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.

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

Marx as critical discourse analyst : the genesis of a critical method and its relevance to the critique of global capital.

In this paper we identify elements in Marx´s economic and political writings that are relevant to contemporary critical discourse analysis (CDA). We argue that Marx can be seen to be engaging in a form of discourse analysis. We identify the elements in Marx´s historical materialist method that support such a perspective, and exemplify these in a longitudinal comparison of Marx´s texts.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Generation in context : an exploratory method for musical enquiry

This paper discusses a method, Generation in Context, for interrogating theories of music analysis and music perception. Given an analytic theory, the method consists of creating a generative process that implements the theory in reverse. Instead of using the theory to create analyses from scores, the theory is used to generate scores from analyses. Subjective evaluation of the quality of the musical output provides a mechanism for testing the theory in a contextually robust fashion. The method is exploratory, meaning that in addition to testing extant theories it provides a general mechanism for generating new theoretical insights. We outline our initial explorations in the use of generative processes for music research, and we discuss how generative processes provide evidence as to the veracity of theories about how music is experienced, with insights into how these theories may be improved and, concurrently, provide new techniques for music creation. We conclude that Generation in Context will help reveal new perspectives on our understanding of music.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

A study into the permeability and compressibility of Australian bagasse pulp

This is an experimental study into the permeability and compressibility properties of bagasse pulp pads. Three experimental rigs were custom-built for this project. The experimental work is complemented by modelling work. Both the steady-state and dynamic behaviour of pulp pads are evaluated in the experimental and modelling components of this project. Bagasse, the fibrous residue that remains after sugar is extracted from sugarcane, is normally burnt in Australia to generate steam and electricity for the sugar factory. A study into bagasse pulp was motivated by the possibility of making highly value-added pulp products from bagasse for the financial benefit of sugarcane millers and growers. The bagasse pulp and paper industry is a multibillion dollar industry (1). Bagasse pulp could replace eucalypt pulp which is more widely used in the local production of paper products. An opportunity exists for replacing the large quantity of mainly generic paper products imported to Australia. This includes 949,000 tonnes of generic photocopier papers (2). The use of bagasse pulp for paper manufacture is the main application area of interest for this study. Bagasse contains a large quantity of short parenchyma cells called ‘pith’. Around 30% of the shortest fibres are removed from bagasse prior to pulping. Despite the ‘depithing’ operations in conventional bagasse pulp mills, a large amount of pith remains in the pulp. Amongst Australian paper producers there is a perception that the high quantity of short fibres in bagasse pulp leads to poor filtration behaviour at the wet-end of a paper machine. Bagasse pulp’s poor filtration behaviour reduces paper production rates and consequently revenue when compared to paper production using locally made eucalypt pulp. Pulp filtration can be characterised by two interacting factors; permeability and compressibility. Surprisingly, there has previously been very little rigorous investigation into neither bagasse pulp permeability nor compressibility. Only freeness testing of bagasse pulp has been published in the open literature. As a result, this study has focussed on a detailed investigation of the filtration properties of bagasse pulp pads. As part of this investigation, this study investigated three options for improving the permeability and compressibility properties of Australian bagasse pulp pads. Two options for further pre-treating depithed bagasse prior to pulping were considered. Firstly, bagasse was fractionated based on size. Two bagasse fractions were produced, ‘coarse’ and ‘medium’ bagasse fractions. Secondly, bagasse was collected after being processed on two types of juice extraction technology, i.e. from a sugar mill and from a sugar diffuser. Finally one method of post-treating the bagasse pulp was investigated. The effects of chemical additives, which are known to improve freeness, were also assessed for their effect on pulp pad permeability and compressibility. Pre-treated Australian bagasse pulp samples were compared with several benchmark pulp samples. A sample of commonly used kraft Eucalyptus globulus pulp was obtained. A sample of depithed Argentinean bagasse, which is used for commercial paper production, was also obtained. A sample of Australian bagasse which was depithed as per typical factory operations was also produced for benchmarking purposes. The steady-state pulp pad permeability and compressibility parameters were determined experimentally using two purpose-built experimental rigs. In reality, steady-state conditions do not exist on a paper machine. The permeability changes as the sheet compresses over time. Hence, a dynamic model was developed which uses the experimentally determined steady-state permeability and compressibility parameters as inputs. The filtration model was developed with a view to designing pulp processing equipment that is suitable specifically for bagasse pulp. The predicted results of the dynamic model were compared to experimental data. The effectiveness of a polymeric and microparticle chemical additives for improving the retention of short fibres and increasing the drainage rate of a bagasse pulp slurry was determined in a third purpose-built rig; a modified Dynamic Drainage Jar (DDJ). These chemical additives were then used in the making of a pulp pad, and their effect on the steady-state and dynamic permeability and compressibility of bagasse pulp pads was determined. The most important finding from this investigation was that Australian bagasse pulp was produced with higher permeability than eucalypt pulp, despite a higher overall content of short fibres. It is thought this research outcome could enable Australian paper producers to switch from eucalypt pulp to bagasse pulp without sacrificing paper machine productivity. It is thought that two factors contributed to the high permeability of the bagasse pulp pad. Firstly, thicker cell walls of the bagasse pulp fibres resulted in high fibre stiffness. Secondly, the bagasse pulp had a large proportion of fibres longer than 1.3 mm. These attributes helped to reinforce the pulp pad matrix. The steady-state permeability and compressibility parameters for the eucalypt pulp were consistent with those found by previous workers. It was also found that Australian pulp derived from the ‘coarse’ bagasse fraction had higher steady-state permeability than the ‘medium’ fraction. However, there was no difference between bagasse pulp originating from a diffuser or a mill. The bagasse pre-treatment options investigated in this study were not found to affect the steady-state compressibility parameters of a pulp pad. The dynamic filtration model was found to give predictions that were in good agreement with experimental data for pads made from samples of pretreated bagasse pulp, provided at least some pith was removed prior to pulping. Applying vacuum to a pulp slurry in the modified DDJ dramatically reduced the drainage time. At any level of vacuum, bagasse pulp benefitted from chemical additives as quantified by reduced drainage time and increased retention of short fibres. Using the modified DDJ, it was observed that under specific conditions, a benchmark depithed bagasse pulp drained more rapidly than the ‘coarse’ bagasse pulp. In steady-state permeability and compressibility experiments, the addition of chemical additives improved the pad permeability and compressibility of a benchmark bagasse pulp with a high quantity of short fibres. Importantly, this effect was not observed for the ‘coarse’ bagasse pulp. However, dynamic filtration experiments showed that there was also a small observable improvement in filtration for the ‘medium’ bagasse pulp. The mechanism of bagasse pulp pad consolidation appears to be by fibre realignment. Chemical additives assist to lubricate the consolidation process. This study was complemented by pulp physical and chemical property testing and a microscopy study. In addition to its high pulp pad permeability, ‘coarse’ bagasse pulp often (but not always) had superior physical properties than a benchmark depithed bagasse pulp.