Numerical solution of buoyancy-driven flow problems in two spatial dimensions is presented. A high-order spectral method is applied for the spatial discretization, while the temporal discretization is done by operator splitting methods. By solving the convection-diffusion equation, which governs the temperature distribution, a thorough description of both the spatial and the temporal discretization methods is given. A fast direct solver for the arising system of algebraic equations is presented, and the expected convergence rates of both the spatial and the temporal discretizations are verified. As a step towards the Navier--Stokes equations, a solution of the Stokes problem is given, where a splitting scheme technique is introduced. An extension of this framework is used to solve the incompressible Navier--Stokes equations, which govern the fluid flow. By solving the Navier-Stokes equations and the convection-diffusion equation as a coupled system, two different buoyancy-driven flow problems in two-dimensional enclosures are studied numerically. In the first problem, emphasis is put on the arising fluid flow and the corresponding thermal distribution, while the second problem mainly consists of determining critical parameters for the onset of convection rolls.

This paper investigates the solution of the free rigid body equations of motion, as well as of the equations governing the torqued rigid body. We will consider two semi-exact methods for the solution of the free rigid body equations, and we discuss the use of both rotation matrices and quaternions to describe the motion of the body; our focus is on the quaternion formulation. The approach to which we give the most attention is based on the Magnus series expansion, and we derive numerical methods of order 2, 4, 6, and 8, which are optimal as they require a minimal number of commutators. The other approach uses Gaussian quadrature to approximate an elliptic integral of the third kind. Both methods rely on the exact solution of the Euler equation which involves the exact computation of the elliptic integral of the first kind. For the solution of the torqued rigid body equations, we divide the equations into two systems where one of them is the free rigid body equations; the solutions of these two systems are then combined in the Störmer-Verlet splitting scheme. We use these methods to solve the so-called marine vessel equations. Our numerical experiments suggest that the methods we present are robust and accurate numerical integrators of both the free and the torqued rigid body.

In kernel density estimation, the most crucial step is to select a proper bandwidth (smoothing parameter). There are two conceptually different approaches to this problem: a subjective and an objective approach. In this report, we only consider the objective approach, which is based upon minimizing an error, defined by an error criterion. The most common objective bandwidth selection method is to minimize some squared error expression, but this method is not without its critics. This approach is said to not perform satisfactory in the tail(s) of the density, and to put too much weight on observations close to the mode(s) of the density. An approach which minimizes an absolute error expression, is thought to be without these drawbacks. We will provide a new explicit formula for the mean integrated absolute error. The optimal mean integrated absolute error bandwidth will be compared to the optimal mean integrated squared error bandwidth. We will argue that these two bandwidths are essentially equal. In addition, we study data-driven bandwidth selection, and we will propose a new data-driven bandwidth selector. Our new bandwidth selector has promising behavior with respect to the visual error criterion, especially in the cases of limited sample sizes.

In this master thesis we have presented a security analysis of the TLS protocol with particular emphasis on the recently discovered renegotiation attack. From our security proof we get that the Handshake protocol with renegotiation, including the fix from IETF, is secure, and hence not vulnerable to the renegotiation attack anymore. We have also analysed the Handshake protocol with session resumption, and the Application data protocol together with the Record protocol. Both of these protocols were deemed secure as well. All the security proofs are based on the UC (Universal Composability) security framework.

In the last years, there has been done research in using topology as a new tool for studying data sets, typically high dimensional data. These studies have brought new methods for qualitative analysis, simplification, and visualization of high dimensional data sets. One good example, where these methods are useful, is in the study of microarray data (DNA data). To be able to use these methods, one needs to acquire knowledge of different topics in topology. In this paper we introduce simplicial homology, persistent homology, Mapper, and some simplicial complex constructions.

A priority based rule for use during the process of scheduling trains oper- ating on a single track railway line was proposed by the Norwegian railway operator and owner, Jernbaneverket. The purpose of this study is to inves- tigate the effect of the suggested scheduling rule on the scheduled waiting times suffered by trains operating on a segment of the railway line. It is shown that the scheduling rule, under certain limiting assumptions, can be studied in the setting of queuing theory and that it has properties in common with a theoretical priority discipline combining two well docu- mented priority rules. The main part of this study is the development and analysis of a threshold based, combined preemptive/non-preemptive priority discipline. Under the combined discipline, preemptions are allowed during the early stage of processing only. Theoretical expressions for flow-times of jobs passing through the queuing system are reached through detailed studies of the non-preemptive and the preemptive priority discipline. The relationship between the suggested priority based scheduling rule and the theoretical, combined priority discipline is finally illustrated by sim- ulations. When adjusted for actual time spent by trains on traversing the line segment, the steady state solution for flow-times obtained from queuing theory yields an accurate expression for the trains’ average scheduled wait- ing times. The scheduling problem can in fact be modeled accurately by an M/G/1 queue under the combined priority discipline.

