permittivity in the lens is varied to achieve the same effect. Two different computational methods were investigated when approaching this problem: physical and geometrical optics. In physical optics the incoming radio waves are treated as waves in contrast to geometrical optics where it is considered as rays. Both methods are used as approximations of Maxwell's equations.

The variation of permittivity in the lens was formulated as an optimisation problem where the lens' focusing abilities were maximised. The optimisation was implemented with an interior point method. Both arbitrary permittivity distributions as well as predetermined distributions were examined in this work. All optimised lens models were then simulated in a full wave commercial simulation software to verify and compare the two.

The simulations showed that both approaches gave promising results as they focused the electromagnetic wave in a satisfying way. However the physical optics approach was more prominent as the focused radio waves had a much higher magnitude than the approach based on geometrical optics. The conclusion was therefore that physical optics is the preferred approach.

This thesis presents some methods of multivariate dimension reduction, with emphasis on methods guided by the work of R.A. Fisher. Some of the methods presented can be traced back to the 20th century, while some are much more recent. For the more recent methods, additional attention will paid to the foundational underpinnings. The presentation for each of the methods contains a brief introduction of its general philosophy, accompanied by some theorems and ends with the connection to the work of Fisher.

The paper is concerned with methods for computing the best low multilinear rank approximation of large and sparse tensors. Krylov-type methods have been used for this problem; here block versions are introduced. For the computation of partial eigenvalue and singular value decompositions of matrices the Krylov-Schur (restarted Arnoldi) method is used. A generalization of this method to tensors is described, for computing the best low multilinear rank approximation of large and sparse tensors. In analogy to the matrix case, the large tensor is only accessed in multiplications between the tensor and blocks of vectors, thus avoiding excessive memory usage. It is proved that if the starting approximation is good enough, then the tensor Krylov-Schur method is convergent. Numerical examples are given for synthetic tensors and sparse tensors from applications, which demonstrate that for most large problems the Krylov-Schur method converges faster and more robustly than higher order orthogonal iteration.

The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-(2,2,lambda) approximation is computed for lambda=1,2,3, and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.

This thesis examines a simple method for detecting a change with respect to the variance in a sequence of independent normally distributed observations with a constant mean. The method filters out observations with extreme values and divides the sequence into equally large subsequences. For each subsequence, the count of extreme values is translated to a binomial random variable which is tested towards the expected number of extremes. The expected number of extremes comes from prior knowledge of the sequence and a specified probability of how common an extreme value should be. Then specifying the significance level of the goodness-of-fit test yields the number of extreme observations needed to detect a change. 

The approach is extended to a sequence of independent multivariate normally distributed observations by transforming the sequence to a univariate sequence with the help of the Mahalanobis distance. Thereafter it is possible to apply the same approach as when working with a univariate sequence. Given that a change has occurred, the distribution of the Mahalanobis distance of a multivariate normally distributed random vector with zero mean is shown to approximately follow a gamma distribution. The parameters for the approximated gamma distribution depend only on Σ₁^−1/2 Σ₂Σ₁^−1/2 with Σ₁ and Σ₂ being the covariance matrices before and after the change has occurred. In addition to the proposed approach, other statistics such as the largest eigenvalue, the Kullback-Leibler divergence, and the Bhattacharyya distance are considered. 

We clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes delta. They reduce to the better-known Markov properties if 8 is the empty set, or, with the exception of the local property, if delta is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes delta(0), all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set delta \ delta(0). We note that many of the above results are new even for reciprocal chains.

As the demand for electricity increases throughout the globe while we want to reduce the use of fossil fuels, the need for renewable energy sources is bigger than ever. In countries where solar power makes up a large part of the total energy production, the overall electricity spot price level has become lower. This thesis investigates the underlying mechanism that drives the energy market, and in specific, how the solar power impacts the electricity spot price. We present results from studies made in other markets, and introduce a Regime Switching model for explaining the impact in Sweden. We show that an increase of photovoltaics power has a price lowering effect on the daily price pattern in price area SE3 and SE4.

For workplaces with a preference or need for staffing around the clock, employees commonly work in shifts, which are work sessions that span different parts of the day. The scheduling of these shifts is a multi-objective optimization problem with both hard and soft constraints. The reduction in the available workforce when employees go on vacation makes the problem especially constrained. We describe a method that uses a genetic algorithm to generate shift schedules, for teams of employees and time periods with vacations. The method supports a staffing demand that can be met with one of multiple combinations of shifts. The genetic algorithm features specialized crossovers, together with a repair step aimed at maintaining staffing that fulfils the staffing requirements. A software implementation of the method is evaluated on three real-life problem instances. For two of them, it can produce schedules that are feasible, but subpar to those constructed manually by an experienced personnel scheduling professional. Several ideas to improve the program are presented.

