This report features the isogeometric finite element method applied to the elastodynamic problem in a brittle medium with a potential for cracking. Griffith's theory for fracturing is used. The development of the model is outlined, complete with the Euler-Lagrange equations. The cracking is described with a phase field supplemented with a history field, contrary to the usual way of building the crack directly into the geometry by modification of the basis, facilitating the use of isogeometric analysis even with simplistic basis functions such as Non-Uniform Rational B-Splines (NURBS). The introduction of the crack-phase field results in non-linearity in the coupled problem. The problem is semi-discretized, upon which the spatial sub-problem is treated with isogeometric analysis. The numerical time-stepping solution routine is built around the Newton-Raphson method, but includes both pre-conditioning and correctors and is known as the predictor/multi-corrector time integration scheme. The Jacobian of the semi-discretized system (needed for the Newton-Raphson iteration) is developed analytically. In addition to the numerical tests with NURBS as our basis, we will also test the method with Locally Refined B-splines (LR B-Splines), ensuring better resolution along the crack path. The LR B-spline represents an alternative to the more commonly used T-Spline.

The topic of this thesis is GPU accelerated sparse linear algebra for subsurface reservoir modeling. Numerical techniques for reservoir sim- ulations are described and we present the open source reservoir simula- tion software toolbox MRST. We discuss some of the challenges related to linear algebra and reservoir simulation. Furthermore, we discuss the possibility GPU-acceleraing the linear algebra for reservoir simulation, and implement a GPU based CG solver preconditioned with AMG for MRST, using the open source linear algebra library CUSP.

An artin algebra $l$ over a commutative, local, artinian ring $R$ was fixed, and with this foundation some topics from representation theory were discussed. A series of functors of module categories were defined, and almost split sequences were introduced with some basic results. An isomorphism $omega_{delta,X} : D delta^* rightarrow delta_*(DTr(X))$ of $Gamma$-modules for an artin $R$-algebra $Gamma$ was constructed. The isomorphism $omega_{delta,X}$ was applied to a special case, yielding a deterministic algorithm for computing almost split sequences in the case that $R$ is a field.

We study the Cauchy problem for the 1-D fractal Burgers equation, which is a non-linear and non-local scalar conservation law used to for instance model overdriven detonation in gases. Properties of classical solutions of this problem are studied using techniques mainly developed for the study of entropy solutions. With this approach we prove several a-priori estimates, using techniques such as Kruzkow doubling of variables. The main theoretical result of this study is a L1-type contraction estimate, where we show the contraction in time of the positive part of solutions of the fractal Burgers equation. This result is used to show several other a priori estimates, as well as the uniqueness and regularity in time of solutions. We also solve our Cauchy problem numerically, by proposing, analyzing and implementing one explicit and one implicit-explicit method, both based on finite volume methods. The methods are proved to be monotone, consistent and conservative under suitable CFL conditions. Subsequently, several a priori estimates for the numerical solutions are established. A discussion on how the numerical methods may be implemented efficiently, as well as discussions of some of the numerical results obtained conclude this study.

Mapper is an algorithm for describing high-dimensional datasets in terms of simple geometric objects. We give a new definition of Mapper, with which we are able to prove that Mapper is a functor and that Mapper is a homotopy equivalence for certain "nice" input data. To establish these results we describe the statistical theory of functorial clustering and the topological machinery of homotopy colimits. At the end of the document we show, by means of numerical experiments, that the functoriality of Mapper is useful in applications.

This thesis is dedicated to the study of Barkley's equation, a stiff diffusion-reaction equation describing waves in excitable media. Several numerical solution methods will be derived and studied, range from the simple explicit Euler method to more complex integrating factor schemes. A C++ application with guided user interface created for performing several of the numerical experiments in this thesis will also be described.

The purpose of modeling a petroleum reservoir consists of finding the underlying reservoir properties based on production data, seismic and other available data. In recent years, progress in technology has made it possible to extract large amount of data from the reservoir frequently. Hence, mathematical models that can rapidly characterize the reservoir as new data become available gained much interest. In this thesis we present a formulation of the first order Hidden Markov Model (HMM) that fits into the description of a reservoir model under production. We use a recursive technique that gives the theoretical solution to the reservoir characterization problem. Further, we introduce the Kalman Filter which serves as the exact solution when certain assumptions about the HMM are made. However, these assumptions are not valid when describing the process of a reservoir under production. Thus, we introduce the Ensemble Kalman Filter (EnKF) which has been shown to give an approximate solution to the reservoir characterization problem. However, the EnKF is depending on multiple realizations from the reservoir model which we obtain from the reservoir production simulator Eclipse. When the number of realizations are kept small for computational purposes, the EnKF has been shown to possibly give unreliable results. Hence, we apply a shrinkage regression technique (DR-EnKF) and a localization technique (Loc-EnKF) that are able to correct the traditional EnKF. Both the traditional EnKF and these corrections are tested on a synthetic reservoir case called the Brugge Field. The results indicate that the traditional EnKF suffers from ensemble collapse when the ensemble size is small. This results in small and unreliable prediction uncertainty in the model variables. The DR-EnKF improves the EnKF in terms of root mean squared error (RMSE) for a small ensemble size, while the Loc-EnKF makes considerable improvements compared to the EnKF and produces model variables that seems reasonable.

A framework for a 10-day ahead probabilistic forecast based on a deterministic model is proposed. The framework is demonstrated on the system price of the Nord Pool electricity market. The framework consists of a two-component mixture model for the error terms (ET) generated by the deterministic model. The components assume the dynamics of “balanced” or “unbalanced” ET respectively. The label of the ET originates from a classification of prices according to their relative difference for consecutive hours. The balanced ET are modeled by a seemingly unrelated model (SUR). For the unbalanced ET we only outline a model. The SUR generates a 240-dimensional Gaussian distribution for the balanced ET. The resulting probabilistic forecast is evaluated by four point-evaluation methods, the Talagrand diagram and the energy score. The probabilistic forecast outperforms the deterministic model both by the standards of point and probabilistic evaluation. The evaluations were performed at four intervals in 2008 consisting of 20 days each. The Talagrand diagram diagnoses the forecasts as under-dispersed and biased. The energy score finds the optimal length of training period and set of explanatory variables of the SUR model to change with time. The proposed framework demonstrates the possibility of constructing a probabilistic forecast based on a deterministic model and that such forecasts can be evaluated in a probabilistic setting. This shows that the implementation and evaluation of probabilistic forecasts as a scenario generating tools in stochastic optimization are possible.

This master’s thesis investigates how well a generalized mixed model fits different dominance data sets. The data sets mainly represent disputes between individuals in a closed group, and the model to be used is an adjusted, intransitive extension of the Bradley-Terry model. Two approaches of model fitting are applied; a frequentist and a Bayesian one. The model is fitted to the data sets both with and without random effects (RE) added. The thesis investigates the relationship between the use of random effects and the accuracy, significance and reliability of the regression coefficients and whether or not the random effects affect the statistical significance of a term modelling intransitivity. The results of the analysis in general suggest that models including random effects better explain the data than models without REs. In general, regression coefficients that appear to be significant in the model excluding REs, seem to remain significant when REs are taken into account. However the underlying variance of the regression coefficients have a clear tendency to increase as REs are included, indicating that the estimates obtained may be less reliable than what is obtained otherwise. Further, data sets fitting to transitive models without REs taken into account also, in general, seem to remain transitive when REs are taken into account.

Sequential value information for Markov random field.