In recent years, stochastic partial differential equations (SPDEs) have been shown to provide a usefulway of specifying some classes of Gaussian random fields. The use of an SPDEallows for the construction of a Gaussian Markov random field (GMRF) approximation, which has verygood computational properties, of the solution.In this thesis this kind of construction is considered for a specificspatial SPDE with non-constant coefficients, a form of diffusion equation driven by Gaussian white noise. The GMRF approximation is derived from the SPDE by a finite volume method. The diffusion matrixin the SPDE provides a way of controlling the covariancestructure of the resulting GMRF.By using different diffusion matrices, itis possible to construct simple homogeneous isotropic and anisotropic fields and more interesting inhomogeneous fields. Moreover, it is possible to introduce random parametersin the coefficients of the SPDE and consider the parametersto be part of a hierarchical model. In this way onecan devise a Bayesian inference scheme for theestimation of the parameters. In this thesis twodifferent parametrizations of the diffusion matrixand corresponding parameter estimations are considered.The results show that the use of an SPDE with non-constant coefficients provides a useful way of creating inhomogeneousspatial GMRFs.

The heterogeneous multiscale method (HMM) was proposed by E and Engquist and is considered to be an efficient method for problems with multiple time scales. We give a short introduction to the HMM for multiscale problems in general, before we restrict our work to HMM schemes developed for stiff ODEs, based on results found by Engquist et al. HMM provides an efficient and systematic way to move between the macroscopic and microscopic model in a problem having multiscale physics. By taking advantage of scale separation in multiscale problems, the HMM approximates the macroscopic variables of the solution without fully resolving the microscopic solution. This introduces computational savings as the total number of evaluations needed for convergence are significantly reduced.We test the features of the HMM on the spinning top. The governing equations of the top produces a highly oscillatory solution as the top spins fast. Despite this fast oscillating nature, we would intuitively expect some slow behavior of the top, for instance the inclination from the vertical axis or the circulation of the top around the vertical axis. We find a set of slow variables of the spinning top, and show that the HMM provides an accurate solution of the macroscopic variables of the top, with a significant gain in computational cost compared to standard solvers.We also study the spinning top subjected to a vibrational external force and find a set of slow variables, which can be approximated accurately with HMM. Finally, we find an averaged equation to the spinning top subjected to a vertical vibrating force. This analysis is based on the Modulated Fourier expansion.The work of this thesis is an extension of my semester project, and we emphasize that the theory part of this thesis is partially from this work.

In this project we have first and foremost been comparing the performance of the ACER method with the POT method in the prediction of extreme values from the heavy tailed distributions; especially for data from the energy markets. The energy market is an exciting dynamic market where small singularities can make large differences in the price. Therefore it is very important and challenging to analyse and make predictions in this market. We have also analysed a dataset which is not from the energy market, to compare and see the main differences between the two markets. We have also taken in consideration of removing the return value for the dates of maturity to see whether this will have any influence on the results.The main concept of the POT method is to find a threshold, $u$, and let the excesses be distributed by the Generalised Pareto Distribution. Whilst for the ACER method, we assume a specific shape of the tail, which in this project was of the kind Fréchet. We have done this analysis for five different data sets where two of them have been considered with and without their expiration dates. We have also filtrated the data sets with an AR-GARCH filter, and then used the POT and ACER on the residuals from the process. We have found out that both methods are not greatly influenced by the filtration, but we see the tendency of the POT method predicting a heavier tail than the ACER method. Further on, we can say that there are no significant large effects of removing the return values for the dates of maturity. Lastly, the data sets from the energy market prove themselves much more heavy tailed than for the data set from Norsk Hydro.

We study persistent homology, methods in discrete differential geometry and discrete Morse theory. Persistent homology is applied to computational biology and range image analysis. Theory from differential geometry is used to define curvature estimates of triangulated hypersurfaces. In particular, a well-known method for triangulated surfacesis generalised to hypersurfaces of any dimension. The thesis concludesby discussing a discrete analogue of Morse theory.

We have constructed four different types of designs for computer experiments. Thedesign types are based on latin hypercube sampling (LHS), orthogonal arrays (OA), ran-dom sampling and the recently proposed multi-level binary replacement (MBR) design.For each type of design we have attempted to find the best possible design out of acertain number of constructed designs using three different optimizing criteria: the alias sum of square criterion (ASSC), the L-criterion and a modified A-criterion. The chosen design has then been tested by fitting an approximate model and calculating maximum error (MAX) and root mean squared error (RMSE) values. We observed that out of the three criteria applied the ASSC performed the best.In addition to comparing criteria for optimizing the design choice, we have alsoconstructed non-optimized designs for comparing the different design types and thedifferent ways of constructing MBR designs. In this setting we observed that OA designsperformed well in general, whereas the MBR designs performed well when restricted toa small number of factors.

