<p>A method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs. The central point is a Neumann-Dirichlet equivalence that allows for evaluation of the output through the bilinear form of the weak formulation of the PDE. Through a comprehensive example related to electrostatics, we consider multiple outputs, a posteriori error estimators and empirical interpolation treatment of the non-affine terms in the bilinear form. Together with the considered Neumann-Dirichlet equivalence, these methods allow for efficient and accurate numerical evaluation of a relationship mu->s(mu), where mu is a parameter vector that determines the geometry of the physical domain and s(mu) is the corresponding flux-type output matrix of interest. As a practical application, we lastly employ the rapid evaluation of s-> s(mu) in solving an inverse (parameter-estimation) problem.</p>

<p>The thesis describes the Average Exceedance Rate (AER) method, which is a method for predicting return levels from sampled time series. The AER method is an alternative to the Peaks over threshold (POT) method, which is based on the assumption that data exceeding a certain threshold will behave asymptotically. The AER methods avoids this assumption by using sub-asymptotic data instead. Also, instead of using declustering to obtain independent data, correlation among the data is dealt with by assuming a Markov-like property. A practical procedure for using the AER method is proposed and tested on two sets of real data. These are a set of wind speed data from Norway and a set of wave height data from the Norwegian continental shelf. From the results, the method appears to give satisfactory results for the wind speed data, but for the wave height data its use appears to be invalid. However, the method itself seems to be robust, and to have certain advantages when compared to the POT method.</p>

<p>This thesis is a study on the effects of varying surface tension along an interface separating two fluids. Varying surface tension leads to tangential forces along the interface. This is often called the Marangoni effect. These forces are discussed in detail, and two test cases are considered to analyse the Marangoni effect, and to verify the present implementation. The first test studies steady-state two-phase flow where the fluids are separated with plane interfaces and the flow is driven by a linear surface-tension gradient. The second case is an analysis of the initial forces on a two-dimensional droplet due to a linear surface-tension gradient. The tests indicate that the present implementation is capable of simulating two-phase flow with a given surface-tension gradient. The underlying model is a two-phase flow model for Newtonian fluids with constant viscosity and density. The two-phase model is based on the Navier-Stokes equations coupled with a singular surface force, which together with the difference in fluid properties induces discontinuities across the interface. The Navier-Stokes equations are solved using a projection method, and a combination of the level-set method for capturing the interface and the ghost-fluid method (GFM) for handling the interface discontinuities. The thesis also discusses the effect of surfactants on an interface. The presence of surfactants reduces the local surface tension, and a non-uniform surfactant distribution results in varying surface tension and the Marangoni effect. A surfactant model is reviewed, where an equation of state couples the surface tension to the surfactant concentration and a transport equation is used to solve the surfactant mass-conservation.</p>

<p>Numerical solution of heat transfer and fluid flow problems in two spatial dimensions is studied. An arbitrary Lagrangian-Eulerian (ALE) formulation of the governing equations is applied to handle time-dependent geometries. A Legendre spectral method is used for the spatial discretization, and the temporal discretization is done with a semi-implicit multi-step method. The Stefan problem, a convection-diffusion boundary value problem modeling phase transition, makes for some interesting model problems. One problem is solved numerically to obtain first, second and third order convergence in time, and another numerical example is used to illustrate the difficulties that may arise with distribution of computational grid points in moving boundary problems. Strategies to maintain a favorable grid configuration for some particular geometries are presented. The Navier-Stokes equations are more complex and introduce new challenges not encountered in the convection-diffusion problems. They are studied in detail by considering different simplifications. Some numerical examples in static domains are presented to verify exponential convergence in space and second order convergence in time. A preconditioning technique for the unsteady Stokes problem with Dirichlet boundary conditions is presented and tested numerically. Free surface conditions are then introduced and studied numerically in a model of a droplet. The fluid is modeled first as Stokes flow, then Navier-Stokes flow, and the difference in the models is clearly visible in the numerical results. Finally, an interesting problem with non-constant surface tension is studied numerically.</p>

<p>In the thesis, a statistical simulation-based framework is presented that is intended for making sample size and power considerations prior to case-control association studies. It reviews biological background and biallelic single- and multiple-SNP disease models, with a focus on single-SNP models. Odds ratios, multiple testing, sample size, statistical power and the genomeSIM package are also reviewed. The framework is tested with the MAX stat method on a dominant disease model, demonstrating that it can be used for assessing whether different sample sizes are sufficient for detecting a causal SNP.</p>

