Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Power management and power flow control with back-to-back converters in a utility connected microgrid

This paper proposes a method for power flow control between utility and microgrid through back-to-back converters, which facilitates desired real and reactive power flow between utility and microgrid. In the proposed control strategy, the system can run in two different modes depending on the power requirement in the microgrid. In mode-1, specified amount of real and reactive power are shared between the utility and the microgrid through the back-to-back converters. Mode-2 is invoked when the power that can be supplied by the DGs in the microgrid reaches its maximum limit. In such a case, the rest of the power demand of the microgrid has to be supplied by the utility. An arrangement between DGs in the microgrid is proposed to achieve load sharing in both grid connected and islanded modes. The back-to-back converters also provide total frequency isolation between the utility and the microgrid. It is shown that the voltage or frequency fluctuation in the utility side has no impact on voltage or power in microgrid side. Proper relay-breaker operation coordination is proposed during fault along with the blocking of the back-to-back converters for seamless resynchronization. Both impedance and motor type loads are considered to verify the system stability. The impact of dc side voltage fluctuation of the DGs and DG tripping on power sharing is also investigated. The efficacy of the proposed control ar-rangement has been validated through simulation for various operating conditions. The model of the microgrid power system is simulated in PSCAD.

Theoretical investigation of mechanisms of formation and interaction of nanoparticles

In this thesis an investigation into theoretical models for formation and interaction of nanoparticles is presented. The work presented includes a literature review of current models followed by a series of five chapters of original research. This thesis has been submitted in partial fulfilment of the requirements for the degree of doctor of philosophy by publication and therefore each of the five chapters consist of a peer-reviewed journal article. The thesis is then concluded with a discussion of what has been achieved during the PhD candidature, the potential applications for this research and ways in which the research could be extended in the future. In this thesis we explore stochastic models pertaining to the interaction and evolution mechanisms of nanoparticles. In particular, we explore in depth the stochastic evaporation of molecules due to thermal activation and its ultimate effect on nanoparticles sizes and concentrations. Secondly, we analyse the thermal vibrations of nanoparticles suspended in a fluid and subject to standing oscillating drag forces (as would occur in a standing sound wave) and finally on lattice surfaces in the presence of high heat gradients. We have described in this thesis a number of new models for the description of multicompartment networks joined by a multiple, stochastically evaporating, links. The primary motivation for this work is in the description of thermal fragmentation in which multiple molecules holding parts of a carbonaceous nanoparticle may evaporate. Ultimately, these models predict the rate at which the network or aggregate fragments into smaller networks/aggregates and with what aggregate size distribution. The models are highly analytic and describe the fragmentation of a link holding multiple bonds using Markov processes that best describe different physical situations and these processes have been analysed using a number of mathematical methods. The fragmentation of the network/aggregate is then predicted using combinatorial arguments. Whilst there is some scepticism in the scientific community pertaining to the proposed mechanism of thermal fragmentation,we have presented compelling evidence in this thesis supporting the currently proposed mechanism and shown that our models can accurately match experimental results. This was achieved using a realistic simulation of the fragmentation of the fractal carbonaceous aggregate structure using our models. Furthermore, in this thesis a method of manipulation using acoustic standing waves is investigated. In our investigation we analysed the effect of frequency and particle size on the ability for the particle to be manipulated by means of a standing acoustic wave. In our results, we report the existence of a critical frequency for a particular particle size. This frequency is inversely proportional to the Stokes time of the particle in the fluid. We also find that for large frequencies the subtle Brownian motion of even larger particles plays a significant role in the efficacy of the manipulation. This is due to the decreasing size of the boundary layer between acoustic nodes. Our model utilises a multiple time scale approach to calculating the long term effects of the standing acoustic field on the particles that are interacting with the sound. These effects are then combined with the effects of Brownian motion in order to obtain a complete mathematical description of the particle dynamics in such acoustic fields. Finally, in this thesis, we develop a numerical routine for the description of "thermal tweezers". Currently, the technique of thermal tweezers is predominantly theoretical however there has been a handful of successful experiments which demonstrate the effect it practise. Thermal tweezers is the name given to the way in which particles can be easily manipulated on a lattice surface by careful selection of a heat distribution over the surface. Typically, the theoretical simulations of the effect can be rather time consuming with supercomputer facilities processing data over days or even weeks. Our alternative numerical method for the simulation of particle distributions pertaining to the thermal tweezers effect use the Fokker-Planck equation to derive a quick numerical method for the calculation of the effective diffusion constant as a result of the lattice and the temperature. We then use this diffusion constant and solve the diffusion equation numerically using the finite volume method. This saves the algorithm from calculating many individual particle trajectories since it is describes the flow of the probability distribution of particles in a continuous manner. The alternative method that is outlined in this thesis can produce a larger quantity of accurate results on a household PC in a matter of hours which is much better than was previously achieveable.

