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.

High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations

The pioneering work of Runge and Kutta a hundred years ago has ultimately led to suites of sophisticated numerical methods suitable for solving complex systems of deterministic ordinary differential equations. However, in many modelling situations, the appropriate representation is a stochastic differential equation and here numerical methods are much less sophisticated. In this paper a very general class of stochastic Runge-Kutta methods is presented and much more efficient classes of explicit methods than previous extant methods are constructed. In particular, a method of strong order 2 with a deterministic component based on the classical Runge-Kutta method is constructed and some numerical results are presented to demonstrate the efficacy of this approach.

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.

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.

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.

Cement-in-cement revision for selected Vancouver Type B1 femoral periprosthetic fractures : A biomechanical analysis

The aim of this study was to perform a biomechanical analysis of the cement-in-cement (c-in-c) technique for fixation of selected Vancouver Type B1 femoral periprosthetic fractures and to assess the degree of cement interposition at the fracture site. Six embalmed cadaveric femora were implanted with a cemented femoral stem. Vancouver Type B1 fractures were created by applying a combined axial and rotational load to failure. The femora were repaired using the c-in-c technique and reloaded to failure. The mean primary fracture torque was 117 Nm (SD 16.6, range 89–133). The mean revision fracture torque was 50 Nm (SD 16.6, range 29–74), which is above the torque previously observed for activities of daily living. Cement interposition at the fracture site was found to be minimal.

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.

Visualization of thermal characteristics around the absorber tube of a standard parabolic trough thermal collector by 3D simulation

Three dimensional conjugate heat transfer simulation of a standard parabolic trough thermal collector receiver is performed numerically in order to visualize and analyze the surface thermal characteristics. The computational model is developed in Ansys Fluent environment based on some simplified assumptions. Three test conditions are selected from the existing literature to verify the numerical model directly, and reasonably good agreement between the model and the test results confirms the reliability of the simulation. Solar radiation flux profile around the tube is also approximated from the literature. An in house macro is written to read the input solar flux as a heat flux wall boundary condition for the tube wall. The numerical results show that there is an abrupt variation in the resultant heat flux along the circumference of the receiver. Consequently, the temperature varies throughout the tube surface. The lower half of the horizontal receiver enjoys the maximum solar flux, and therefore, experiences the maximum temperature rise compared to the upper part with almost leveled temperature. Reasonable attributions and suggestions are made on this particular type of conjugate thermal system. The knowledge that gained so far from this study will be used to further the analysis and to design an efficient concentrator photovoltaic collector in near future.

Knowledge sharing leveraging new product development activities to derive enhanced business performance : mixed method study

Knowledge has been widely recognised as a determinant of business performance. Business capabilities require an effective share of resource and knowledge. Specifically, knowledge sharing (KS) between different companies and departments can improve manufacturing processes since intangible knowledge plays an enssential role in achieving competitive advantage. This paper presents a mixed method research study into the impact of KS on the effectiveness of new product development (NPD) in achieving desired business performance (BP). Firstly, an empirical study utilising moderated regression analysis was conducted to test whether and to what extent KS has leveraging power on the relationship between NPD and BP constructs and variables. Secondly, this empirically verified hypothesis was validated through explanatory case studies involving two Taiwanese manufacturing companies using a qualitative interaction term pattern matching technique. The study provides evidence that knowledge sharing and management activities are essential for deriving competitive advantage in the manufacturing industry.

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.

ACE research vignette 024 : New firm formation and job creation by type of firm an type of region.

This series of research vignettes is aimed at sharing current and interesting research findings from our team of international Entrepreneurship researchers. This vignette, written by Mr. Darren Kavanagh and Professor Per Davidsson, takes a closer look at job creation by new firms.

Does experience with one type of two-wheeler affect behaviours and attitudes to other types?

Failure to give way by motor vehicles is a factor in many collisions with both powered and unpowered two wheelers (TWs). Motor vehicle drivers often report that they did not see the TW, but research has shown that motor vehicle drivers who have experience riding a motorcycle are less likely to fail to detect motorcycles. The research reported here examines whether this phenomenon extends to detection of bicycles and whether car drivers who have experience with one mode of TW show improved detection of the other mode. A driving simulator study was conducted in an Australian urban setting which incorporated some of the most common car-TW crash scenarios. Participants with car-only, car plus motorcycle, car plus bicycle, and car plus bicycle plus motorcycle experience operated a car simulator. Their interactions with both types of TWs were measured in terms of visual detection, lateral distance and speed when approaching and passing. The effects of different levels of colour and lighting of the TWs on driver responses were also examined. The attitudes of participants towards TWs were measured in a questionnaire.

Perceived barriers to healthy lifestyle activities in midlife and older Australian women with type 2 diabetes

Purpose: Type 2 diabetes is a leading cause of morbidity and mortality in midlife and older Australian women with known modifiable risk factors for type 2 diabetes including smoking, nutrition, physical activity and obesity. In Australia little research has been done to investigate the perceived barriers to healthy lifestyle activities in midlife and older women with type 2 diabetes. The primary aim of this study was to explore the level and type of perceived barriers to health promotion activities. The secondary aim was to explore the relationship of perceived barriers to smoking behaviour, fruit and vegetable intake, physical activity, and body mass index. Methods: The study was a cross sectional survey of women, aged over 45 with type 2 diabetes, attending metropolitan community health clinics (N = 41). Data was collected from self-report questionnaire and analysed using descriptive and inferential statistics. Results: Women in the study had average total barriers scores similar to those reported in the literature for women with a range of physical disabilities and illnesses. The leading barriers for this group of women were: lack of interest, concern about safety, too tired, lack of money and feeling what they do does not help. There was no association between total barriers scores and body mass index, physical activity, fruit and vegetable intake or socio-demographic variables. Conclusion: This study contributes to understanding the perceptions of midlife and older women with type 2 diabetes about the level and type of barriers to healthy lifestyle activities that they experience. Evidence from this study can be applied to inform health promotion for lifestyle risk factor reduction in women with type 2 diabetes.