A thesis submitted to the Faculty of Science, School of Computational and Applied Mathematics University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, South Africa, July 2015. / Reinforcement learning agents solve tasks by finding policies that maximise their reward over time. The policy can be found from the value function, which represents the value of each state-action pair. In continuous state spaces, the value function must be approximated. Often, this is done using a fixed linear combination of functions across all dimensions. We introduce and demonstrate the wavelet basis for reinforcement learning, a basis function scheme competitive against state of the art fixed bases. We extend two online adaptive tiling schemes to wavelet functions and show their performance improvement across standard domains. Finally we introduce the Multiscale Adaptive Wavelet Basis (MAWB), a wavelet-based adaptive basis scheme which is dimensionally scalable and insensitive to the initial level of detail. This scheme adaptively grows the basis function set by combining across dimensions, or splitting within a dimension those candidate functions which have a high estimated projection onto the Bellman error. A number of novel measures are used to find this estimate. i

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Univ., Habil.-Schr.--Gießen. / Literaturverz. S. [127] - 133.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 112-120). Available also in a digital version from Dissertation Abstracts.

Thesis (M.S. in Math.)--School of Mathematics, Georgia Institute of Technology, 2004. Directed by Yang Wang. / Includes bibliographical references (leaves 37-38).

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

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We consider the application of the wavelet transform for solving sparse matrix systems and partial differential equations. The first part is devoted to the theory and algorithms of wavelets. The second part is concerned with the sparse representation of matrices and well-known operators. The third part is directed to the application of wavelets to partial differential equations, and to sparse linear systems resulting from differential equations. We present several numerical examples and simulations for the above cases. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.

Hydrological variables such as rainfall and streamfiow vary at a range of temporal scales, from short term (diurnal and seasonal) to the inter annual time scales associated with the El Nino - Southern Oscillation (ENSO) and Indian Ocean Dipole (IOD) phenomena, to even longer time scales such as those linked to the Pacific (inter-) Decadal Oscillation (PDO). This temporal variability poses a significant challenge to hydrologists and water resource managers, since a failure to take such variability into account can lead to an underestimation of the likelihood of droughts and sequences of above average rainfall, which in turn has important implications for the design and operation of reservoirs for hydroelectricity generation, irrigation and municipal water supply. Understanding and accounting for this variability through well designed prediction systems is thus an important part of improving the planning, management and operation of complex water resources systems. This thesis outlines the application of two statistical techniques: wavelets and independent component analysis, to identify sources of hydrological variability, and then use this information to probabilistically generate multivariate seasonal forecasts or develop extended synthetic sequences of hydrological time series. The research is divided into four main parts. The first part outlines an application of the method of wavelets to analyse sources of Australian rainfall variability, and shows that there are coherent regions of variability in addition to the ENSO phenomenon that should be considered when developing seasonal forecasts. The second part examines the capability of three component extraction techniques: principal component analysis (PCA), Varimax and independent component analysis (ICA), in identifying and interpreting modes of variability in the global sea surface temperature dataset. The third part outlines a new technique that uses ICA to factorise multivariate reservoir inflow time series into a set of independent univariate time series, so that univariate methods can be used to develop multivariate synthetic sequences and probabilistic seasonal forecasts. Finally, the fourth part synthesises the previous three parts by demonstrating a wavelets- and correlation-based methodology for assessing sources of climate variability, and then using ICA to generate probabilistic multivariate seasonal forecasts of reservoir inflows that form part of Sydney's water supply system.