Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

Stratospheric dust collections : valuable resources for space and atmospheric scientists

Collections of solid particles from the Earths' stratosphere have been a significant part of atmospheric research programs since 1965 [1], but it has only been in the past decade that space-related disciplines have provided the impetus for a continued interest in these collections. Early research on specific particle types collected from the stratosphere established that interplanetary dust particles (IDP's) can be collected efficiently and in reasonable abundance using flat-plate collectors [2-4]. The tenacity of Brownlee and co-workers in this subfield of cosmochemistry has led to the establishment of a successful IDP collection and analysis program (using flat-plate collectors on high-flying aircraft) based on samples available for distribution from Johnson Space Center [5]. Other stratospheric collections are made, but the program at JSC offers a unique opportunity to study well-documented, individual particles (or groups of particles) from a wide variety of sources [6]. The nature of the collection and curation process, as well as the timeliness of some sampling periods [7], ensures that all data obtained from stratospheric particles is a valuable resource for scientists from a wide range of disciplines. A few examples of the uses of these stratospheric dust collections are outlined below.

A novel numerical approximation for the space fractional advection-dispersion equation

In this paper, we consider a space fractional advection–dispersion equation. The equation is obtained from the standard advection–diffusion equation by replacing the first- and second-order space derivatives by the Riesz fractional derivatives of order β1 ∈ (0, 1) and β2 ∈ (1, 2], respectively. The fractional advection and dispersion terms are approximated by using two fractional centred difference schemes. A new weighted Riesz fractional finite-difference approximation scheme is proposed. When the weighting factor θ equals 12, a second-order accuracy scheme is obtained. The stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

A comparison of finite difference and finite volume methods for solving the space fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Time and space in biogeography : response to Parenti & Ebach (2013)

A recent Guest Editorial by Parenti & Ebach (2013, Journal of Biogeography, 40, 813–820) disagrees with the methods or interpretations in two of our recent papers. In addition, the authors open a debate on biogeographical concepts, and present an alternative philosophy for biogeographical research in the context of their recently described biogeographical subregion called ‘Pandora’. We disagree with their approach and conclusions, and comment on several issues related to our differing conceptual approaches for biogeographical research; namely, our use of molecular phylogenetic analyses, including time estimates; and Parenti & Ebach's reliance on taxon/general area cladograms. Finally, we re-examine their ‘tests’ supporting the existence of ‘Pandora’.

Public Space and the Marginalisation of Children and Young People

Throughout much of the world, urban and rural public spaces may be said to be under attack by property developers, commercial interests and also attempts by civic authorities to regulate, restrict, reframe and rebrand these spaces. A consequence of the increasingly security driven, privatised, commercial and surveilled nature of public space is the exclusion and displacement of those considered ‘flawed’ and unwelcome in the ‘spectacular’ consumption spaces of many major urban centres. In the name of urban regeneration, processes of securitisation, ‘gentrification’ and creative cities initiatives can act to refashion public space as sites of selective inclusion and exclusion. The use of surveillance and other control technologies as deployed in and around the UK ‘Riots’ of 2011 may help to promote and encourage a passing sense of personal safety and confidence in using public space. Through systems of social sorting, the same surveillance assemblages can also further the physical, emotional and psychological exclusion of certain groups and individuals, deemed to be both ‘out of time and out of place’ in major zones of urban, conspicuous, consumption. In this harsh environment of monitoring and control procedures, children and young people’s use of public spaces and places in parks, neighbourhoods, shopping malls and streets is often viewed as a threat to social order, requiring various forms of punitive and/or remedial action. Much of this civic action actively excludes some children and young people from participation and as a consequence, their trust in local processes and communities is eroded. This paper discusses worldwide developments in the surveillance, governance and control of the public space environments used by children and young people in particular and the capacity for their displacement and marginality, diminishing their sense of belonging, wellbeing and rights to public space as an expression of their social, political and civil citizenship(s).

