Computational aspects of the numerical solution of SDEs

In the last 30 to 40 years, many researchers have combined to build the knowledge base of theory and solution techniques that can be applied to the case of differential equations which include the effects of noise. This class of ``noisy'' differential equations is now known as stochastic differential equations (SDEs). Markov diffusion processes are included within the field of SDEs through the drift and diffusion components of the Itô form of an SDE. When these drift and diffusion components are moderately smooth functions, then the processes' transition probability densities satisfy the Fokker-Planck-Kolmogorov (FPK) equation -- an ordinary partial differential equation (PDE). Thus there is a mathematical inter-relationship that allows solutions of SDEs to be determined from the solution of a noise free differential equation which has been extensively studied since the 1920s. The main numerical solution technique employed to solve the FPK equation is the classical Finite Element Method (FEM). The FEM is of particular importance to engineers when used to solve FPK systems that describe noisy oscillators. The FEM is a powerful tool but is limited in that it is cumbersome when applied to multidimensional systems and can lead to large and complex matrix systems with their inherent solution and storage problems. I show in this thesis that the stochastic Taylor series (TS) based time discretisation approach to the solution of SDEs is an efficient and accurate technique that provides transition and steady state solutions to the associated FPK equation. The TS approach to the solution of SDEs has certain advantages over the classical techniques. These advantages include their ability to effectively tackle stiff systems, their simplicity of derivation and their ease of implementation and re-use. Unlike the FEM approach, which is difficult to apply in even only two dimensions, the simplicity of the TS approach is independant of the dimension of the system under investigation. Their main disadvantage, that of requiring a large number of simulations and the associated CPU requirements, is countered by their underlying structure which makes them perfectly suited for use on the now prevalent parallel or distributed processing systems. In summary, l will compare the TS solution of SDEs to the solution of the associated FPK equations using the classical FEM technique. One, two and three dimensional FPK systems that describe noisy oscillators have been chosen for the analysis. As higher dimensional FPK systems are rarely mentioned in the literature, the TS approach will be extended to essentially infinite dimensional systems through the solution of stochastic PDEs. In making these comparisons, the advantages of modern computing tools such as computer algebra systems and simulation software, when used as an adjunct to the solution of SDEs or their associated FPK equations, are demonstrated.

Ring strain in boroxine rings: computational and experimental considerations

B3LYP/6-311+G(d) calculations indicate that (HBO)₃ (4) and (HBO)₄ (5) possess (zero-point energy corrected) strain enthalpies of 11.4 and 31.6 kJ mol⁻¹, respectively. The absence of eight-membered (RBO)₄ rings is attributed to a combination of ring strain and the lability of the B---O bond. The synthesis, characterization and molecular structure of (PhBO)₃·pyridine (1) are described and chemical phenomena related to the addition of amines to triorganoboroxine rings are rationalized in terms of relief of ring strain in 4.

A comparative study of some computer algebra packages which determine the lie point symmetries of differential equations

The study seeks to determine which of five computer algebra packages is best at finding the Lie point symmetries of systems of partial differential equations with minimal user intervention. The chosen packages are LIEPDE and DIMSYM for REDUCE, LIE and BIGLIE for MUMATH, DESOLV for MAPLE, and MATHLIE for MATHEMATICA. A series of systems of partial differential equations are used in the study. The paper concludes that while all of the computer packages are useful, DESOLV appears to be the most successful system at determining the complete set of Lie point symmetries of systems of partial differential equations.

Adopting a computational approach to identifying and modelling future development initiatives within the urban context

In 2002 the State Government of Victoria, in partnership with the City of Greater Geelong and Deakin University, initiated and urban design framework to explore development opportunities of a series of redundant industrial sites adjacent to Geelong's central activities area (CAA). One of the objectives of the framework was to explore a range of initiatives based on the expansion of Deakin University Waterfront campus from 700 students to an expected cohort of 5000 students. Working with a set of predefined constructs for different types of city-based user groups, such as net floor space requirements and time/travel distances between facilities, the following paper presents a simple computational approach to assist in identifying, mapping, and spatially exploring different development scenarios.

Algebra skills and traps and diagnostic teaching for the future

Explores on some research about teaching and learning algebra and related classroom issues. Diagnostic instruments that may be used by senior secondary teachers in teaching algebra to senior classes; Strategies for remediating algebraic difficulties and misconceptions; Impact of technology on the algebra curriculum; Usefulness of copying algebraic expressions while using Computer Algebra Systems or mathematics processing software in a calculus class.

A computational approach to the reconstruction of surface geometry from early temple superstructures

Recovering the control or implicit geometry underlying temple architecture requires bringing together fragments of evidence from field measurements, relating these to mathematical and geometric descriptions in canonical texts and proposing "best-fit" constructive models. While scholars in the field have traditionally used manual methods, the innovative application of niche computational techniques can help extend the study of artefact geometry. This paper demonstrates the application of a hybrid computational approach to the problem of recovering the surface geometry of early temple superstructures. The approach combines field measurements of temples, close-range architectural photogrammetry, rule-based generation and parametric modelling. The computing of surface geometry comprises a rule-based global model governing the overall form of the superstructure, several local models for individual motifs using photogrammetry and an intermediate geometry model that combines the two. To explain the technique and the different models, the paper examines an illustrative example of surface geometry reconstruction based on studies undertaken on a tenth century stone superstructure from western India. The example demonstrates that a combination of computational methods yields sophisticated models of the constructive geometry underlying temple form and that these digital artefacts can form the basis for in depth comparative analysis of temples, arising out of similar techniques, spread over geography, culture and time.

