Theory, Analysis and Implementation of Wavelet Monte Carlo.

Theory of Wavelet Monte Carlo (WMC) - a novel sampling algorithm is presented and analysed. It is shown how Wavelet theory and Survival analysis can be combined together, producing a method that is able to generate independent samples from a non-standard multimodal distribution when a direct sampling approach is not viable. It is demonstrated that due to the way the algorithm is constructed it could be easily parallelised, to boost the execution time. Several issues regarding the implementation of WMC are presented and discussed. In particular, the choice of the wavelet family, curse of dimensionality and computation of wavelet coefficients is investigated in detail revealing critical problems with certain wavelet families. Two possible modifications to the original WMC are outlined with their strengths and weaknesses highlighted. Finally, an important connection between Besov spaces and WMC theory is established, revealing intriguing implications of the implicit assumptions made in WMC theory.