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

We present a finite volume method to solve the time-space two-sided fractional advection-dispersion equation on a one-dimensional domain. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes. We demonstrate how the finite volume formulation provides a natural, convenient and accurate means of discretising this equation in conservative form, compared to using a conventional finite difference approach. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non- classical diffusion or dispersion which is not adequately described by the classical theory of Brownian motion and Fick’s law. We consider a space-fractional advection-dispersion equation based on a fractional Fick’s law. Zhang et al. [Water Resources Research, 43(5)(2007)] considered such an equation with variable coefficients, which they dis- cretised using the finite difference method proposed by Meerschaert and Tadjeran [Journal of Computational and Applied Mathematics, 172(1):65-77 (2004)]. For this method the presence of variable coef- ficients necessitates applying the product rule before discretising the Riemann–Liouville fractional derivatives using standard and shifted Gru ̈nwald formulas, depending on the fractional order. As an alternative, we propose using a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Gru ̈nwald formulas are used to discretise the Riemann–Liouville fractional derivatives at control volume faces, eliminating the need for product rule expansions. We compare the two methods for several case studies, highlighting the convenience of the finite volume approach.

An arteriovenous loop in a protected space generates a permanent, highly vascular, tissue-engineered construct

A major obstacle to 3-dimensional tissue engineering is incorporation of a functional vascular supply to support the expanding new tissue. This is overcome in an in vivo intrinsic vascularization model where an arteriovenous loop (AVL) is placed in a noncollapsible space protected by a polycarbonate chamber. Vascular development and hypoxia were examined from 3 days to 112 days by vascular casting, morphometric, and morphological techniques to understand the model's vascular growth and remodeling parameters for tissue engineering purposes. At 3 days a fibrin exudate surrounded the AVL, providing a scaffold to migrating inflammatory, endothelial, and mesenchymal cells. Capillaries formed between 3 and 7 days. Hypoxia and cell proliferation were maximal at 7 days, followed by a peak in percent vascular volume at 10 days (23.20±3.14% compared with 3.59±2.68% at 3 days, P<0.001). Maximal apoptosis was observed at 112 days. The protected space and spontaneous microcirculatory development in this model suggest it would be applicable for in vivo tissue engineering. A temporal window in a period of intense angiogenesis at 7 to 10 days is optimal for exogenous cell seeding and survival in the chamber, potentially enabling specific tissue outcomes to be achieved.

The QUT innovation space : a trans-disciplinary learning environment for entrepreneurship education : final report

In the current climate of global economic volatility, there are increasing calls for training in enterprising skills and entrepreneurship to underpin the systemic innovation required for even medium-term business sustainability. The skills long-recognised as the essential for entrepreneurship now appear on the list of employability skills demanded by industry. The QUT Innovation Space (QIS) was an experiment aimed at delivering entrepreneurship education (EE), as an extra-curricular platform across the university, to the undergraduate students of an Australian higher education institute. It was an ambitious project that built on overseas models of EE studied during an Australian Learning and Teaching Council (ALTC) Teaching Fellowship (Collet, 2011) and implemented those approaches across an institute. Such EE approaches have not been attempted in an Australian university. The project tested resonance not only with the student population, from the perspective of what worked and what didn’t work, but also with every level of university operations. Such information is needed to inform the development of EE in the Australian university landscape. The QIS comprised a physical co-working space, virtual sites (web, Twitter and Facebook) and a network of entrepreneurial mentors, colleagues, and students. All facets of the QIS enabled connection between like-minded individuals that underpins the momentum needed for a project of this nature. The QIS became an innovation community within QUT. This report serves two purposes. First, as an account of the QIS project and its evolution, the report serves to identify the student demand for skills and training as well as barriers and facilitators of the activities that promote EE in an Australian university context. Second, the report serves as a how-to manual, in the tradition of many tomes on EE, outlining the QIS activities that worked as well as those that failed. The activities represent one measure of QIS outcomes and are described herein to facilitate implementation in other institutes. The QIS initially aimed to adopt an incubation model for training in EE. The ‘learning by doing’ model for new venture creation is a highly successful and high profile training approach commonly found in overseas contexts. However, the greatest demand of the QUT student population was not for incubation and progression of a developed entrepreneurial intent, but rather for training that instilled enterprising skills in the individual. These two scenarios require different training approaches (Fayolle and Gailly, 2008). The activities of the QIS evolved to meet that student demand. In addressing enterprising skills, the QIS developed the antecedents of entrepreneurialism (i.e., entrepreneurial attitudes, motivation and behaviours) including high-level skills around risk-taking, effective communication, opportunity recognition and action-orientation. In focusing on the would-be entrepreneur and not on the (initial) idea per se, the QIS also fostered entrepreneurial outcomes that would never have gained entry to the rigid stage-gated incubation model proposed for the original QIS framework. Important lessons learned from the project for development of an innovation community include the need to: 1. Evaluate the context of the type of EE program to be delivered and the student demand for the skills training (as noted above). 2. Create a community that builds on three dimensions: a physical space, a virtual environment and a network of mentors and partners. 3. Supplement the community with external partnerships that aid in delivery of skills training materials. 4. Ensure discovery of the community through the use of external IT services to deliver advertising and networking outlets. 5. Manage unrealistic student expectations of billion dollar products. 6. Continuously renew and rebuild simple activities to maintain student engagement. 7. Accommodate the non-university end-user group within the community. 8. Recognise and address the skills bottlenecks that serve as barriers to concept progression; in this case, externally provided IT and programming skills. 9. Use available on-line and published resources rather than engage in constructing project-specific resources that quickly become obsolete. 10. Avoid perceptions of faculty ownership and operate in an increasingly competitive environment. 11. Recognise that the continuum between creativity/innovation and entrepreneurship is complex, non-linear and requires different training regimes during the different phases of the pipeline. One small entity, such as the QIS, cannot address them all. The QIS successfully designed, implemented and delivered activities that included events, workshops, seminars and services to QUT students in the extra-curricular space. That the QIS project can be considered successful derives directly from the outcomes. First, the QIS project changed the lives of emerging QUT student entrepreneurs. Also, the QIS activities developed enterprising skills in students who did not necessarily have a business proposition, at the time. Second, successful outcomes of the QIS project are evidenced as the embedding of most, perhaps all, of the QIS activities in a new Chancellery-sponsored initiative: the Leadership Development and Innovation Program hosted by QUT Student Support Services. During the course of the QIS project, the Brisbane-based innovation ecosystem underwent substantial change. From a dearth of opportunities for the entrepreneurially inclined, there is now a plethora of entities that cater for a diversity of innovation-related activities. While the QIS evolved with the landscape, the demand endpoint of the QIS activities still highlights a gap in the local and national innovation ecosystems. The freedom to experiment and to fail is not catered for by the many new entities seeking to build viable businesses on the back of the innovation push. The onus of teaching the enterprising skills, which are the employability skills now demanded by industry, remains the domain of the higher education sector.

