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 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.

Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives

In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville and Grunwald-Letnikov fractional derivatives, two modified alternating direction methods: a modified alternating direction implicit Euler method and a modified Peaceman-Rachford method, are proposed for solving the 2D-NCSF-FD in uniform media. The stability and consistency, thus convergence of the two methods in a bounded domain are discussed. Finally, numerical results are given.

Flexural behaviour and design of cold-formed steel beams with rectangular hollow flanges

Until recently, the hot-rolled steel members have been recognized as the most popular and widely used steel group, but in recent times, the use of cold-formed high strength steel members has rapidly increased. However, the structural behavior of light gauge high strength cold-formed steel members characterized by various buckling modes is not yet fully understood. The current cold-formed steel sections such as C- and Z-sections are commonly used because of their simple forming procedures and easy connections, but they suffer from certain buckling modes. It is therefore important that these buckling modes are either delayed or eliminated to increase the ultimate capacity of these members. This research is therefore aimed at developing a new cold-formed steel beam with two torsionally rigid rectangular hollow flanges and a slender web formed using intermittent screw fastening to enhance the flexural capacity while maintaining a minimum fabrication cost. This thesis describes a detailed investigation into the structural behavior of this new Rectangular Hollow Flange Beam (RHFB), subjected to flexural action The first phase of this research included experimental investigations using thirty full scale lateral buckling tests and twenty two section moment capacity tests using specially designed test rigs to simulate the required loading and support conditions. A detailed description of the experimental methods, RHFB failure modes including local, lateral distortional and lateral torsional buckling modes, and moment capacity results is presented. A comparison of experimental results with the predictions from the current design rules and other design methods is also given. The second phase of this research involved a methodical and comprehensive investigation aimed at widening the scope of finite element analysis to investigate the buckling and ultimate failure behaviours of RHFBs subjected to flexural actions. Accurate finite element models simulating the physical conditions of both lateral buckling and section moment capacity tests were developed. Comparison of experimental and finite element analysis results showed that the buckling and ultimate failure behaviour of RHFBs can be simulated well using appropriate finite element models. Finite element models simulating ideal simply supported boundary conditions and a uniform moment loading were also developed in order to use in a detailed parametric study. The parametric study results were used to review the current design rules and to develop new design formulae for RHFBs subjected to local, lateral distortional and lateral torsional buckling effects. Finite element analysis results indicate that the discontinuity due to screw fastening has a noticeable influence only for members in the intermediate slenderness region. Investigations into different combinations of thicknesses in the flange and web indicate that increasing the flange thickness is more effective than web thickness in enhancing the flexural capacity of RHFBs. The current steel design standards, AS 4100 (1998) and AS/NZS 4600 (1996) are found sufficient to predict the section moment capacity of RHFBs. However, the results indicate that the AS/NZS 4600 is more accurate for slender sections whereas AS 4100 is more accurate for compact sections. The finite element analysis results further indicate that the current design rules given in AS/NZS 4600 is adequate in predicting the member moment capacity of RHFBs subject to lateral torsional buckling effects. However, they were inadequate in predicting the capacities of RHFBs subject to lateral distortional buckling effects. This thesis has therefore developed a new design formula to predict the lateral distortional buckling strength of RHFBs. Overall, this thesis has demonstrated that the innovative RHFB sections can perform well as economically and structurally efficient flexural members. Structural engineers and designers should make use of the new design rules and the validated existing design rules to design the most optimum RHFB sections depending on the type of applications. Intermittent screw fastening method has also been shown to be structurally adequate that also minimises the fabrication cost. Product manufacturers and builders should be able to make use of this in their applications.

Evolutionary optimisation methods with uncertainty for modern multidisciplinary design in aeronautical engineering

One of the new challenges in aeronautics is combining and accounting for multiple disciplines while considering uncertainties or variability in the design parameters or operating conditions. This paper describes a methodology for robust multidisciplinary design optimisation when there is uncertainty in the operating conditions. The methodology, which is based on canonical evolution algorithms, is enhanced by its coupling with an uncertainty analysis technique. The paper illustrates the use of this methodology on two practical test cases related to Unmanned Aerial Systems (UAS). These are the ideal candidates due to the multi-physics involved and the variability of missions to be performed. Results obtained from the optimisation show that the method is effective to find useful Pareto non-dominated solutions and demonstrate the use of robust design techniques.

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.

