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The Genetic Algorithm (Holland, 1975) is a powerful search technique based upon the principles of Darwinian evolution. In its simplest form the GA consists of three main operators - crossover, mutation and selection. The principal theoretical treatment of the Genetic Algorithm (GA) is provided by the Schema Theorem and building block hypothesis (Holland, 1975). The building block hypothesis describes the GA search process as the combination, sampling and recombination of fragments of solutions known as building blocks. The crossover operator is responsible for the combination of building blocks, whilst the selection operator allocates increasing numbers of samples to good building blocks. Thus the GA constructs the optimal (or near-optimal) solution from those fragments of solutions which are, in some sense, optimal. The first part of this thesis documents the development of a technique for the isolation of building blocks from the populations of the GA. This technique is shown to extract exactly those building blocks of interest - those which are sampled most regularly by the GA. These building blocks are used to empirically investigate the validity of the building block hypothesis. It is shown that good building blocks do not combine to form significantly better solution fragments than those resulting from the addition of randomly generated building blocks to good building blocks. This results casts some doubt onto the value of the building block hypothesis as an account of the GA search process (at least for the functions used during these experiments). The second part of this thesis describes an alternative account of the action of crossover. This account is an approximation of the geometric effect of crossover upon the population of samples maintained by the GA. It is shown that, for a simple function, this description of the crossover operator is sufficiently accurate to warrant further investigation. A pair of performance models for the GA upon this function are derived and shown to be accurate for a wide range of crossover schemes. Finally, the GA search process is described in terms of this account of the crossover operator and parallels are drawn with the search process of the simulated annealing algorithm (Kirkpatrick et al, 1983). The third and final part of this thesis describes a technique for the visualisation of high dimensional surfaces, such as are defined by functions of many parameters. This technique is compared to the statistical technique of projection pursuit regression (Friedman & Tukey, 1974) and is shown to compare favourably both in terms of computational expense and quantitative accuracy upon a wide range of test functions. A fundamental flaw of this technique is that it may produce poor visualisations when applied to functions with a small high frequency (or order) components.

Generalized Linear Models (GLMs) (McCullagh and Nelder, 1989) provide a unified framework for fixed effect models where response data arise from exponential family distributions. Much recent research has attempted to extend the framework to include random effects in the linear predictors. Different methodologies have been employed to solve different motivating problems, for example Generalized Linear Mixed Models (Clayton, 1994) and Multilevel Models (Goldstein, 1995). A thorough review and classification of this and related material is presented. In Item Response Theory (IRT) subjects are tested using banks of pre-calibrated test items. A useful model is based on the logistic function with a binary response dependent on the unknown ability of the subject. Item parameters contribute to the probability of a correct response. Within the framework of the GLM, a latent variable, the unknown ability, is introduced as a new component of the linear predictor. This approach affords the opportunity to structure intercept and slope parameters so that item characteristics are represented. A methodology for fitting such GLMs with latent variables, based on the EM algorithm (Dempster, Laird and Rubin, 1977) and using standard Generalized Linear Model fitting software GLIM (Payne, 1987) to perform the expectation step, is developed and applied to a model for binary response data. Accurate numerical integration to evaluate the likelihood functions is a vital part of the computational process. A study of the comparative benefits of two different integration strategies is undertaken and leads to the adoption, unusually, of Gauss-Legendre rules. It is shown how the fitting algorithms are implemented with GLIM programs which incorporate FORTRAN subroutines. Examples from IRT are given. A simulation study is undertaken to investigate the sampling distributions of the estimators and the effect of certain numerical attributes of the computational process. Finally a generalized latent variable model is developed for responses from any exponential family distribution.

This thesis is concerned with the numerical solution of boundary integral equations and the numerical analysis of iterative methods. In the first part, we assume the boundary to be smooth in order to work with compact operators; while in the second part we investigate the problem arising from allowing piecewise smooth boundaries. Although in principle most results of the thesis apply to general problems of reformulating boundary value problems as boundary integral equations and their subsequent numerical solutions, we consider the Helmholtz equation arising from acoustic problems as the main model problem. In Chapter 1, we present the background material of reformulation of Helmhoitz boundary value problems into boundary integral equations by either the indirect potential method or the direct method using integral formulae. The problem of ensuring unique solutions of integral equations for exterior problems is specifically discussed. In Chapter 2, we discuss the useful numerical techniques for solving second kind integral equations. In particular, we highlight the superconvergence properties of iterated projection methods and the important procedure of Nystrom interpolation. In Chapter 3, the multigrid type methods as applied to smooth boundary integral equations are studied. Using the residual correction principle, we are able to propose some robust iterative variants modifying the existing methods to seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease, faster convergence of multigrid type methods and fixed step convergence of the conjugate gradient method. In the case of non-smooth integral boundaries, we first derive the singular forms of the solution of boundary integral solutions for Dirichlet problems and then discuss the numerical solution in Chapter 5. Iterative methods such as two grid methods and the conjugate gradient method are successfully implemented in Chapter 6 to solve the non-smooth integral equations. The study of two grid methods in a general setting and also much of the results on the conjugate gradient method are new. Chapters 3, 4 and 5 are partially based on publications [4], [5] and [35] respectively.

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A flat twistor space is a complex 3 - manifold having the property that every point of the manifold has a neighbourhood which is biholomorphic to a neighbourhood of a complex projective line in complex projective 3 - space. The Penrose transform provides an isomorphism between holomorphic structures on twistor spaces and certain field equations on (Riemannian or Lorentzian) space - times. The initial examples studied by Penrose were solutions to zero rest mass equations and, amongst these, the elementary states were of particular interest. These were elements of a sheaf cohomology group having a singularity on a particular complex projective line, with a codimension-2 structure similar, in some sense, to a Laurent series with a pole of finite order. In this work we extend this idea to the notion of codimension-2 poles for analytic cohomology classes on a punctured flat twistor space, by which we mean a general, compact, flat, twistor space with a finite number of non-intersecting complex, projective lines removed. We define a holomorphic line bundle on the blow-up of the compact flat twistor space along these lines and show that elements of the first cohomology group with coefficients in the line bundle, when restricted to the punctured twistor space, are cohomology classes with singularities on the removed lines which have precisely the kind of codimension - 2 structure which we define as codimension-2 poles. The dimension of this cohomology group on the blown-up manifold is then calculated for the twistor space of a compact, Riemannian, hyperbolic 4-manifold. The calculation uses the Hirzebruch - Riemann - Roch theorem to find the holomorphic Euler characteristic of the line bundle, (in chapter 3) together with vanishing theorems. In chapter 4 we show that it is sufficient to find vanishing theorems for the compact flat - twistor space. In chapter 5 we prove a number of vanishing theorems to be used. The technique uses the Penrose transform to convert the theorem to a vanishing theorem for spinor fields. These are then proved by using Penrose's Spinor calculus.

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