Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Order conditions of stochastic Runge-Kutta methods by B-series

In this paper, general order conditions and a global convergence proof are given for stochastic Runge Kutta methods applied to stochastic ordinary differential equations ( SODEs) of Stratonovich type. This work generalizes the ideas of B-series as applied to deterministic ordinary differential equations (ODEs) to the stochastic case and allows a completely general formalism for constructing high order stochastic methods, either explicit or implicit. Some numerical results will be given to illustrate this theory.

Time-aware topic recommendation based on micro-blogs

Topic recommendation can help users deal with the information overload issue in micro-blogging communities. This paper proposes to use the implicit information network formed by the multiple relationships among users, topics and micro-blogs, and the temporal information of micro-blogs to find semantically and temporally relevant topics of each topic, and to profile users' time-drifting topic interests. The Content based, Nearest Neighborhood based and Matrix Factorization models are used to make personalized recommendations. The effectiveness of the proposed approaches is demonstrated in the experiments conducted on a real world dataset that collected from Twitter.com.

On robust face recognition via sparse encoding : the good, the bad, and the ugly

In the field of face recognition, Sparse Representation (SR) has received considerable attention during the past few years. Most of the relevant literature focuses on holistic descriptors in closed-set identification applications. The underlying assumption in SR-based methods is that each class in the gallery has sufficient samples and the query lies on the subspace spanned by the gallery of the same class. Unfortunately, such assumption is easily violated in the more challenging face verification scenario, where an algorithm is required to determine if two faces (where one or both have not been seen before) belong to the same person. In this paper, we first discuss why previous attempts with SR might not be applicable to verification problems. We then propose an alternative approach to face verification via SR. Specifically, we propose to use explicit SR encoding on local image patches rather than the entire face. The obtained sparse signals are pooled via averaging to form multiple region descriptors, which are then concatenated to form an overall face descriptor. Due to the deliberate loss spatial relations within each region (caused by averaging), the resulting descriptor is robust to misalignment & various image deformations. Within the proposed framework, we evaluate several SR encoding techniques: l1-minimisation, Sparse Autoencoder Neural Network (SANN), and an implicit probabilistic technique based on Gaussian Mixture Models. Thorough experiments on AR, FERET, exYaleB, BANCA and ChokePoint datasets show that the proposed local SR approach obtains considerably better and more robust performance than several previous state-of-the-art holistic SR methods, in both verification and closed-set identification problems. The experiments also show that l1-minimisation based encoding has a considerably higher computational than the other techniques, but leads to higher recognition rates.

The relationship between CFH and ARMS2 genotypes and neuroretinal function in persons without age-related macular degeneration

Purpose: To determine whether neuroretinal function differs in healthy persons with and without common risk gene variants for age- related macular degeneration (AMD) and no ophthalmoscopic signs of AMD, and to compare those findings in persons with manifest early AMD. Methods and Participants: Neuroretinal function was assessed with the multifocal electroretinogram (mfERG) (VERIS, Redwood City, CA,) in 32 participants (22 healthy persons with no clinical signs of AMD and 10 early AMD patients). The 22 healthy participants with no AMD were risk genotypes for either the CFH (rs380390) and/or ARMS2 (rs10490920). We used a slow flash mfERG paradigm (3 inserted frames) and a 103 hexagon stimulus array. Recordings were made with DTL electrodes; fixation and eye movements were monitored online. Trough N1 to peak P1 (N1P1) response densities and P1-implicit times (IT) were analysed in 5 concentric rings. Results: N1P1 response densities (mean ± SD) for concentric rings 1-3 were on average significantly higher in at-risk genotypes (ring 1: 17.97 nV/deg2 ± 1.9, ring 2: 11.7 nV/deg2 ±1.3, ring 3: 8.7 nV/deg2 ± 0.7) compared to those without risk (ring 1: 13.7 nV/deg2 ± 1.9, ring 2: 9.2 nV/deg2 ±0.8, ring 3: 7.3 nV/deg2 ± 1.1) and compared to persons with early AMD (ring 1: 15.3 nV/deg2 ± 4.8, ring 2: 9.1 nV/deg2 ±2.3, ring 3 nV/deg2: 7.3± 1.3) (p<0.5). The group implicit times, P1-ITs for ring 1 were on average delayed in the early AMD patients (36.4 ms ± 1.0) compared to healthy participants with (35.1 ms ± 1.1) or without risk genotypes (34.8 ms ±1.3), although these differences were not significant. Conclusion: Neuroretinal function in persons with normal fundi can be differentiated into subgroups based on their genetics. Increased neuroretinal activity in persons who carry AMD risk genotypes may be due to genetically determined subclinical inflammatory and/or histological changes in the retina. Assessment of neuroretinal function in healthy persons genetically susceptible to AMD may be a useful early biomarker before there is clinical manifestation of AMD.