In this thesis I have studied how to estimate parameters in an extreme value model with Markov Chain Monte Carlo (MCMC) given a data set. This is done with synthetic Gaussian time series generated by spectral densities, called spectrums, with a "box" shape. Three different spectrums have been used. In the acceptance probability in the MCMC algorithm, the likelihood have been built up by dividing the time series into blocks consisting of a constant number of points. In each block, only the maximum value, i.e. the extreme value, have been used. Each extreme value will then be interpreted as independent. Since the time series analysed are generated the way they are, there exists theoretical values for the parameters in the extreme value model. When the MCMC algorithm is tested to fit a model to the generated data, the true parameter values are already known. For the first and widest spectrum, the method is unable to find estimates matching the true values for the parameters in the extreme value model. For the two other spectrums, I obtained good estimates for some block lengths, others block lengths gave poor estimates compared to the true values. Finally, it looked like an increasing block length gave more accurate estimates as the spectrum became more narrow banded. A final simulation on a time series generated by a narrow banded spectrum, disproved this hypothesis.

Analyzing latent Gaussian models by using approximate Bayesian inference methods has proven to be a fast and accurate alternative to running time consuming Markov chain Monte Carlo simulations. A crucial part of these methods is the use of a Gaussian approximation, which is commonly found using an asymptotic expansion approximation. This study considered an alternative method for making a Gaussian approximation, the expectation propagation (EP) algorithm, which is known to be more accurate, but also more computationally demanding. By assuming that the latent field is a Gaussian Markov random field, specialized algorithms for factorizing sparse matrices was used to speed up the EP algorithm. The approximation methods were then compared both with regards to computational complexity and accuracy in the approximations. The expectation propagation algorithm was shown to provide some improvements in accuracy compared to the asymptotic expansion approximation when tested on a binary logistic regression model. However, tests of computational time requirement for computing approximations in simple examples show that the EP algorithm is as much as 15-20 times slower than the alternative method.

Today there is an immense production of data, and the need for better methods to analyze data is ever increasing. Topology has many features and good ideas which seem favourable in analyzing certain datasets where statistics is starting to have problems. For example, we see this in datasets originating from microarray experiments. However, topological methods cannot be directly applied on finite point sets coming from such data, or atleast it will not say anything interesting. So, we have to modify the data sets in some way such that we can work on them with the topological machinery. This way of applying topology may be viewed as a kind of discrete version of topology. In this thesis we present some ways to construct simplicial complexes from a finite point cloud, in an attempt to model the underlying space. Together with simplicial homology and persistent homology and barcodes, we obtain a tool to uncover topological features in finite point clouds. This theory is tested with a Java software package called JPlex, which is an implementation of these ideas. Lastly, a method called Mapper is covered. This is also a method for creating simplicial complexes from a finite point cloud. However, Mapper is mostly used to create low dimensional simplicial complexes that can be easily visualized, and structures are then detected this way. An implementation of the Mapper method is also tested on a self made data set.

The main task of this assignment is to filter out noise from a series of radar images and to carry out short term precipitation forecasts. It is important that the final routine is performed online, yielding new forecasts as radar images arrive with time. The data available is a time series arriving at a one hour ratio, from the Rissa radar located in Sør Trøndelag. Gaussian radial basis functions are introduced to create the precipitation field, whose movement is solely governed by its velocity field, called advection. By performing discretization forward in time, from the basis given by the differential advection equation, prior distributions can be obtained for both basis functions and advection. Assuming normal distributed radar errors, the basis functions and advection are conditioned on associating radar images, which in turn can be taken into the prior distributions, yielding new forecasts. A modification to the model, labeling the basis functions either active or inactive, enable the process of birth and death of new rain showers. The preferred filtering technique is a joint MCMC sampler, but we make some approximations, sampling from a single MCMC sampler, to successfully implement an online routine. The model yield good results on synthetic data. In the real data situation the filtered images are satisfying, and the forecast images are approximately predicting the forthcoming precipitation. The model removes statistical noise efficiently and obtain satisfying predictions. However, due to the approximation in the MCMC algorithm used, the variance is somewhat underestimated. With some further work with the MCMC update scheme, and given a higher frequency of incoming data, it is the authors belief that the model can be a very useful tool in short term precipitation forecasting. Using gauge data to estimate the radar errors, and merging online gauge data with incoming radar images using block-Kriging, will further improve the estimates.