This work is concerned with the numerical solution of Friedrichs systems by discontinuous Galerkin finite element methods (DGFEMs). Friedrichs systems are boundary value problems with symmetric, positive, linear first-order partial differential operators and allow the unified treatment of a wide range of elliptic, parabolic, hyperbolic and mixed-type equations. We do not assume that the exact solution of a Friedrichs system belongs to a Sobolev space, but only require that it is contained in the associated graph space, which amounts to differentiability in the characteristic direction. We show that the numerical approximations to the solution of a Friedrichs system by the DGFEM converge in the energy norm under hierarchical h- and p- refinement. We introduce a new compatibility condition for the boundary data, from which we can deduce, for instance, the validity of the integration-by-parts formula. Consequently, we can admit domains with corners and allow changes of the inertial type of the boundary, which corresponds in special cases to the componentwise transition from in- to outflow boundaries. To establish the convergence result we consider in equal parts the theory of graph spaces, Friedrichs systems and DGFEMs. Based on the density of smooth functions in graph spaces over Lipschitz domains, we study trace and extension operators and also investigate the eigensystem associated with the differential operator. We pay particular attention to regularity properties of the traces, that limit the applicability of energy integral methods, which are the theoretical underpinning of Friedrichs systems. We provide a general framework for Friedrichs systems which incorporates a wide range of singular boundary conditions. Assuming the aforementioned compatibility condition we deduce well-posedness of admissible Friedrichs systems and the stability of the DGFEM. In a separate study we prove hp-optimality of least-squares stabilised DGFEMs.

Quantification of flood risk and flood inundation requires accurate numerical simulations, both in terms of the mathematical theory that underpins the methods used and the manner in which the meteorological phenomena that cause flooding are coupled to such systems. Through our research, we have demonstrated how rainfall and infiltration effects can be incorporated into existing flood models in a rigorous and mathematically consistent manner; this approach departs from preceding methods, which neglect terms representing such phenomena in the conservation or balancing of momentum. We demonstrate how the omission of these terms means the solution derived from such models cannot a priori be assumed to be the correct one, which is in contrast to solutions from the extended system we have developed which respect the energetic consistency of the problem. The second issue we address is determining how we can model these meteorological phenomena that lead to flooding, with a specific interest in how existing observation data from rain gauges can be incorporated into our modelling approach. To capture the random nature of the precipitation, we use stochastic processes to model the complex meteorological interactions, and demonstrate how an accurate representation of the precipitation can be built. Given the specific industrial applications we have mind in regards to flood modelling and prediction, there will be a high computational cost associated with any such simulations, and so we consider techniques which can be used to reduce the computational cost whilst maintaining the accuracy of our solutions. Having such an accurate flood model, coupled with a stochastic weather model designed for efficient computational modelling, will enable us to make useful predictions on how future climate change and weather patterns will impact flood risk and flood damage.

In this thesis, I study the performance behaviour of hedge funds and mutual funds. I study a basket of various risk statistics that are widely used to measure the fluctuation of asset prices. Those risk statistics are used to rank the performance of the assets. The linear dependence relation of these risk measures in ranking assets is investigated and the set of risk measures is reduced by excluding risk measures that produce linearly dependent ranking vectors to other risk measures. The ranks within each of the selected remaining risk statistics are standardised and then linearly transformed into a new set of linearly independent factors where principal component analysis is carried out as a variable reduction technique to remove the noise while preserve the main variation of the original data. The transformed factors are sorted in descending order according to their contribution to the variation of the original data. The factor loadings of the first two principal components PC1 and PC2 are reviewed and interpreted as styles (PC1 as consistency and PC2 as aggression). The universe of a set of hedge funds is classified according to these styles as BL=(low consistency, low aggression), BR=(high consistency, low aggression), TL=(low consistency, high aggression) and TR=(high consistency, high aggression). I examine the performance behaviour of the four different classified classes whereby this classification method provides an indication on returns and management styles of hedge funds. A three-factor prediction model for asset returns is introduced by regressing 12 weeks' forward rank of return on the historical ranks of risk statistics. The first few principal components, which explain the main variation of information captured by risk statistics, are used in the prediction model. The robustness of the model is tested by applying the model to the following 12-week period using the set of independent factors. An investment strategy is constructed based on the prediction model using the set of independent factors. I discover high evidence of predictability and I test for out-of-sample forecasting performance. I then examine the use of subsets of risk statistics from the basket rather than using the set of all risk statistics. I further study the use of the so-called σ2/μ risk measure in predicting the market “turning point” of performance of a portfolio of hedge funds. Risk measure quantity σ2/μ replaces the traditional variance σ2 in the Black-Scholes option valuation formula when it is evaluated for hedge funds.