Nanoparticle melting as a Stefan moving boundary problem

The melting of spherical nanoparticles is considered from the perspective of heat flow in a pure material and as a moving boundary (Stefan) problem. The dependence of the melting temperature on both the size of the particle and the interfacial tension is described by the Gibbs-Thomson effect, and the resulting two-phase model is solved numerically using a front-fixing method. Results show that interfacial tension increases the speed of the melting process, and furthermore, the temperature distribution within the solid core of the particle exhibits behaviour that is qualitatively different to that predicted by the classical models without interfacial tension.

The moderating role of self-efficacy on the relationship between emotional exhaustion and job strain for boundary-spanning service employees

Appropriate behaviours toward customers often requires employees to suppress some genuine emotions and/or express other emotions; genuine or contrived. Managing one's emotions in this way gives rise to emotional exhaustion. This can have consequences for psychological ill health, in the form of work place strain, and ultimately employee's desire to leave. This student examines the relationships between emotional management, emotional exhaustion and turnover intentions amongst diversional therapy professionals. We find that some forms of emotional management have a significant impact on emotional exhaustion and that this predicts workplace strain. Furthermore, the deleterious effects of emotional exhaustion are mitigated somewhat for employees who have strong beliefs in their ability to provide good service, compared to employees with lower self efficacy beliefs.

Linkage and heritability analysis of migraine symptom groupings: A comparison of three different clustering methods on twin data

Migraine is a painful disorder for which the etiology remains obscure. Diagnosis is largely based on International Headache Society criteria. However, no feature occurs in all patients who meet these criteria, and no single symptom is required for diagnosis. Consequently, this definition may not accurately reflect the phenotypic heterogeneity or genetic basis of the disorder. Such phenotypic uncertainty is typical for complex genetic disorders and has encouraged interest in multivariate statistical methods for classifying disease phenotypes. We applied three popular statistical phenotyping methods—latent class analysis, grade of membership and grade of membership “fuzzy” clustering (Fanny)—to migraine symptom data, and compared heritability and genome-wide linkage results obtained using each approach. Our results demonstrate that different methodologies produce different clustering structures and non-negligible differences in subsequent analyses. We therefore urge caution in the use of any single approach and suggest that multiple phenotyping methods be used.

Industrial playwriting : forms, strategies, and methods for creative production

This study, in its exploration of the attached play scripts and their method of development, evaluates the forms, strategies, and methods of an organised model of formalised playwriting. Through the examination, reflection and reaction to a perceived crisis in playwriting in the Australian theatre sector, the notion of Industrial Playwriting is arrived at: a practice whereby plays are designed and constructed, and where the process of writing becomes central to the efficient creation of new work and the improvement of the writer’s skill and knowledge base. Using a practice-led methodology and action research the study examines a system of play construction appropriate to and addressing the challenges of the contemporary Australian theatre sector. Specifically, using the action research methodology known as design-based research a conceptual framework was constructed to form the basis of the notion of Industrial Playwriting. From this two plays were constructed using a case study method and the process recorded and used to create a practical, step-by-step system of Industrial Playwriting. In the creative practice of manufacturing a single authored play, and then a group-devised play, Industrial Playwriting was tested and found to also offer a valid alternative approach to playwriting in the training of new and even emerging playwrights. Finally, it offered insight into how Industrial Playwriting could be used to greatly facilitate theatre companies’ ongoing need to have access to new writers and new Australian works, and how it might form the basis of a cost effective writer development model. This study of the methods of formalised writing as a means to confront some of the challenges of the Australian theatre sector, the practice of playwriting and the history associated with it, makes an original and important contribution to contemporary playwriting practice.

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.

Multivariate methods to identify cancer-related symptom clusters

Multivariate methods are required to assess the interrelationships among multiple, concurrent symptoms. We examined the conceptual and contextual appropriateness of commonly used multivariate methods for cancer symptom cluster identification. From 178 publications identified in an online database search of Medline, CINAHL, and PsycINFO, limited to articles published in English, 10 years prior to March 2007, 13 cross-sectional studies met the inclusion criteria. Conceptually, common factor analysis (FA) and hierarchical cluster analysis (HCA) are appropriate for symptom cluster identification, not principal component analysis. As a basis for new directions in symptom management, FA methods are more appropriate than HCA. Principal axis factoring or maximum likelihood factoring, the scree plot, oblique rotation, and clinical interpretation are recommended approaches to symptom cluster identification.