Regression testing is the process of testing software to make sure changes to the software will not change the functionality. With growing test suites theneed to prioritize arises. This thesis explores how to weigh factors such as the number of fails detected, days since latest test case execution, and coverage. The prioritization is done over multiple test systems, software branches, and over many test sessions where the software can change in-between. With data provided by an industrial partner, we evaluate different ways to prioritize. The developed mathematical model could not cope with the size of the problem, whereas a simulated annealing approach based on said model proved highly successful. We also found that prioritizing test cases related to recent codechanges was effective.

In this paper we study the Growth Curve model with orthogonal covariance structure and derive estimators for all parameters. The orthogonal covariance structure is a generalization of many known structures, e.g., compound symmetry covariance structure. Hence, we compare our estimators with earlier results found in the literature.

In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwells equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.

The valuation of weather derivatives is greatly dependent on accurate modeling and forecasting of the underlying temperature indices. The complexity and uncertainty in such modeling has led to several temperature processes being developed for the Monte Carlo simulation of daily average temperatures. In this report, we aim to compare the results of two recently developed models by Gyamerah et al. (2018) and Evarest, Berntsson, Singull, and Yang (2018).

The paper gives a thorough introduction to option theory, Lévy and Wiener processes, and generalized hyperbolic distributions frequently used in temperature modeling. Implementations of maximum likelihood estimation and the expectation-maximization algorithm with Kim's smoothed transition probabilities are used to fit the Lévy process distributions and both models' parameters, respectively. Later, the use of both models is considered for the pricing of European HDD and CDD options by Monte Carlo simulation.

The evaluation shows a tendency toward the shifted temperature regime over the base regime, in contrast to the two articles, when evaluated for three data sets. Simulation is successfully demonstrated for the model of Evarest, however Gyamerah's model was unable to be replicated. This is concluded to be due to the two articles containing several incorrect derivations, why the thesis is left unanswered and the articles' conclusions are questioned. We end by proposing further validation of the two models and summarize the alterations required for a correct implementation.

A Reynolds equation governing the steady flow of a fluid through a curvilinear, narrow tube, with its derivation from Navier-Stokes equations through asymptotic methods is presented. The channel considered may have a rather large curvature and torsion. Approximations of the velocity and the pressure of the fluid inside the channel are constructed by artificially imposing appropriate boundary conditions at the inlet and the outlet. A justification for the approximations is provided along with a comparison with a simpler case.

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. © 2023 The Author(s)

Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.

Logic-based Benders decomposition (LBBD) is a strategy for solving discrete optimisation problems. In LBBD, the optimisation problem is divided into a master problem and a subproblem and each part is solved separately. LBBD methods that combine mixed-integer programming and constraint programming have been successfully applied to solve large-scale scheduling and resource allocation problems. Such combinations typically solve an assignment-type master problem and a scheduling-type subproblem. However, a challenge with LBBD methods that have feasibility subproblems are that they do not provide a feasible solution until an optimal solution is found. 

In this thesis, we show that feasible solutions can be obtained by finding and combining feasible parts of an infeasible master problem assignment. We use these insights to develop an acceleration technique for LBBD that solves a series of subproblems, according to algorithms for constructing a maximal feasible subset of constraints (MaFS). Using a multiprocessor scheduling problem as a benchmark, we study the computational impact from using this technique. We evaluate three variants of LBBD schemes. The first uses MaFS, the second uses irreducible subset of constraints (IIS) and the third combines MaFS with IIS.

Computational tests were performed on an instance set of multiprocessor scheduling problems. In total, 83 instances were tested, and their number of tasks varied between 2794 and 10,661. The results showed that when applying our acceleration technique in the decomposition scheme, the pessimistic bounds were strong, but the convergence was slow. The decomposition scheme combining our acceleration technique with the acceleration technique using IIS showed potential to accelerate the method.

This study provides numerical solutions to the two-dimensional linearized shallow water equations (SWE) using a high-order finite difference scheme in Summation By Parts (SBP) form. In addition to the SBP operators for the discretizations, penalty terms, Simultaneous Approximation Terms (SAT) are applied to impose well-posed open boundary conditions. The conventional SWE with height and velocities as the prognostic variables, and a new type of the vorticity–divergence SWE with wave height gradients, vorticity and divergence as the prognostic variables were investigated. It was shown that the solution in all numerical tests enter and exit the domain without instabilities. The convergence rates were correct for all orders of the SBP operators in both the entrance and exit tests. Interestingly, the error norm of the wave height were orders of magnitude lower in the vorticity–divergence solutions compared to the conventional SWE solutions.