Performance tests for visual servo control systems, with application to partitioned approaches to visual servo control

Visual servoing has been a viable method of robot manipulator control for more than a decade. Initial developments involved positionbased visual servoing (PBVS), in which the control signal exists in Cartesian space. The younger method, image-based visual servoing (IBVS), has seen considerable development in recent years. PBVS and IBVS offer tradeoffs in performance, and neither can solve all tasks that may confront a robot. In response to these issues, several methods have been devised that partition the control scheme, allowing some motions to be performed in the manner of a PBVS system, while the remaining motions are performed using an IBVS approach. To date, there has been little research that explores the relative strengths and weaknesses of these methods. In this paper we present such an evaluation. We have chosen three recent visual servo approaches for evaluation in addition to the traditional PBVS and IBVS approaches. We posit a set of performance metrics that measure quantitatively the performance of a visual servo controller for a specific task. We then evaluate each of the candidate visual servo methods for four canonical tasks with simulations and with experiments in a robotic work cell.

Fingerprint analysis using wavelet transform with application to compression and feature extraction

The main goal of this research is to design an efficient compression al~ gorithm for fingerprint images. The wavelet transform technique is the principal tool used to reduce interpixel redundancies and to obtain a parsimonious representation for these images. A specific fixed decomposition structure is designed to be used by the wavelet packet in order to save on the computation, transmission, and storage costs. This decomposition structure is based on analysis of information packing performance of several decompositions, two-dimensional power spectral density, effect of each frequency band on the reconstructed image, and the human visual sensitivities. This fixed structure is found to provide the "most" suitable representation for fingerprints, according to the chosen criteria. Different compression techniques are used for different subbands, based on their observed statistics. The decision is based on the effect of each subband on the reconstructed image according to the mean square criteria as well as the sensitivities in human vision. To design an efficient quantization algorithm, a precise model for distribution of the wavelet coefficients is developed. The model is based on the generalized Gaussian distribution. A least squares algorithm on a nonlinear function of the distribution model shape parameter is formulated to estimate the model parameters. A noise shaping bit allocation procedure is then used to assign the bit rate among subbands. To obtain high compression ratios, vector quantization is used. In this work, the lattice vector quantization (LVQ) is chosen because of its superior performance over other types of vector quantizers. The structure of a lattice quantizer is determined by its parameters known as truncation level and scaling factor. In lattice-based compression algorithms reported in the literature the lattice structure is commonly predetermined leading to a nonoptimized quantization approach. In this research, a new technique for determining the lattice parameters is proposed. In the lattice structure design, no assumption about the lattice parameters is made and no training and multi-quantizing is required. The design is based on minimizing the quantization distortion by adapting to the statistical characteristics of the source in each subimage. 11 Abstract Abstract Since LVQ is a multidimensional generalization of uniform quantizers, it produces minimum distortion for inputs with uniform distributions. In order to take advantage of the properties of LVQ and its fast implementation, while considering the i.i.d. nonuniform distribution of wavelet coefficients, the piecewise-uniform pyramid LVQ algorithm is proposed. The proposed algorithm quantizes almost all of source vectors without the need to project these on the lattice outermost shell, while it properly maintains a small codebook size. It also resolves the wedge region problem commonly encountered with sharply distributed random sources. These represent some of the drawbacks of the algorithm proposed by Barlaud [26). The proposed algorithm handles all types of lattices, not only the cubic lattices, as opposed to the algorithms developed by Fischer [29) and Jeong [42). Furthermore, no training and multiquantizing (to determine lattice parameters) is required, as opposed to Powell's algorithm [78). For coefficients with high-frequency content, the positive-negative mean algorithm is proposed to improve the resolution of reconstructed images. For coefficients with low-frequency content, a lossless predictive compression scheme is used to preserve the quality of reconstructed images. A method to reduce bit requirements of necessary side information is also introduced. Lossless entropy coding techniques are subsequently used to remove coding redundancy. The algorithms result in high quality reconstructed images with better compression ratios than other available algorithms. To evaluate the proposed algorithms their objective and subjective performance comparisons with other available techniques are presented. The quality of the reconstructed images is important for a reliable identification. Enhancement and feature extraction on the reconstructed images are also investigated in this research. A structural-based feature extraction algorithm is proposed in which the unique properties of fingerprint textures are used to enhance the images and improve the fidelity of their characteristic features. The ridges are extracted from enhanced grey-level foreground areas based on the local ridge dominant directions. The proposed ridge extraction algorithm, properly preserves the natural shape of grey-level ridges as well as precise locations of the features, as opposed to the ridge extraction algorithm in [81). Furthermore, it is fast and operates only on foreground regions, as opposed to the adaptive floating average thresholding process in [68). Spurious features are subsequently eliminated using the proposed post-processing scheme.