<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>

<p>In this thesis an open-source software for permeability upscaling is developed. The software is based on DUNE, an open-source C++ framework for finding numerical solutions of partial differential equations (PDEs). It provides functionality used in finite elements, finite volumes and finite differences methods. Permeability is a measure of the ability of a material to transmit fluids, and determines the flow characteristics in reservoir models. Permeability upscaling is a technique to include fine-scale variations of the permeability field in a coarse-scale reservoir model. The upscaling technique used in this thesis involves solving an elliptic partial differential equation. This is solved with mixed and hybrid finite element methods. The mixed method transforms the original second order PDE into a system of two linear equations. The great advantage with these methods compared with standard finite element methods is continuity of the variable of interest in the upscaling problem. The hybrid method was introduced for being able to solve larger problems. The resulting system of equations from the hybrid method can be transformed into a symmetric positive definite system, which again can be solved with efficient iterative methods. Efficiency of the implementation is important, and as for most implementations of PDE solvers, the computational time is dominated by solving a system of linear equations. In this implementation it is used an algebraic multigrid (AMG) preconditioner provided with DUNE. This is known to be efficient on system arising from elliptic PDEs. The efficiency of the AMG preconditioner is compared with other alternatives, and is superior to the others. On the largest problem investigated, the AMG based solver is almost three times faster than the next best alternative. The performance of the implementation based on DUNE is also compared with an existing implementation by Sintef. Sintef's implementation is based on a mimetic finite difference method, but on the grid type investigated in this thesis, the methods are equivalent. Sintef's implementation uses the proprietary SAMG solver developed by Fraunhofer SCAI to solve the linear system of equations. SAMG is 58% faster than DUNE's solver on a test case consisting of 322 200 unknowns. The scalability of SAMG seem to be better than DUNE's AMG as the problem size increases. However, a great advantage with DUNE's solver is 50% lower memory usage measured on a problem consisting of approx. 3 million unknowns. Another advantage is the licensing of the software. Both DUNE and the upscaling software developed in this thesis is GPL licensed which means that anyone is free to improve or adjust the software.</p>

<p>We discuss nonholonomic systems in general and numerical methods for solving them. Two different approaches for obtaining numerical methods are considered; discretization of the Lagrange-d'Alembert equations on the one hand, and using the discrete Lagrange-d'Alembert principle to obtain nonholonomic integrators on the other. Among methods using the first approach, we focus on the super partitioned additive Runge-Kutta (SPARK) methods. Among nonholonomic integrators, we focus on a reversible second order method by McLachlan and Perlmutter. Through several numerical experiments the methods we present are compared by considering error-growth, conservation of energy, geometric properties of the solution and how well the constraints are satisfied. Of special interest is the comparison of the 2-stage SPARK Lobatto IIIA-B method and the nonholonomic integrator by McLachlan and Perlmutter, which both are reversible and of second order. We observe a clear connection between energy-conservation and the geometric properties of the numerical solution. To preserve energy in long-time integrations is seen to be important in order to get solutions with the correct qualitative properties. Our results indicate that the nonholonomic integrator by McLachlan and Perlmutter sometimes conserves energy better than the 2-stage SPARK Lobatto IIIA-B method. In a recent work by Jay, however, the same two methods are compared and are found to conserve energy equally well in long-time integrations.</p>

<p>Numerical solution of buoyancy-driven flow problems in two spatial dimensions is presented. A high-order spectral method is applied for the spatial discretization, while the temporal discretization is done by operator splitting methods. By solving the convection-diffusion equation, which governs the temperature distribution, a thorough description of both the spatial and the temporal discretization methods is given. A fast direct solver for the arising system of algebraic equations is presented, and the expected convergence rates of both the spatial and the temporal discretizations are verified. As a step towards the Navier--Stokes equations, a solution of the Stokes problem is given, where a splitting scheme technique is introduced. An extension of this framework is used to solve the incompressible Navier--Stokes equations, which govern the fluid flow. By solving the Navier-Stokes equations and the convection-diffusion equation as a coupled system, two different buoyancy-driven flow problems in two-dimensional enclosures are studied numerically. In the first problem, emphasis is put on the arising fluid flow and the corresponding thermal distribution, while the second problem mainly consists of determining critical parameters for the onset of convection rolls.</p>

<p>The main task of this assignment is to filter out noise from a series of radar images and to carry out short term precipitation forecasts. It is important that the final routine is performed online, yielding new forecasts as radar images arrive with time. The data available is a time series arriving at a one hour ratio, from the Rissa radar located in Sør Trøndelag. Gaussian radial basis functions are introduced to create the precipitation field, whose movement is solely governed by its velocity field, called advection. By performing discretization forward in time, from the basis given by the differential advection equation, prior distributions can be obtained for both basis functions and advection. Assuming normal distributed radar errors, the basis functions and advection are conditioned on associating radar images, which in turn can be taken into the prior distributions, yielding new forecasts. A modification to the model, labeling the basis functions either active or inactive, enable the process of birth and death of new rain showers. The preferred filtering technique is a joint MCMC sampler, but we make some approximations, sampling from a single MCMC sampler, to successfully implement an online routine. The model yield good results on synthetic data. In the real data situation the filtered images are satisfying, and the forecast images are approximately predicting the forthcoming precipitation. The model removes statistical noise efficiently and obtain satisfying predictions. However, due to the approximation in the MCMC algorithm used, the variance is somewhat underestimated. With some further work with the MCMC update scheme, and given a higher frequency of incoming data, it is the authors belief that the model can be a very useful tool in short term precipitation forecasting. Using gauge data to estimate the radar errors, and merging online gauge data with incoming radar images using block-Kriging, will further improve the estimates.</p>

<p>For small error the isogeometric analysis is more efficient than the finite element method. The condition number is lower in the isogeometric analysis than the finite element method. The isogeometric analysis produce general and robust implementation methods. The isogeometric analysis basis has higher continuity than the finite element method basis, and is more suitable to represent different geometries.</p>