A competing risks situation arises when a unit can fail due to several distinct failure types. In a competing risk situation, standard techniques from survival analysis may lead to incorrect and biased results. In this thesis, the theory of competing risks is used to identify possible risk factors for breast, uterine and ovarian cancer. This has been done by regression on the cause specific hazard functions, the subdistribution hazard functions and two approximate methods. Cox regression is used for a complete analysis of the medical data.By following 61457 women over approximately 50 years, it has been observed 3407 cases of breast cancer, 934 of uterine cancer and 843 of ovarian cancer. Summarized, it has been found that several births decrease the risk of breast, uterine and ovarian cancer. Obesity is associated with increasing risk of ovarian cancer for postmenopausal women, but not premenopausal. A long reproductive period (early menarche and/or late menopause) and high BMI increases the risk of breast and uterine cancer. Late first and last birth decreases the risk of uterine cancer, while it increases the risk of breast cancer. The data used in the analysis is selected from a screening program organized by the Norwegian Cancer Society for early diagnosis of breast cancer. postmenopausale women, but not premenopausale. A long reproductive period (early menarche and/or late menopause) and high BMI increases the risk of breast and uterine cancer. Late first and last birth decreases the risk of uterine cancer, while it increases the risk of breast cancer. The data used in the analysis is selected from a screening program organized by the Norwegian Cancer Society for early diagnosis of breast cancer.

This thesis is concerned with 4D AVA seismic inversion problems. By comparing two seismic surveys done over the same area, but at different times, one hopes to discover untapped pockets of oil or gas. Using the full likelihood to analyse 4D AVA seismic data is impossible in practice due to memory and computational restrictions. The goal of the thesis is to find a useful framework for parameter estimation and predictions for 4D AVA seismic data, and the composite likelihood is introduced as a possible solution. The composite likelihood method takes in pairs of data points and sums over them instead of taking in all the data as is the case for the full likelihood. This makes calculations fast while avoiding matrix operations on large matrices.The composite likelihood method is tested on a data set from the Norne field for parameter estimations and predictions. Eight variations of the model are tested, the variations being the exponential or Matern correlation function, one or two data columns used as a data point in the composite likelihood, and a simple or wavelet convoluted noise term. The composite likelihood method is shown to perform well; it is fast and the estimates found agree well with previous experience. Comparison of the different models indicate that the choice of correlation function has little effect on the results, that the noise term should be kept simple, and that it is sufficient to use one data column.

Flexible data regression is an important tool for capturing complicated trends in data. One approach is penalized smoothing splines, where there are several mainstream methods. A weakness is, however, the quantification of uncertainty. We will in thesis present two mainstream smoothing spline methods, P-splines and O'Sullivan splines, and the RW2 model; a Bayesian hierarchical model based on a latent field. The Bayesian priors are specified by a stochastic Poisson equation, and spline estimates are approximated along a finite element Galerkin approach. We evaluate the three methods using integrated nested Laplace approximations (INLA) for a full Bayesian analysis supplying credible bands. The methods give fairly similar results and we investigate the theoretical motivates behind the methods. As an extension of the Bayesian models, the smoothing parameter is incorporated in latent field. This gives an adaptive smoothing method, which better estimates jumps and quick curvature changes. Further, the close relationship between O'Sullivan splines smoothing splines is discussed, revealing O'Sullivan splines to be a finite element Petrov-Galerkin approximation of smoothing splines. The main results are the possibility of credible bands, the extension to adaptive smoothing and the finite element understanding of O'Sullivan splines.

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.