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In general the numerical solution of boundary integral equations leads to full coefficient matrices. The discrete system can be solved in O(N2) operations by iterative solvers of the Conjugate Gradient type. Therefore, we are interested in fast methods such as fast multipole and wavelets, that reduce the computational cost to O(N lnp N). In this thesis we are concerned with wavelet methods. They have proved to be very efficient and effective basis functions due to the fact that the coefficients of a wavelet expansion decay rapidly for a large class of functions. Due to the multiresolution property of wavelets they provide accurate local descriptions of functions efficiently. For example in the presence of corners and edges, the functions can still be approximated with a linear combination of just a few basis functions. Wavelets are attractive for the numerical solution of integral equations because their vanishing moments property leads to operator compression. However, to obtain wavelets with compact support and high order of vanishing moments, the length of the support increases as the order of the vanishingmoments increases. This causes difficulties with the practical use of wavelets particularly at edges and corners. However, with multiwavelets, an increase in the order of vanishing moments is obtained not by increasing the support but by increasing the number of mother wavelets. In chapter 2 we review the methods and techniques required for these reformulations, we also discuss how these boundary integral equations may be discretised by a boundary element method. In chapter 3, we discuss wavelet and multiwavelet bases. In chapter 4, we consider two boundary element methods, namely, the standard and non-standard Galerkin methods with multiwavelet basis functions. For both methods compression strategies are developed which only require the computation of the significant matrix elements. We show that they are O(N logp N) such significant elements. In chapters 5 and 6 we apply the standard and non-standard Galerkin methods to several test problems.

There has been great interest in recent years in the development of wavelet methods for estimating an unknown function observed in the presence of noise, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, a novel empirical Bayes block (EBB) shrinkage procedure is proposed and the performance of this approach with both independent identically distributed (IID) noise and correlated noise is thoroughly explored. The first part of this thesis develops a Bayesian methodology involving the non-central X[superscript]2 distribution to simultaneously shrink wavelet coefficients in a block, based on the block sum of squares. A useful (and to the best of our knowledge, new) identity satisfied by the non-central X[superscript]2 density is exploited. This identity leads to tractable posterior calculations for suitable families of prior distributions. Also, the families of prior distribution we work with are sufficiently flexible to represent various forms of prior knowledge. Furthermore, an efficient method for finding the hyperparameters is implemented and simulations show that this method has a high degree of computational advantage. The second part relaxes the assumption of IID noise considered in the first part of this thesis. A semi-parametric model including a parametric component and a nonparametric component is presented to deal with correlated noise situations. In the parametric component, attention is paid to the covariance structure of the noise. Two distinct parametric methods (maximum likelihood estimation and time series model identification techniques) for estimating the parameters in the covariance matrix are investigated. Both methods have been successfully implemented and are believed to be new additions to smoothing methods.

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