A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

Fractional differential equations have been increasingly used as a powerful tool to model the non-locality and spatial heterogeneity inherent in many real-world problems. However, a constant challenge faced by researchers in this area is the high computational expense of obtaining numerical solutions of these fractional models, owing to the non-local nature of fractional derivatives. In this paper, we introduce a finite volume scheme with preconditioned Lanczos method as an attractive and high-efficiency approach for solving two-dimensional space-fractional reaction–diffusion equations. The computational heart of this approach is the efficient computation of a matrix-function-vector product f(A)bf(A)b, where A A is the matrix representation of the Laplacian obtained from the finite volume method and is non-symmetric. A key aspect of our proposed approach is that the popular Lanczos method for symmetric matrices is applied to this non-symmetric problem, after a suitable transformation. Furthermore, the convergence of the Lanczos method is greatly improved by incorporating a preconditioner. Our approach is show-cased by solving the fractional Fisher equation including a validation of the solution and an analysis of the behaviour of the model.

The role of active locomotion in space perception

It has been shown that active control of locomotion increases accuracy and precision of nonvisual space perception, but psychological mechanisms of this enhancement are poorly understood. The present study explored a hypothesis that active control of locomotion enhances space perception by facilitating crossmodal interaction between visual and nonvisual spatial information. In an experiment, blindfolded participants walked along a linear path under one of the following two conditions: (1) They walked by themselves following a guide rope; and (2) they were led by an experimenter. Subsequently, they indicated the walked distance by tossing a beanbag to the origin of locomotion. The former condition gave participants greater control of their locomotion, and thus represented a more active walking condition. In addition, before each trial, half the participants viewed the room in which they performed the distance perception task. The other half remained blindfolded throughout the experiment. Results showed that although the room was devoid of any particular cues for walked distances, visual knowledge of the surroundings improved the precision of nonvisual distance perception. Importantly, however, the benefit of preview was observed only when participants walked more actively. This indicates that active control of locomotion allowed participants to better utilize their visual memory of the environment for perceiving nonvisually encoded distance, suggesting that active control of locomotion served as a catalyst for integrating visual and nonvisual information to derive spatial representations of higher quality.