<p>Simulations of the nuclear magnetic resonance relaxation method is an important part of a digital laboratory developed by Numerical Rocks. The laboratory is used to model petrophysical properties and simulating fluid flow in the pore scale of reservoir rocks. The nuclear magnetic resonance relaxation method can be simulated on a computer using a method involving random walkers. This computer simulation can be parallelized to reduce computational time. The aim of this study has been to examine how overlapping boundaries affects speed-up and communication in a parallel simulation of random walkers. Several parallel algorithms have been proposed and implemented. It was found that an overlapping partitioning of the problem is recommended, and that communication decreases exponentially with increasing overlap. However, increased overlap resulted only in a small negative impact on memory usage and speed-up.</p>

<p>Safety instrumented systems (SISs) are of prime importance to the process industry to avoid catastrophic consequences or even loss of human life. The dangerous situations that any equipment may face should be analysed in order to quantify the associated risk and to choose a design of the SIS that reduces the risk to a tolerable level. The safe failure fraction (SFF) is a parameter defined in the standards IEC 61508 and IEC 61511, and is used to determine the need for additional channels that can activate the safety function if a failure is present. The standards consider a high SFF as an indicator of a safe design, and by increasing SFF, one may allow a lower redundancy level for a SIS and therefore reduce costs. Safety engineers discuss the suitability of this parameter, and some argue that the negative effects of safe failures on the reliability are so significant that the parameter should not be used. For a safety shutdown valve installed to prevent overpressure, a safe failure is defined as a spurious closure where the source of high pressure is isolated. This thesis examines the effects of safe failures on the reliability of such systems by using a Markov model. According to IEC 61508 and IEC 61511 the system reliability of a safety shutdown system is measured by the probability of failure on demand (PFD). From the results it can be concluded that the time needed to restore the system back to initial state after a safe failure does not have a significant effect on PFD. A long restoration time after a safe failure in combination with a high frequency of safe failures is negative with respect to production downtime. The main contributor to PFD is the long run probability of being in a state where a dangerous undetected (DU) failure is present. DU failures are normally detected by function tests or sometimes upon demand, but they can also be revealed by a spurious closure. This effect is based on the assumption of perfect repair of safe failures, which means that all possible failure modes are detected and the failed items are repaired or replaced after restoration of safe failures. The ability to reveal DU failures is clearly dependent on the frequency of a DU failure and safe failure occurring in the same test interval. This thesis demonstrates that safe failures only have significant effect when the dangerous failure rate is high. Other parameters affect the PFD to a greater extent, and the importance of exact parameter estimation is crucial and more important than the positive effects of safe failures. The SFF must be close to 100% to have a significant effect on the PFD, and since it is always aimed at minimising the number of dangerous failures, the alternative is to add safe failures. This is probably not the intent of SFF and is negative with respect to production downtime. Safe failures does not justify a lower degree of redundancy. On the other hand, the positive effects of safe failures show a satisfactory reason for adopting a longer test interval. This is an optimisation of PFD and can reduce costs or even the frequency of dangerous situations during start-up and shutdown. This thesis demonstrates that the PFD is not affected by safe failures, and indicates no reason to be in doubt about this parameter as a measure of reliability. The SFF gives hardly any information and the choice of SIS architecture should not be based on SFF alone. An alternative parameter that considers different means of revealing DU failures seems to be a better choice.</p>

<p>The last two years have seen the most volatile financial markets for decades with steep losses in asset values and a deteriorating world economy. The insolvency of several banks and their negative impact on the economy has led to criticism of their risk management systems for not being adequate and lacking foresight. This thesis will study the performance of two broadly adopted portfolio risk measures before and during the current financial turbulence to examine their accuracy and reliability. The study will be carried out on a case portfolio consisting of American and European fixed income and equity. The portfolio uses a dynamic asset allocation scheme to maximize the ratio between expected return and portfolio risk. The market risk of the portfolio will be calculated on a daily basis using both Value-at-Risk (VaR) and expected shortfall (ES) in a Monte Carlo framework. These risk measures are then compared with prior measurements and the actual loss over the period. The results from the study indicate that the implemented risk model do not give totally reliable estimates, with more frequent and larger real losses than predicted. Nevertheless, the study sees a significant worsening in the performance of the risk measures during the current financial crisis from June 2007 to December 2008 compared with the previous years. This thesis argues that VaR and ES are useful risk measures, but that users should be well aware of the pitfalls in the underlying models and take appropriate precautions.</p>

