Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, 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. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we 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 finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

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 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 fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients

In this paper, we derive a new nonlinear two-sided space-fractional diffusion equation with variable coefficients from the fractional Fick’s law. A semi-implicit difference method (SIDM) for this equation is proposed. The stability and convergence of the SIDM are discussed. For the implementation, we develop a fast accurate iterative method for the SIDM by decomposing the dense coefficient matrix into a combination of Toeplitz-like matrices. This fast iterative method significantly reduces the storage requirement of O(n2)O(n2) and computational cost of O(n3)O(n3) down to n and O(nlogn)O(nlogn), where n is the number of grid points. The method retains the same accuracy as the underlying SIDM solved with Gaussian elimination. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation

In this paper, we consider a two-sided space-fractional diffusion equation with variable coefficients on a finite domain. Firstly, based on the nodal basis functions, we present a new fractional finite volume method for the two-sided space-fractional diffusion equation and derive the implicit scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the implicit fractional finite volume method and conclude that the method is unconditionally stable and convergent. Finally, some numerical examples are given to show the effectiveness of the new numerical method, and the results are in excellent agreement with theoretical analysis.

Internet-enabled international entrepreneurship : a conceptual exploration of the link between a firm’s resource base, dynamic capabilities and international market performance

It has been suggested that the accumulation of valuable resources and capabilities, such as Internet application, is not enough to support a firm’s sustainable competitive advantage, especially for high technology-mediated firms; which often operate in fast changing dynamic environments. While the idea of ‘Internet-enabled resources and capabilities’ has been recognised by RBV theorists, the notion has largely been ignored in conceptual and empirical studies. Given this finding, a conceptual framework is constructed and research issues are then developed in order to focus attention on the relationship between, the Internet and a firm’s resource base, dynamic capabilities and international market performance. We postulate that successful Internet-enabled market performance arises from those international entrepreneurial-oriented firms which encompass: international vision, international business experience, Internet-international marketing capabilities and international networking capabilities. Recommendations for future theory development are presented, along with the implications for international entrepreneurial managers in Australian small and medium sized firms

Small firm internet adoption: A market oriented approach

Fundamental to the development of new customer value offerings via web-based commerce is a small firm's ability to strategically acquire and exploit knowledge. The focus of this paper is the empirical testing of a normative web-based commerce adoption model developed from a review of the extant literature related to electronic marketing, the Internet and the diffusion of new innovations. A preliminary test of the model's theoretical contentions lent support to its overall focus, but found that the firm's existing learning capabilities were diminished during the adoption of web-based commerce. Consequently, sub-optimal adoption outcomes were associated with insufficient knowledge development.

Small firm internet adoption: opportunities forgone, a journey not begun

This paper explores the endeavours of five small firms to develop web-based commerce capabilities within their existing operations. The focus is upon the strategic acquisition and exploitation of knowledge which underpins new value creating activates related to web-based commerce. A normative web-based commerce adoption model developed from a review of the extant literature related to electronic marketing, entrepreneurship, and the diffusion of new innovations was empirically tested. A multiple case study design enabled the exploration of contemporary marketing and entrepreneurship issues within the real life context of five small firms. The model aimed to emphasis best-practice adoption methods emphasizing the value of a firm's market orientation and entrepreneurial capabilities. A preliminary test of the model's theoretical contentions lent support to its overall focus, but found that the firm's existing learning capabilities were diminished during the adoption of web-based commerce, and that a lack of vision and prior knowledge produced sub-optimal adoption outcomes.

Dacomitinib versus erlotinib in patients with EGFR-mutated advanced nonsmall-cell lung cancer (NSCLC): Pooled subset analyses from two randomized trials