Thermal control of high pressure die casting using computational fluid dynamics

The quality of high pressure die castings is a function of many interdependent parameters. It has been observed that many defects detected in the HPDC castings can be tracked back to poor die temperature distribution in the critical areas. It has therefore been recommended that the development of a technique to directly control the critical features - making them less sensitive to thermal related parameters - be very beneficial to the HPDC industry. From the information obtained from thermal image (processing), computational fluid dynamics has been applied to design the layout of internal cooling system and assign the flow conditions such as flow rate and pressure of the cooling water. it is observed that CFD prediction provides an excellent insight into the thermal balance of the high pressure die casting.

A computational framework for nucleic acid sub-sequence identification

Identification of nucleic acid sub-sequences within larger background sequences is a fundamental need of the biology community. The applicability correlates to research studies looking for homologous regions, diagnostic purposes and many other related activities. This paper serves to detail the approaches taken leading to sub-sequence identification through the use of hidden Markov models and associated scoring optimisations. The investigation of techniques for locating conserved basal promoter elements correlates to promoter thus gene identification techniques. The case study centred on the TATA box basal promoter element, as such the background is a gene sequence with the TATA box the target. Outcomes from the research conducted, highlights generic algorithms for sub-sequence identification, as such these generic processes can be transposed to any case study where identification of a target sequence is required. Paths extending from the work conducted in this investigation have led to the development of a generic framework for the future applicability of hidden Markov models to biological sequence analysis in a computational context.

Computational simulation of heat transfer from alternate heating sources

The alternant heat transfer induced by particle packet and gas bubbles on an object surface in a gas fluidised bed is computationally studied. The particle packet and bubble are modelled by a DPPM (double particle-layer and Porous Medium) model and a hemispherical model, respectively. Different meshing schemes are applied and different mesh sizes are used in meshing particle packet and heated object and a very large geometrical size difference between them was considered. Two parallel solver processes were proposed to perform the simulation of heat transfer for different purposes and implemented with the Fluent CFD package.

A comparative study of computational value in visual cues used in positioning 3D objects

This paper explores the comparative computational values of using a variety of visual cues in 3D environments. The authors reflect upon the possible repercussions of computationally less expensive visual cues on users' ability to efficiently and accurately interact with three-dimensional images. This study compares the effectiveness of expensive soft shadows against less expensive hard shadows and expensive partial-occlusion (obtained by semi-transparent surface) against less expensive occlusion on users' ability to accurately position objects in imminent contact with other objects in a three-dimensional environment.

Computational simulation of gas flow and heat transfer near an immersed object in fluidized beds

To improve the understanding of the heat transfer mechanism and to find a reliable and simple heat-transfer model, the gas flow and heat transfer between fluidized beds and the surfaces of an immersed object is numerically simulated based on a double particle-layer and porous medium model. The velocity field and temperature distribution of the gas and particles are analysed during the heat transfer process. The simulation shows that the change of gas velocity with the distance from immersed surface is consistent with the variation of bed voidage, and is used to validate approximately dimensional analysing result that the gas velocity between immersed surface and particles is 4.6Umf/εmf. The effects of particle size and particle residence time on the thermal penetration depth and the heat-transfer coefficients are also discussed.

Comparison of the flexibility of eight-membered tetrasiloxane and stannasiloxane rings: a crystallographic and computational study

The syntheses and crystal structures of the eight-membered cyclo-stannasiloxanes cyclo-[t-Bu(OH)Si(OSnt-Bu₂O)₂Si(OH)t-Bu] (1) and cyclo-{t-Bu₂Si[OSn(CH₂SiMe₃)₂O]₂Sit-Bu₂} (2) as well as the synthesis of the six-membered cyclo-stannasiloxane cyclo-{t-Bu₂Si[OSn(CH₂SiMe₃)₂]₂O} (3) are reported. Compound 1 crystallizes as its trans isomer, but the cis isomer dominates in solution. In agreement with the experimentally obtained results, ab initio and DFT calculations on the model compounds cyclo-(H₂SiO)₄ (4), cyclo-[H₂Si(OSnH₂)OSiH₂] (5), cyclo-O(H₂SiOSnH₂)₂O (6), and cyclo-[H₂Si(OSiH₂)OSnH₂] (7) indicate that the energetic preference to adopt puckered structures increases and the ring flexibility decreases with an increasing number of tin atoms in the ring. The rich diversity of puckered conformations is attributed to the steric demand of the different organic substituents.

Computational simulation of chassis design and manufacturing

Australian Universities including Deakin University are entering the Formula One SAE car competition this year which was initialised in USA many years ago and has attracted many universities to participate all over the world. There are many categories to be competed for novelty of the design. To achieve optimised design of the car, computational techniques have widely been used in all aspects of the design and manufacturing of the car for the team at Deakin University. In this work, the design and manufacturing of the chassis was computationally simulated to optimise the structure for stress and natural frequencies. The chassis is designed using solid modelling code IDEAS with the geometry exported to ABAQUS, a FEM software, for optimisation.

Self-adaptive clock synchronization for computational grid

This paper presents an innovative method to synchronize physical clocks for a computational grid, in particular for a computational grid linked through the asynchronous Intranet or Internet environments. The method discussed is an asynchronous self-adaptive clock synchronization mechanism. Two strategies for clock synchronisation are introduced. (1) Use continuous time intervals to calculate the precision of clocks, which can reduce the effect of network delay efficiently. (2) Every node synchronizes its clock with its leader actively. In addition, a node self-adaptive model is presented, and the relationship between the clock precision and synchronization time is induced, hence a node can predict when it should begin the synchronization process. Detailed simulation and extension of this issue are provided at the end of the paper. The presented model is both practical and feasible.