'Pretend I was Mummy': Children's Production of Authority and Subordinance in Their Pretend Play Interaction During Disputes

Purpose – This chapter examines an episode of pretend play amongst a group of young girls in an elementary school in Australia, highlighting how they interact within the membership categorization device ‘family’ to manage their social and power relationships. Approach – Using conversation analysis and membership categorization analysis, an episode of video-recorded interaction that occurs amongst a group of four young girls is analyzed. Findings – As disputes arise amongst the girls, the mother category is produced as authoritative through authoritative actions by the girl in the category of mother, and displays of subordination on the part of the other children, in the categories of sister, dog and cat. Value of paper – Examining play as a social practice provides insight into the social worlds of children. The analysis shows how the children draw upon and co-construct family-style relationships in a pretend play context, in ways that enable them to build and organize peer interaction. Authority is highlighted as a joint accomplishment that is part of the social and moral order continuously being negotiated by the children. The authority of the mother category is produced and oriented to as a means of managing the disputes within the pretend frame of play.

Exponential integrators and a dual-scale model for wood drying

For the timber industry, the ability to simulate the drying of wood is invaluable for manufacturing high quality wood products. Mathematically, however, modelling the drying of a wet porous material, such as wood, is a diffcult task due to its heterogeneous and anisotropic nature, and the complex geometry of the underlying pore structure. The well{ developed macroscopic modelling approach involves writing down classical conservation equations at a length scale where physical quantities (e.g., porosity) can be interpreted as averaged values over a small volume (typically containing hundreds or thousands of pores). This averaging procedure produces balance equations that resemble those of a continuum with the exception that effective coeffcients appear in their deffnitions. Exponential integrators are numerical schemes for initial value problems involving a system of ordinary differential equations. These methods differ from popular Newton{Krylov implicit methods (i.e., those based on the backward differentiation formulae (BDF)) in that they do not require the solution of a system of nonlinear equations at each time step but rather they require computation of matrix{vector products involving the exponential of the Jacobian matrix. Although originally appearing in the 1960s, exponential integrators have recently experienced a resurgence in interest due to a greater undertaking of research in Krylov subspace methods for matrix function approximation. One of the simplest examples of an exponential integrator is the exponential Euler method (EEM), which requires, at each time step, approximation of φ(A)b, where φ(z) = (ez - 1)/z, A E Rnxn and b E Rn. For drying in porous media, the most comprehensive macroscopic formulation is TransPore [Perre and Turner, Chem. Eng. J., 86: 117-131, 2002], which features three coupled, nonlinear partial differential equations. The focus of the first part of this thesis is the use of the exponential Euler method (EEM) for performing the time integration of the macroscopic set of equations featured in TransPore. In particular, a new variable{ stepsize algorithm for EEM is presented within a Krylov subspace framework, which allows control of the error during the integration process. The performance of the new algorithm highlights the great potential of exponential integrators not only for drying applications but across all disciplines of transport phenomena. For example, when applied to well{ known benchmark problems involving single{phase liquid ow in heterogeneous soils, the proposed algorithm requires half the number of function evaluations than that required for an equivalent (sophisticated) Newton{Krylov BDF implementation. Furthermore for all drying configurations tested, the new algorithm always produces, in less computational time, a solution of higher accuracy than the existing backward Euler module featured in TransPore. Some new results relating to Krylov subspace approximation of '(A)b are also developed in this thesis. Most notably, an alternative derivation of the approximation error estimate of Hochbruck, Lubich and Selhofer [SIAM J. Sci. Comput., 19(5): 1552{1574, 1998] is provided, which reveals why it performs well in the error control procedure. Two of the main drawbacks of the macroscopic approach outlined above include the effective coefficients must be supplied to the model, and it fails for some drying configurations, where typical dual{scale mechanisms occur. In the second part of this thesis, a new dual{scale approach for simulating wood drying is proposed that couples the porous medium (macroscale) with the underlying pore structure (microscale). The proposed model is applied to the convective drying of softwood at low temperatures and is valid in the so{called hygroscopic range, where hygroscopically held liquid water is present in the solid phase and water exits only as vapour in the pores. Coupling between scales is achieved by imposing the macroscopic gradient on the microscopic field using suitably defined periodic boundary conditions, which allows the macroscopic ux to be defined as an average of the microscopic ux over the unit cell. This formulation provides a first step for moving from the macroscopic formulation featured in TransPore to a comprehensive dual{scale formulation capable of addressing any drying configuration. Simulation results reported for a sample of spruce highlight the potential and flexibility of the new dual{scale approach. In particular, for a given unit cell configuration it is not necessary to supply the effective coefficients prior to each simulation.