<p>This thesis presents and considers several different methods for computation of availability and production availability for production system. It is assumed that the system handles flow of a fluid. The thesis presents two software programs for computation of reliability measures, MIRIAM Regina and Relex Reliability Studio, and several analytical methods, among them one especially adapted to computation of production availability. For the methods not able to compute production availability, a method is presented which makes it possible to estimate production availability from computation of availability. Among the methods, Relex and three of the analytical computation methods are made to compute availability of the system. The analytical methods considered are standard availability computation based on the structure function of the system and the definitions of availability and computation based on renewal and quasi renewal processes. Relex makes it possible to compute availability both by simulation and, if the system is simple enough, by analytical methods. The usefulness of the analytical methods is to an extent limited by the assumptions laid on the system. Relex makes it possible to take into account more features one would expect to have in a real life system, but for analytical methods to be employed in the computations, the system must be quite simple. Two methods especially made for computing production availability are presented. These are the software program MIRIAM Regina, which combines a sophisticated flow algorithm with Monte Carlo simulation, and a method based on using Markov chains for computing the probability distribution for flow through subsystems of the system under consideration, and then employing simple merging rules to compute the flow through the entire system. These methods are both very flexible, and makes it possible to take into account many different aspects of a real life system. The most important source of uncertainty in the results form a computation, lies in the relation between the real life system and the model system computations are made on. A model system will always be significantly simplified. When choosing a computation method and interpreting results, it is important to keep in mind all assumptions made regarding the system, both explicitly when making the model, and implicit in the computation method. Another source of uncertainty is uncertainty in the input data. A method for propagation of uncertainty through computations is presented and employed on some of the methods. For simulation, one will in addition have the uncertainty due to simulation being an way of making a statistical sample. The size of the sample, given by the number of simulation iterations done, will decide the accuracy of the result.</p>

<p>In this study we consider 2D seismic lithology/fluid inversion constrained by rock physics depth trends and a prior lithology/fluid Markov random field. A stochastic relation from porosity and lithology/fluid to seismic observations is established. The inversion is done in a Bayesian framework with an approximate posterior distribution. Block Gibbs samplers are used to estimate the approximate posterior distribution. Two different inversion algorithms are established, one with the support of well observations and one without. Both inversion algorithms are tested on a synthetic reservoir and the algorithm with well observations is also tested on a data set from the North Sea. The classification results with both algorithms are good. Without the support of well observations it is problematic to estimate the level of the porosity trends, however the classification results are approximately translation invariant with respect to porosity trends.</p>

<p>This is a study of the risk/return characteristics of large equity portfolios, consisting of different option contracts. In times when there is nervousness present in the financial market, and the future prospects of the market are highly uncertain, the importance of appropriate mathematical models is emphasized by traders. Tools to implement option strategies in different markets will be investigated, in order to minimize the risk and maximize the return of the portfolios. Using various options and statistical simulation methods, the risk/return characteristics of options will be studied. Monte Carlo simulation is used, in order to obtain a thorough understanding of the risk/return characteristics of various option strategies, applied on different indices. As the strategies investigated are meant to be applied by traders, the daily changes in portfolio value are the basis from which the risk of the strategies is estimated. There are various methods of estimating the risk available. In order to examine the risk characteristics, the probability of extreme outcomes is of interest. The Value-at-Risk and the expected shortfall are to be analyzed, when examining the risk of the strategies evaluated. Analyzing the properties of the expected return of the portfolios, the tails of the densities of the portfolio values, for the different strategies and indices, will be of interest.</p>

<p>This study will focus on the pricing of interest rate derivatives within the framework of the LIBOR Market Model. First we introduce the mathematical and financial foundations behind the basic theory. Then we give a rather rigouros introduction to the LIBOR Market Model and show how to calibrate the model to a real data set. We use the model to price a basic swaption contract before we choose to concentrate on a more exotic Bermudan swaption. We use the Least Squares Monte Carlo (LSM) algorithm to handle the early exercise features of the Bermuda swaption. All major results are vizualised and the C++ implementation code is enclosed in appendix B.</p>

<p>We prove that if two hyperfunctions on the unit circle have disjoint support, then the convolution of their Fourier coefficients multiplied with a weight is zero when the weight goes to 1. We prove this by using the Fourier-Borel transform and the G-transform of analytic functionals. The proof is inspired by an article by Yngve Domar. In the end of his article he proves the existence of a translation-invariant subspace of a certain weighted l^p-space. This proof has similarities to our proof, so we compare them. We also look at other topics related to Domar's article, for example the existence of entire functions of order less than or equal to 1 under certain restrictions on the axes. We will see how the Beurling-Malliavin theorem gives some answers to this question. Finally, we prove that if two hyperfunctions on the real line have compact and disjoint support, then the convolution of their Fourier transforms multiplied with a weight is zero when the weight goes to 1.</p>