<p>Properties of the empirical interpolation (EI) method is investigated by solving selected model problems. Also, a more challenging example with deformed geometry is solved within the online/offline computational framework of the reduced basis method.</p>

<p>It is often very difficult, particularly in higher dimensions, to find a good multivariate model that describes both marginal behavior and dependence structure of data efficiently. The copula approach to multivariate models has been found to fit this purpose particularly well, and since it is a relatively new concept in statistical modeling, it is under frequent development. In this thesis we focus on the decomposition of a multivariate model into pairwise copulas rather than the usual multivariate copula approach. We account for the theory behind the decomposition of a multivariate model into pairwise copulas, and apply the theory on both daily and intra day financial returns. The results are compared with the usual multivariate copula approach, and problems applying the theory are accounted for. The multivariate copula is rejected in favor of the pairwise decomposed model on daily returns with a level of significance less than 1%, while our decomposed models on intra day data does not lead to a rejection of the models with multivariate copulas. On daily returns a pairwise decomposition with Student copulas is preferable to multivariate copulas, while the decomposed models on intra day data need more development before outperforming multivariate copulas.</p>

<p>Shape analysis and recognition is a field ripe with creative solutions and innovative algorithms. We give a quick introduction to several different approaches, before basing our work on a representation introduced by Klassen et. al., considering shapes as equivalence classes of closed curves in the plane under reparametrization, and invariant under translation, rotation and scaling. We extend this to a definition for nonclosed curves, and prove a number of results, mostly concerning under which conditions on these curves the set of shapes become manifolds. We then motivate the study of geodesics on these manifolds as a means to compute a shape metric, and present two methods for computing such geodesics: the shooting method from Klassen et. al. and the ``direct'' method, new to this paper. Some numerical experiments are performed, which indicate that the direct method performs better for realistically chosen parameters, albeit not asymptotically.</p>

<p>In this thesis we study market risk in turbulent markets over different risk horizons. We construct portfolios which represent possible investments for a life assurance fund. The portfolios consist of equities, fixed income instruments, cash positions and interest rate derivatives. Today, the most commonly used metrics for market risk are Value-at-Risk (VaR) and Expected Shortfall (ES), and they will be central. We introduce necessary theory from quantitative finance related to asset price dynamics and security pricing. Further, interest rate related instruments are handled by the LIBOR Market Model (LMM), while equity prices are modeled as geometric Brownian motions. We use implied volatilities for instruments where they are available, and historical for the rest. We implement a risk model and make daily and quarterly market risk estimates between 2000-2008 for the portfolios. We choose some central events from the last quarter of 2008, a critical phase of the ongoing financial crisis, and analyze how the portfolios and the corresponding risk estimates are affected. Comparison of the portfolio losses against risk estimates allows us to evaluate the reliability of the broadly adopted model.</p>

<p>Compare the Normal Inverse Gaussian market model against empirical financial market data, and price exotic options using the numerical path integration approach.</p>

<p>The governing equations for waves propagating in water are derived by use of conservation laws. The equations are then cast onto dimensionless form and two important parameters are obtained. Approximations by use of asymptotic expansions in one or both of the parameters are then applied on the governing equations and we show that several different completely integrable equations, with different scaling transformations and at different order of approximations, can be derived. More precisely, the Korteweg-de Vries, Kadomtsev-Petviashvili and Boussinesq are obtained at first order, while the Camassa-Holm, Degasperis-Procesi, nonlinear Schrödinger and the Davey-Stewartson equations are obtained at second order. We discuss shortly some of the properties for each of the obtained equations.</p>

<p>In this text, I will look at some new approaches that may shed some light on the Kadison Singer problem, mainly one instigated by Vern Paulsen using dynamical systems in the Stone-Cech compactification of a discrete group. In order to do this, I will try to develop the theory in a crossed product setting, and look at some aspects of it that may hold interest of their own.</p>