Numerical solutions of stochastic differential equations – implementation and stability issues

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Urban renewal, migration and memories : the affordances of place-based pedagogies for developing immigrant students’ literate repertoires

This paper reports on the new literacy demands in the middle years of schooling project in which the affordances of placed-based pedagogy are being explored through teacher inquiries and classroom-based design experiments. The school is located within a large-scale urban renewal project in which houses are being demolished and families relocated. The original school buildings have recently been demolished and replaced by a large ‘superschool’ which serves a bigger student population from a wider area. Drawing on both quantitative and qualitative data, the teachers reported that the language literacy learning of students (including a majority of students learning English as a second language) involved in the project exceeded their expectations. The project provided the motivation for them to develop their oral language repertoires, by involving them in processes such as conducting interviews with adults for their oral histories, through questioning the project manager in regular meetings, and through reporting to their peers and the wider community at school assemblies. At the same time students’ written and multimodal documentation of changes in the neighbourhood and the school grounds extended their literate and semiotic repertoires as they produced books, reports, films, powerpoints, visual designs and models of structures.

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.

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 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.

Exploring subscription renewal intention of operational cloud enterprise systems - a stakeholder perspective

Retaining customers is a relevant topic throughout all service industries. However, only limited attention has been directed towards studying the antecedents of subscription renewal in the context of operational cloud enterprise systems. Cloud services have historically been offered as subscription-based services with the (theoretical) possibility of seamless service cancellation, in contrast to classical IT-Outsourcing contracts or license-based software installations of on-premise enterprise systems. In this work, we investigate the central concept of subscription renewal by focusing on different facets of IS success and their relevance for distinct employee cohorts. Analyzing inter-cohort differences has strong practical implications, as it helps IT vendors to focus on specific IT-related factors when trying to retain customers. Therefore an empirical study was undertaken. The hypotheses were developed on an individual level and tested using survey responses of IT decision makers within companies which adopted cloud enterprise systems. Gathered data was then analyzed using PLS. The results show that subscription renewal intention of the strategic cohort is mainly based on perceived system quality, whereas information quality explains most of the variance of subscription renewal in the management cohort. Beneath the cloud enterprise systems specific contributions, the work adds to the theoretical body of research related to IS success and IS continuation, as well as stakeholder perspectives.

A micro perspective analysis of capability renewal through knowledge integration within product innovation projects

The aim of this paper is examine how firms renew their organisational capabilities based on micro organisational processes. Organisational capability development literature points to firms’ failure in capability renewal process. To overcome this inefficiency, it is proposed to integrate dynamic capability and ambidexterity perspectives by studying knowledge integration within product innovation. In this relation, applying micro perspective in studying technology diffusion within Iranian Auto industry revealed micro co-evolutionary relationships between knowledge integration within product innovation and capability development. Furthermore, based on near decomposability principals, the analysis suggested relationships among modularity of product architecture, modularity of organisational modularity and modularity of industry architecture in downstream and upstream value chain. Based on these micro-macro co evolutionary effects, capability development process underlying successful corporate entrepreneurship may be verified.

A simplified approach for calculating the moments of action for linear reaction-diffusion equations

The mean action time is the mean of a probability density function that can be interpreted as a critical time, which is a finite estimate of the time taken for the transient solution of a reaction-diffusion equation to effectively reach steady state. For high-variance distributions, the mean action time under-approximates the critical time since it neglects to account for the spread about the mean. We can improve our estimate of the critical time by calculating the higher moments of the probability density function, called the moments of action, which provide additional information regarding the spread about the mean. Existing methods for calculating the nth moment of action require the solution of n nonhomogeneous boundary value problems which can be difficult and tedious to solve exactly. Here we present a simplified approach using Laplace transforms which allows us to calculate the nth moment of action without solving this family of boundary value problems and also without solving for the transient solution of the underlying reaction-diffusion problem. We demonstrate the generality of our method by calculating exact expressions for the moments of action for three problems from the biophysics literature. While the first problem we consider can be solved using existing methods, the second problem, which is readily solved using our approach, is intractable using previous techniques. The third problem illustrates how the Laplace transform approach can be used to study coupled linear reaction-diffusion equations.

Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise

There has been considerable recent work on the development of energy conserving one-step methods that are not symplectic. Here we extend these ideas to stochastic Hamiltonian problems with additive noise and show that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep. Some numerical simulations illustrate the performance of these methods.

Curriculum renewal in legal education

This project investigated ways in which the learning experience for students in Australian law schools could be enhanced by renewing final year legal curriculum through the design of effective capstone experiences to close the loop on tertiary legal studies and better prepare students for a smooth transition into the world of work and professional practice. Key project outcomes are a set of final year curriculum design principles and a transferable model for an effective final year program – a final year Toolkit comprising a range of templates, models and specific capstone examples for adoption or adaptation by legal educators. The project found that the efficacy of capstone experiences is affected by the curriculum context within which they are offered. For this reason, a number of ‘favourable conditions’, which promote the effectiveness of capstone experiences, have also been identified. The project’s final year principles and Toolkit promote program coherence and integration, should increase student satisfaction and levels of engagement with their experience of legal education and make a valuable contribution to assurance of learning in the new Tertiary Education Quality and Standards Agency (TEQSA) environment. From the point of view of the student experience, the final year principles and models address the current fragmented approach to final year legal curricula design and delivery. The knowledge and research base acquired under the auspices of this project is of both discipline and national importance as the project’s outcomes are transferable and have the potential to significantly influence the quality and coherence of the program experience of final year students in other tertiary disciplines, both within Australia and beyond. Project outcomes and deliverables are available on both the project’s website http://wiki.qut.edu.au/display/capstone/Home and on the Law Capstone Experience Forum website http://www.lawcapstoneexperience.com/. In the course of developing its deliverables, the project found that the design of capstone experiences varies significantly within and across disciplines; different frameworks may be used (for example, a disciplinary or inter-disciplinary focus, or to satisfy professional accreditation requirements), rationales and objectives may differ, and a variety of models utilised (for example, an integrated final year program, a single subject, a suite of subjects, or modules within several subjects). Broadly however, capstone experiences should provide final year students with an opportunity both to look back over their academic learning, in an effort to make sense of what they have accomplished, and to look forward to their professional and personal futures that build on that foundational learning.

Development and validation of anthropometric prediction equations for estimation of lean body mass and appendicular lean soft tissue in Indian men and women

Lean body mass (LBM) and muscle mass remains difficult to quantify in large epidemiological studies due to non-availability of inexpensive methods. We therefore developed anthropometric prediction equations to estimate the LBM and appendicular lean soft tissue (ALST) using dual energy X-ray absorptiometry (DXA) as a reference method. Healthy volunteers (n= 2220; 36% females; age 18-79 y) representing a wide range of body mass index (14-44 kg/m2) participated in this study. Their LBM including ALST was assessed by DXA along with anthropometric measurements. The sample was divided into prediction (60%) and validation (40%) sets. In the prediction set, a number of prediction models were constructed using DXA measured LBM and ALST estimates as dependent variables and a combination of anthropometric indices as independent variables. These equations were cross-validated in the validation set. Simple equations using age, height and weight explained > 90% variation in the LBM and ALST in both men and women. Additional variables (hip and limb circumferences and sum of SFTs) increased the explained variation by 5-8% in the fully adjusted models predicting LBM and ALST. More complex equations using all the above anthropometric variables could predict the DXA measured LBM and ALST accurately as indicated by low standard error of the estimate (LBM: 1.47 kg and 1.63 kg for men and women, respectively) as well as good agreement by Bland Altman analyses. These equations could be a valuable tool in large epidemiological studies assessing these body compartments in Indians and other population groups with similar body composition.

Exploring influencing factors of short-lived buildings in China's urban renewal : evidence from Chongqing

With the fast development of urban sprawl and renewal in China, many buildings are “non-nature” short-lived, i.e. demolished after only a few years. For this concern, this research explores the influencing factors of short-lived buildings and provides the scientific foundation for sustainable urban management and planning. Cases for this research are 1734 buildings demolished in Jiangbei district, the middle region of Chongqing City. Internal and external factors for the short-lived buildings are identified by applying logistic analysis. The results indicate that nine factors have significant influence on short-lived buildings. This research also find that buildings with low density, utilization and compensation while high land development potential are more likely to become short-lived buildings.