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.

Computational simulation of gas flow and heat transfer near immersed object in fluidised bed

To improve the understanding of the heat transfer mechanism and find a reliable and simple heat-transfer model, the gas flow and heat transfer between fluidised beds and immersed object surfaces was numerically simulated based on a double particlelayer and porous medium model. The velocity field and temperature distribution of gas were discussed to analyse the heat transfer process.

Computational thermal analysis of a continuous flow micro Polymerase Chain Reaction (PCR) chip

The first continuous flow micro PCR introduced in 1998 has attracted considerable attention for the past several years because of its ability to amplify DNA at much faster rate than the conventional PCR and micro chamber PCR method. The amplification is obtained by moving the sample through 3 different fixed temperature zones. In this paper, the thermal behavior of a continuous flow PCR chip is studied using commercially available finite element software. We study the temperature uniformity and temperature gradient on the chip’s top surface, the cover plate and the interface of the two layers. The material for the chip body and cover plate is glass. The duration for the PCR chip to achieve equilibrium temperature is also studied.

Computational immunology for anomaly detection

The thesis makes a significant contribution to the issue of anomaly detection by introducing a computational immunology approach. Immunity-based anomaly detection in high dimensional space is systematically investigated and the proposed hybrid method (combining data mining techniques and computational immunology) improves both accuracy and efficiency.

Computational investigations on periodic trends in Group 14 chemistry

Describes the use of computer-aided molecular modelling to investigate trends in the chemistry of the Group 14 elements, namely carbon, silicon, germanium, tin and lead. The chemical behaviour of two classes of molecules containing Group 14 elements was related to trends in the fundamental properties of these elements.

Dots versus antidots : computational exploration of structure, magnetism, and half-metallicity in boron-nitride nanostructures

Triangle-shaped nanohole, nanodot, and lattice antidot structures in hexagonal boron-nitride (h-BN) monolayer sheets are characterized with density functional theory calculations utilizing the local spin density approximation. We find that such structures may exhibit very large magnetic moments and associated spin splitting. N-terminated nanodots and antidots show strong spin anisotropy around the Fermi level, that is, half-metallicity. While B-terminated nanodots are shown to lack magnetism due to edge reconstruction, B-terminated nanoholes can retain magnetic character due to the enhanced structural stability of the surrounding two-dimensional matrix. In spite of significant lattice contraction due to the presence of multiple holes, antidot super lattices are predicted to be stable, exhibiting amplified magnetism as well as greatly enhanced half-metallicity. Collectively, the results indicate new opportunities for designing h-BN-based nanoscale devices with potential applications in the areas of spintronics, light emission, and photocatalysis. © 2009 American Chemical Society.

A brief review of recent research trends on applications of computational and statistical techniques in financial and business intelligence

Artificial neural networks and statistical techniques like decision trees, discriminant analysis, logistic regression and survival analysis play a crucial role in Business Intelligence. These predictive analytical tools exploit patterns found in historical data to make predictions about future events. In this paper we have shown some recent developments of a few of these techniques in financial and business intelligence applications like fraud detection, bankruptcy prediction and credit rating scoring.

Computational exploration of the biological basis of Black-Scholes expected utility function

It has often been argued that there exists an underlying biological basis of utility functions. Taking this line of argument a step further in this paper, we have aimed to computationally demonstrate the biological basis of the Black-Scholes functional form as applied to classical option pricing and hedging theory. The evolutionary optimality of the classical Black-Scholes function has been computationally established by means of a haploid genetic algorithm model. The objective was to minimize the dynamic hedging error for a portfolio of assets that is built to replicate the payoff from a European multi-asset option. The functional form that is seen to evolve over successive generations which best attains this optimization objective is the classical Black-Scholes function extended to a multiasset scenario.

Computational modeling aspects of photon transport equations in biological media

This paper presents an overview of modeling light propagation through biological media by solving the photon transport equation. Different variants of the photon transport equation (PTE) are discussed. Several methods for modeling static distributions and the transient response are presented. A discussion on how to mix and match electromagnetic problems with the PTE is also provided.