Modelling water droplet movement on a leaf surface

The central aim for the research undertaken in this PhD thesis is the development of a model for simulating water droplet movement on a leaf surface and to compare the model behavior with experimental observations. A series of five papers has been presented to explain systematically the way in which this droplet modelling work has been realised. Knowing the path of the droplet on the leaf surface is important for understanding how a droplet of water, pesticide, or nutrient will be absorbed through the leaf surface. An important aspect of the research is the generation of a leaf surface representation that acts as the foundation of the droplet model. Initially a laser scanner is used to capture the surface characteristics for two types of leaves in the form of a large scattered data set. After the identification of the leaf surface boundary, a set of internal points is chosen over which a triangulation of the surface is constructed. We present a novel hybrid approach for leaf surface fitting on this triangulation that combines Clough-Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a continuously turning normal. The accuracy of the hybrid technique is assessed using numerical experimentation. The hybrid CT-RBF method is shown to give good representations of Frangipani and Anthurium leaves. Such leaf models facilitate an understanding of plant development and permit the modelling of the interaction of plants with their environment. The motion of a droplet traversing this virtual leaf surface is affected by various forces including gravity, friction and resistance between the surface and the droplet. The innovation of our model is the use of thin-film theory in the context of droplet movement to determine the thickness of the droplet as it moves on the surface. Experimental verification shows that the droplet model captures reality quite well and produces realistic droplet motion on the leaf surface. Most importantly, we observed that the simulated droplet motion follows the contours of the surface and spreads as a thin film. In the future, the model may be applied to determine the path of a droplet of pesticide along a leaf surface before it falls from or comes to a standstill on the surface. It will also be used to study the paths of many droplets of water or pesticide moving and colliding on the surface.

Digital kids, analogue students : a mixed methods study of students' engagement with a school-based Web 2.0 learning innovation

The inquiry documented in this thesis is located at the nexus of technological innovation and traditional schooling. As we enter the second decade of a new century, few would argue against the increasingly urgent need to integrate digital literacies with traditional academic knowledge. Yet, despite substantial investments from governments and businesses, the adoption and diffusion of contemporary digital tools in formal schooling remain sluggish. To date, research on technology adoption in schools tends to take a deficit perspective of schools and teachers, with the lack of resources and teacher ‘technophobia’ most commonly cited as barriers to digital uptake. Corresponding interventions that focus on increasing funding and upskilling teachers, however, have made little difference to adoption trends in the last decade. Empirical evidence that explicates the cultural and pedagogical complexities of innovation diffusion within long-established conventions of mainstream schooling, particularly from the standpoint of students, is wanting. To address this knowledge gap, this thesis inquires into how students evaluate and account for the constraints and affordances of contemporary digital tools when they engage with them as part of their conventional schooling. It documents the attempted integration of a student-led Web 2.0 learning initiative, known as the Student Media Centre (SMC), into the schooling practices of a long-established, high-performing independent senior boys’ school in urban Australia. The study employed an ‘explanatory’ two-phase research design (Creswell, 2003) that combined complementary quantitative and qualitative methods to achieve both breadth of measurement and richness of characterisation. In the initial quantitative phase, a self-reported questionnaire was administered to the senior school student population to determine adoption trends and predictors of SMC usage (N=481). Measurement constructs included individual learning dispositions (learning and performance goals, cognitive playfulness and personal innovativeness), as well as social and technological variables (peer support, perceived usefulness and ease of use). Incremental predictive models of SMC usage were conducted using Classification and Regression Tree (CART) modelling: (i) individual-level predictors, (ii) individual and social predictors, and (iii) individual, social and technological predictors. Peer support emerged as the best predictor of SMC usage. Other salient predictors include perceived ease of use and usefulness, cognitive playfulness and learning goals. On the whole, an overwhelming proportion of students reported low usage levels, low perceived usefulness and a lack of peer support for engaging with the digital learning initiative. The small minority of frequent users reported having high levels of peer support and robust learning goal orientations, rather than being predominantly driven by performance goals. These findings indicate that tensions around social validation, digital learning and academic performance pressures influence students’ engagement with the Web 2.0 learning initiative. The qualitative phase that followed provided insights into these tensions by shifting the analytics from individual attitudes and behaviours to shared social and cultural reasoning practices that explain students’ engagement with the innovation. Six indepth focus groups, comprising 60 students with different levels of SMC usage, were conducted, audio-recorded and transcribed. Textual data were analysed using Membership Categorisation Analysis. Students’ accounts converged around a key proposition. The Web 2.0 learning initiative was useful-in-principle but useless-in-practice. While students endorsed the usefulness of the SMC for enhancing multimodal engagement, extending peer-topeer networks and acquiring real-world skills, they also called attention to a number of constraints that obfuscated the realisation of these design affordances in practice. These constraints were cast in terms of three binary formulations of social and cultural imperatives at play within the school: (i) ‘cool/uncool’, (ii) ‘dominant staff/compliant student’, and (iii) ‘digital learning/academic performance’. The first formulation foregrounds the social stigma of the SMC among peers and its resultant lack of positive network benefits. The second relates to students’ perception of the school culture as authoritarian and punitive with adverse effects on the very student agency required to drive the innovation. The third points to academic performance pressures in a crowded curriculum with tight timelines. Taken together, findings from both phases of the study provide the following key insights. First, students endorsed the learning affordances of contemporary digital tools such as the SMC for enhancing their current schooling practices. For the majority of students, however, these learning affordances were overshadowed by the performative demands of schooling, both social and academic. The student participants saw engagement with the SMC in-school as distinct from, even oppositional to, the conventional social and academic performance indicators of schooling, namely (i) being ‘cool’ (or at least ‘not uncool’), (ii) sufficiently ‘compliant’, and (iii) achieving good academic grades. Their reasoned response therefore, was simply to resist engagement with the digital learning innovation. Second, a small minority of students seemed dispositionally inclined to negotiate the learning affordances and performance constraints of digital learning and traditional schooling more effectively than others. These students were able to engage more frequently and meaningfully with the SMC in school. Their ability to adapt and traverse seemingly incommensurate social and institutional identities and norms is theorised as cultural agility – a dispositional construct that comprises personal innovativeness, cognitive playfulness and learning goals orientation. The logic then is ‘both and’ rather than ‘either or’ for these individuals with a capacity to accommodate both learning and performance in school, whether in terms of digital engagement and academic excellence, or successful brokerage across multiple social identities and institutional affiliations within the school. In sum, this study takes us beyond the familiar terrain of deficit discourses that tend to blame institutional conservatism, lack of resourcing and teacher resistance for low uptake of digital technologies in schools. It does so by providing an empirical base for the development of a ‘third way’ of theorising technological and pedagogical innovation in schools, one which is more informed by students as critical stakeholders and thus more relevant to the lived culture within the school, and its complex relationship to students’ lives outside of school. It is in this relationship that we find an explanation for how these individuals can, at the one time, be digital kids and analogue students.