A general introduction to conformal maps and the Riemann mapping theorem is given. Three methods for numerically approximating conformal maps from arbitrary domains to the unit disc are presented: the Schwarz-Christoffel method, the geodesic algorithm and the circle packing method. Basic implementations of the geodesic algorithm and the circle packing method were made, and program code is presented. Applications of these numerical methods to problems in physics and mathematical research are briefly discussed.

In reservoir simulations, one of the biggest challenges is solving large modelswith complex geological properties. Because reservoirs can be several kilome-ters long, and still be geologically inhomogeneous over centimeters, the com-putational power required to solve a full set of mass balance equations can beimmense. Several methods for overcoming this challenge has been proposed,including various upscaling and multiscale methods.One of these approaches is the Multiscale Finite Volume (MsFV) method, whichaims to create a set of basis functions for the pressure which can be computedin parallel and reused for different boundary conditions. This thesis aims togive a thorough study of the MsFV-method itself, before extending it to threedimensional, unstructured grids. An implementation was done as a modulefor the MATLAB Reservoir Simulation Toolbox developed by SINTEF AppliedMathematics. A new variant of the method designed to overcome some of thecomputational challenges arising from an extension to 3D was also formulated.The implementation was then applied to both synthetic and realistic gridsand permeabilities, and compared against a full two point flux approximation(TPFA) solver.

In Genome-Wide Association Studies (GWAS) the aim is to look for associationbetween genetic markers and phenotype (disease). For each genetic marker weperform an hypothesis test. Since the number of markers is high (in the order of hundred thousands), we use multiple hypothesis tests. One popular strategy in multippel testing is to estimate an effective number of independent tests, and then use methods based on independent tests to control the total type I error. The focus of this thesis has been to study different methods for estimating the effective number of independent tests. The methods are applied to a large data set on bipolar disorder and schizophrenia in Norwegian individuals from the TOP study at the University of Oslo and Oslo University Hospital (OUS). A key featureof these methods is the correlation between the genetic markers. The methodsconsidered in this thesis are based on either haplotype or genotype correlation andone focus of this thesis has been to study the difference between haplotype andgenotype correlation.

The objective of this study is to classify lithology/fluid(LF) variables along depth profiles. The classification is done by a Bayesian inversion method to obtain the posterior probability density functions(PDFs) for the LF classes at every depth, given data in form of petrophysical variables or elastic properties. In this way we determine the most probable lithology/fluid profile. A stationary Markov chain prior model will be used to model the continuity of the LF classes a priori. The likelihood relates the LF classes to data. A statistical rock-physics forward model is used to relate the petrophysical variables to elastic attributes. This will be done for synthetic test data inspired by a North Sea sandstone reservoir and for real test data in form of a well log from the North Sea. Data for the synthetic case is either the petrophysical variables or the elastic properties. For the real data is only the elastic properties considered.

This thesis aims to implement time integrators in the FEniCS framework. More specifically, the thesis focuses on selecting suitable time integrators, implement these and verify that the implementation works by applying them to various relevant test problems. This work resulted in a module for FEniCS, named Gryphon. The thesis is divided into four parts.The first part builds a theoretical framework which will motivate why singly diagonally implicit Runge-Kutta methods with an explicit first stage (ESDIRKs) should be considered for solving stiff ordinary differential equations (ODEs). It will also be shown how an ESDIRK method can be utilized to solve time dependent partial differential equations (PDEs) by solving the semidiscretized system arising from first applying a finite element method. We will restrict our attention to PDEs which either give rise to a pure ODE system or a DAE (differential-algebraic equation) system of index 1.The second part discusses the implementation of Gryphon, focusing on why such a module is useful and how the source code is structured.The third part is devoted to numerical experiments on the ESDIRK solvers implemented in Gryphon. The experiments will establish convergence and give some run-time statistics for various ESDIRK schemes. We will also see that L-stability is a favorable trait when working with stiff equations, by comparing an ESDIRK method to the trapezoidal rule. It will also be verified that the step size selectors implemented in Gryphon behaves as expected. As test problems we consider the heat equation, the Fisher-Kolmogorov equation, the Gray-Scott equations, the Fitzhugh-Nagumo equations and the Cahn-Hilliard equations.The fourth part is a user manual for Gryphon. All the parameters which can be changed by the user are explained. The manual also includes example code for solving the heat equation, the Gray-Scott equations and the Cahn-Hilliard equation, to get the reader starting on solving their own problems.