The enormous computing resources that large-scale simulations in Lattice QCD require will continue to test the limits of even the largest supercomputers into the foreseeable future. The efficiency of such simulations will therefore concern practitioners of lattice QCD for some time to come. I begin with an introduction to those aspects of lattice QCD essential to the remainder of the thesis, and follow with a description of the Wilson fermion matrix M, an object which is central to my theme. The principal bottleneck in Lattice QCD simulations is the solution of linear systems involving M, and this topic is treated in depth. I compare some of the more popular iterative methods, including Minimal Residual, Corij ugate Gradient on the Normal Equation, BI-Conjugate Gradient, QMR., BiCGSTAB and BiCGSTAB2, and then turn to a study of block algorithms, a special class of iterative solvers for systems with multiple right-hand sides. Included in this study are two block algorithms which had not previously been applied to lattice QCD. The next chapters are concerned with a generalised Hybrid Monte Carlo algorithm (OHM C) for QCD simulations involving dynamical quarks. I focus squarely on the efficient and robust implementation of GHMC, and describe some tricks to improve its performance. A limited set of results from HMC simulations at various parameter values is presented. A treatment of the non-hermitian Lanczos method and its application to the eigenvalue problem for M rounds off the theme of large-scale matrix computations.

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We consider the Cauchy problem for a second order parabolic PDE in half spaces, arising from the stochastic modelling of a multidimensional European financial option. To improve generality, the asset price drift and volatility in the underlying stochastic model are taken time and space-dependent and the payoff function is not specified. The numerical methods and possible approximation results are strongly linked to the theory on the solvability of the PDE. We make use of two theories: the theory of linear PDE in Hölder spaces and the theory of linear PDE in Sobolev spaces. First, instead of the problem in half spaces, we consider the correspondent problem in domains. This localised PDE problem is solvable in Hölder spaces. The solution is numerically approximated, using finite differences (with both the explicit and implicit schemes) and the rate of convergence of the time-space finite differences scheme is estimated. Finally, we estimate the localization error. Then, using the L² theory of solvability in Sobolev spaces and in weighted Sobolev spaces, the solution of the PDE problem is approximated in space, also using finite differences. The approximation in time is considered in abstract spaces for evolution equations (making use of both the explicit and implicit schemes) and then specified to the second-order parabolic PDE problem. The rates of convergence are estimated for the approximation in space and in time.

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This research is concerned with the study of bulk queueing models. A number of M/G/l queues with bulk-arrival and/or bulk-service are studied via the imbedded Markov chain technique to derive steady-state models. Results from these models are compared with simulation results after the appropriate numerical analysis, needed to implement these solutions, has been considered. Similar queueing models are considered from a time-dependent viewpoint. Because of the difficulties of obtaining time-dependent solutions, the emphasis is placed upon the development of approximations to the queueing results. Finally, to demonstrate the practicability of these models, two real-life problems are considered, namely the Gatwick Rapid Transit systems and the proposed Channel Tunnel freight shuttle operation. It is demonstrated that the former is an under-utilised system, while the latter is likely to operate very close to its capacity conditions.

In this thesis, we have given out a comprehensive approach to estimate the pairwise interaction in a bivariate additive model when the additive assumption is untenable. A new "nested backfilling' model fitting approach has been proposed and compared with some earlier approaches. The explicit estimators of each individual term were derived for the hierarchical bivariate additive model fitted by this "nested backfilling' approach. The convergence of the "nested backfilling" approach and the existence of these estimators have been shown depending on the ratio of the bandwidths that were used in the estimation of the effect of the same variable but in different terms. The mean average square error properties of these proposed explicit estimators were investigated. A discussion the pattern left in this bias and variance expressions derived for these estimators, such as mean corrected, Gauss-Seidel style etc., were provided to facilitate the understanding these properties. Unlike in the pure additive model case, the mean average square error of our model cannot be attributed to each individual variable. The four optimal bandwidths used need to be selected simultaneously to minimize the mean average square error. These estimators were shown worked reasonably well in simulated datasets, regardless of the level of dependence of the covariates.

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