Near-infrared spectroscopic studies of human scalp hair in a forensic context

Human hair is a relatively inert biopolymer and can survive through natural disasters. It is also found as trace evidence at crime scenes. Previous studies by FTIRMicrospectroscopy and – Attenuated Total Reflectance (ATR) successfully showed that hairs can be matched and discriminated on the basis of gender, race and hair treatment, when interpreted by chemometrics. However, these spectroscopic techniques are difficult to operate at- or on-field. On the other hand, some near infrared spectroscopic (NIRS) instruments equipped with an optical probe, are portable and thus, facilitate the on- or at –field measurements for potential application directly at a crime or disaster scene. This thesis is focused on bulk hair samples, which are free of their roots, and thus, independent of potential DNA contribution for identification. It explores the building of a profile of an individual with the use of the NIRS technique on the basis of information on gender, race and treated hair, i.e. variables which can match and discriminate individuals. The complex spectra collected may be compared and interpreted with the use of chemometrics. These methods can then be used as protocol for further investigations. Water is a common substance present at forensic scenes e.g. at home in a bath, in the swimming pool; it is also common outdoors in the sea, river, dam, puddles and especially during DVI incidents at the seashore after a tsunami. For this reason, the matching and discrimination of bulk hair samples after the water immersion treatment was also explored. Through this research, it was found that Near Infrared Spectroscopy, with the use of an optical probe, has successfully matched and discriminated bulk hair samples to build a profile for the possible application to a crime or disaster scene. Through the interpretation of Chemometrics, such characteristics included Gender and Race. A novel approach was to measure the spectra not only in the usual NIR range (4000 – 7500 cm-1) but also in the Visible NIR (7500 – 12800 cm-1). This proved to be particularly useful in exploring the discrimination of differently coloured hair, e.g. naturally coloured, bleached or dyed. The NIR region is sensitive to molecular vibrations of the hair fibre structure as well as that of the dyes and damage from bleaching. But the Visible NIR region preferentially responds to the natural colourants, the melanin, which involves electronic transitions. This approach was shown to provide improved discrimination between dyed and untreated hair. This thesis is an extensive study of the application of NIRS with the aid of chemometrics, for matching and discrimination of bulk human scalp hair. The work not only indicates the strong potential of this technique in this field but also breaks new ground with the exploration of the use of the NIR and Visible NIR ranges for spectral sampling. It also develops methods for measuring spectra from hair which has been immersed in different water media (sea, river and dam)