Non-Newtonian natural convection flow along an isothermal horizontal circular cylinder using modified power-law model

Laminar two-dimensional natural convection boundary-layer flow of non-Newtonian fluids along an isothermal horizontal circular cylinder has been studied using a modified power-law viscosity model. In this model, there are no unrealistic limits of zero or infinite viscosity. Therefore, the boundary-layer equations can be solved numerically by using marching order implicit finite difference method with double sweep technique. Numerical results are presented for the case of shear-thinning as well as shear thickening fluids in terms of the fluid velocity and temperature distributions, shear stresses and rate of heat transfer in terms of the local skin-friction and local Nusselt number respectively.

I know, but I don’t care : how awareness of Queensland’s drug driving testing methods impact upon perceptions of deterrence and offending behaviours

An increasing body of research is highlighting the involvement of illicit drugs in many road fatalities. Deterrence theory has been a core conceptual framework underpinning traffic enforcement as well as interventions designed to reduce road fatalities. Essentially the effectiveness of deterrence-based approaches is predicated on perceptions of certainty, severity, and swiftness of apprehension. However, much less is known about how the awareness of legal sanctions can impact upon the effectiveness of deterrence mechanisms and whether promoting such detection methods can increase the deterrent effect. Nevertheless, the implicit assumption is that individuals aware of the legal sanctions will be more deterred. This study seeks to explore how awareness of the testing method impacts upon the effectiveness of deterrence-based interventions and intentions to drug drive again in the future. In total, 161 participants who reported drug driving in the previous six months took part in the current study. The results show that awareness of testing had a small effect upon increasing perceptions of the certainty of apprehension and severity of punishment. However, awareness was not a significant predictor of intentions to drug drive again in the future. Importantly, higher levels of drug use were a significant predictor of intentions to drug drive in the future. Whilst awareness does have a small effect on deterrence variables, the influence of levels of drug use seems to reduce any deterrent effect.

Integrating collaborative filtering and matching-based search for product recommendation

Currently, recommender systems (RS) have been widely applied in many commercial e-commerce sites to help users deal with the information overload problem. Recommender systems provide personalized recommendations to users and thus help them in making good decisions about which product to buy from the vast number of product choices available to them. Many of the current recommender systems are developed for simple and frequently purchased products like books and videos, by using collaborative-filtering and content-based recommender system approaches. These approaches are not suitable for recommending luxurious and infrequently purchased products as they rely on a large amount of ratings data that is not usually available for such products. This research aims to explore novel approaches for recommending infrequently purchased products by exploiting user generated content such as user reviews and product click streams data. From reviews on products given by the previous users, association rules between product attributes are extracted using an association rule mining technique. Furthermore, from product click streams data, user profiles are generated using the proposed user profiling approach. Two recommendation approaches are proposed based on the knowledge extracted from these resources. The first approach is developed by formulating a new query from the initial query given by the target user, by expanding the query with the suitable association rules. In the second approach, a collaborative-filtering recommender system and search-based approaches are integrated within a hybrid system. In this hybrid system, user profiles are used to find the target user’s neighbour and the subsequent products viewed by them are then used to search for other relevant products. Experiments have been conducted on a real world dataset collected from one of the online car sale companies in Australia to evaluate the effectiveness of the proposed recommendation approaches. The experiment results show that user profiles generated from user click stream data and association rules generated from user reviews can improve recommendation accuracy. In addition, the experiment results also prove that the proposed query expansion and the hybrid collaborative filtering and search-based approaches perform better than the baseline approaches. Integrating the collaborative-filtering and search-based approaches has been challenging as this strategy has not been widely explored so far especially for recommending infrequently purchased products. Therefore, this research will provide a theoretical contribution to the recommender system field as a new technique of combining collaborative-filtering and search-based approaches will be developed. This research also contributes to a development of a new query expansion technique for infrequently purchased products recommendation. This research will also provide a practical contribution to the development of a prototype system for recommending cars.

Critical attributes of public-private partnerships : a case study in vocational education

The term ‘partnership’ is increasingly used by governments, industry, community organisations and schools in supporting their daily businesses. Similar to the terms ‘ICT’ and ‘learning’, ‘partnerships’ are now ubiquitous in policy discourse. Yet, the term remains ill-defined and ambiguous. This study reviews and reflects on a government led industry-school partnership initiative in the state of Queensland, Australia, to understand how the term is used in this initiative. Given the frequent use of Public Private Partnership (PPP) language, PPP was used as a framework to review this initiative. The methodology of this qualitative case study involved consultations with stakeholders and an analysis of Gateway schools documents, policy documents, and literature. The review suggests that despite the use of terminology akin to PPP projects in Gateway school and policy documents, the implicit suggestion that this initiative is a public-private partnership is untenable. The majority of principles shaping a PPP have not been considered to a significant extent in the Gateway project. Although the review recognises the legitimate and sincere purpose of the Gateway schools initiative, the adoption of a PPP framework during the design, monitoring, or evaluation stages could have strengthened the initiative in terms of outcomes, benefits, and sustainability.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusionwave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

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.