Background: The irreversible epidermal growth factor receptor (EGFR) inhibitors have demonstrated efficacy in NSCLC patients with activating EGFR mutations, but it is unknown if they are superior to the reversible inhibitors. Dacomitinib is an oral, small-molecule irreversible inhibitor of all enzymatically active HER family tyrosine kinases. Methods: The ARCHER 1009 (NCT01360554) and A7471028 (NCT00769067) studies randomized patients with locally advanced/metastatic NSCLC following progression with one or two prior chemotherapy regimens to dacomitinib or erlotinib. EGFR mutation testing was performed centrally on archived tumor samples. We pooled patients with exon 19 deletion and L858R EGFR mutations from both studies to compare the efficacy of dacomitinib to erlotinib. Results: One hundred twenty-one patients with any EGFR mutation were enrolled; 101 had activating mutations in exon 19 or 21. For patients with exon19/21 mutations, the median progression-free survival was 14.6 months [95% confidence interval (CI) 9.0–18.2] with dacomitinib and 9.6 months (95% CI 7.4–12.7) with erlotinib [unstratified hazard ratio (HR) 0.717 (95% CI 0.458–1.124), two-sided log-rank, P = 0.146]. The median survival was 26.6 months (95% CI 21.6–41.5) with dacomitinib versus 23.2 months (95% CI 16.0–31.8) with erlotinib [unstratified HR 0.737 (95% CI 0.431–1.259), two-sided log-rank, P = 0.265]. Dacomitinib was associated with a higher incidence of diarrhea and mucositis in both studies compared with erlotinib. Conclusions: Dacomitinib is an active agent with comparable efficacy to erlotinib in the EGFR mutated patients. The subgroup with exon 19 deletion had favorable outcomes with dacomitinib. An ongoing phase III study will compare dacomitinib to gefitinib in first-line therapy of patients with NSCLC harboring common activating EGFR mutations (ARCHER 1050; NCT01774721). Clinical trials number: ARCHER 1009 (NCT01360554) and A7471028 (NCT00769067).

On the Classification of Recursive Languages

A one-sided classifier for a given class of languages converges to 1 on every language from the class and outputs 0 infinitely often on languages outside the class. A two-sided classifier, on the other hand, converges to 1 on languages from the class and converges to 0 on languages outside the class. The present paper investigates one-sided and two-sided classification for classes of recursive languages. Theorems are presented that help assess the classifiability of natural classes. The relationships of classification to inductive learning theory and to structural complexity theory in terms of Turing degrees are studied. Furthermore, the special case of classification from only positive data is also investigated.

Risk of Human Papillomavirus-Associated Cancers Among Persons With AIDS

Background Although risk of human papillomavirus (HPV)–associated cancers of the anus, cervix, oropharynx, penis, vagina, and vulva is increased among persons with AIDS, the etiologic role of immunosuppression is unclear and incidence trends for these cancers over time, particularly after the introduction of highly active antiretroviral therapy in 1996, are not well described. Methods Data on 499 230 individuals diagnosed with AIDS from January 1, 1980, through December 31, 2004, were linked with cancer registries in 15 US regions. Risk of in situ and invasive HPV-associated cancers, compared with that in the general population, was measured by use of standardized incidence ratios (SIRs) and 95% confidence intervals (CIs). We evaluated the relationship of immunosuppression with incidence during the period of 4–60 months after AIDS onset by use of CD4 T-cell counts measured at AIDS onset. Incidence during the 4–60 months after AIDS onset was compared across three periods (1980–1989, 1990–1995, and 1996–2004). All statistical tests were two-sided. Results Among persons with AIDS, we observed statistically significantly elevated risk of all HPV-associated in situ (SIRs ranged from 8.9, 95% CI = 8.0 to 9.9, for cervical cancer to 68.6, 95% CI = 59.7 to 78.4, for anal cancer among men) and invasive (SIRs ranged from 1.6, 95% CI = 1.2 to 2.1, for oropharyngeal cancer to 34.6, 95% CI = 30.8 to 38.8, for anal cancer among men) cancers. During 1996–2004, low CD4 T-cell count was associated with statistically significantly increased risk of invasive anal cancer among men (relative risk [RR] per decline of 100 CD4 T cells per cubic millimeter = 1.34, 95% CI = 1.08 to 1.66, P = .006) and non–statistically significantly increased risk of in situ vagina or vulva cancer (RR = 1.52, 95% CI = 0.99 to 2.35, P = .055) and of invasive cervical cancer (RR = 1.32, 95% CI = 0.96 to 1.80, P = .077). Among men, incidence (per 100 000 person-years) of in situ and invasive anal cancer was statistically significantly higher during 1996–2004 than during 1990–1995 (61% increase for in situ cancers, 18.3 cases vs 29.5 cases, respectively; RR = 1.71, 95% CI = 1.24 to 2.35, P < .001; and 104% increase for invasive cancers, 20.7 cases vs 42.3 cases, respectively; RR = 2.03, 95% CI = 1.54 to 2.68, P < .001). Incidence of other cancers was stable over time. Conclusions Risk of HPV-associated cancers was elevated among persons with AIDS and increased with increasing immunosuppression. The increasing incidence for anal cancer during 1996–2004 indicates that prolonged survival may be associated with increased risk of certain HPV-associated cancers.