Removing snow in a city is an unavoidable task in Nordic countries like Sweden. A number of streets in an area need to be cleared of snow by a limited number of vehicles and the tours for the vehicles must be planned in order to minimize the time and/or cost. Since the amount of snow can vary significantly from one year to another, the plans/tours of one year cannot be used for the next year. Hence, new tours need to be planned each time. Snow removal can be done in rural or urban areas and in addition during snowfall or after a snowfall. In this thesis, we study urban snow removal after a snowfall. 

There are different relevant specifics of the urban snow removal problem. For instance, there are different types of streets which need different numbers of sweeps in order to remove the snow. In addition, some tasks must be done before other tasks can be started. This leads to precedence constraints. Furthermore, each vehicle needs a certain time to switch from a task to another task. The problem can be formulated as a huge time-indexed mixed integer programming which often is not directly solvable in practice. 

The contributions of this thesis include the study of different relaxations and heuristics to find feasible solutions and improve the bounds on the optimal objective function values which are discussed in five papers. Paper I deals with single vehicle snow removal. A branch-and-dive heuristic based on branch-and-bound principles is given in order to improve the solutions and bounds. In Paper II, feasible solutions for the snow removal problem with a limited number of identical vehicles are obtained. First, the work is broken down into smaller parts, one for each vehicle. Based on the obtained allocation, a feasible tour for each single vehicle snow removal is obtained. Finally, combined solution approaches and co-ordination of the vehicles to find a feasible solution for the original problem are discussed. 

In order to improve the computational efficiency, one can take advantage of the tree structure, since modern real life city networks often contain parts that are trees. In Paper III, tree parts are studied and a tree elimination procedure is given for the snow removal problem, to be used before searching for optimal tours. Two variations encountered in practice for normal streets are compared in Paper IV. The first variant is doing a middle sweep before the two side sweeps and the second one is doing only side sweeps. Paper V studies the problem from modeling perspective. The problem is formulated as a mixed integer programming model and different relaxations of it are investigated. Finally, Lagrangian relaxation of the problem is studied in Paper VI. Different possibilities for Lagrangian relaxations are investigated and subgradient optimization is used to solve the Lagrangian dual. 

Snow removal is an unavoidable problem in Nordic countries like Sweden. A number of streets in a city need to be cleared of snow by a limited number of vehicles. The problem can be formulated as a very large mixed integer programming model, which is practically unsolvable. In order to find a feasible solution, first we break done the work into smaller parts, one for each vehicle. To find which streets a vehicle shall take care of, we solve a weighted k-Chinese postman problem. Based on the allocation obtained, we consider snow removal problems for single vehicles, where details such as turning penalties and precedences are included. These problems can be reformulated to asymmetric traveling salesman problems in extended graphs, and we have a heuristic for finding feasible solution of those. In this paper, we discuss combined solution approaches and coordination of the vehicles to find a feasible solution for the whole original problem including all details. We use an iterative procedure to combine the tours, based on the tools mentioned above, and a procedure for constructive coordination of the tours. We also have new improvement procedures for the combined solution. We have implemented the methods and applied them to real life city networks. The numerical results show that the methods obtain feasible tours for large problems within a reasonable time.

Snow removal is, in Sweden, an infrequently occurring challenge. Doing the snow removal more efficiently could give much benefits for society. Since the amounts of snow vary a lot from day to day, and from year to year, fixed plans are not the best. Optimization of the snow removal tours could save much money. In this paper, we study the multi-vehicle urban snow removal problem from a mixed integer programming perspective. It is a very hard problem, and obtaining the exact optimum seems to be out of reach. Therefore, we study relaxations of the problem. Our goal is simply to find the best bounds for the optimal objective function value that is possible in limited time. We present some promising possibilities, verified by extensive computational tests.  

Snow removal problem in cities is a challenging task in Nordic countries. The problem is finding optimal tours for a certain number of vehicles with some circumstances in order to clear a number of streets in a city. We have formulated the urban snow removal problem as a time-indexed mixed integer linear programming model which is huge and complicated. In our previous work, we studied the model and its different relaxations which show that the problem is not solvable in practice. Since the problem has many sets of constraints with complicated structures, relaxing them with Lagrangian relaxation might be beneficial. In this paper, we discuss different possibilities of relaxing sets of constraints and develop a Lagrangian heuristic which consists of a suitable Lagrangian relaxation of the problem, a subgradient optimization method for solving the Lagrangian dual, and procedures for obtaining feasible solutions. The heuristic has been implemented and applied to artificial and real life city networks. The results show that the bounds have been improved. 

We study designing a tour for a snow removal vehicle. Several sweeps are required to clear a street of snow. We compare two variations for normal streets, the first is doing a middle sweep before the two side sweeps, and the second is not doing the middle sweep. We apply a previously developed method called branch-and-dive, and show that it yields very good results if the middle sweep is not used.