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.

A new solution method for homogenization of effective properties of electromagnetic honeycombs

In this paper, an analytical model and its new numerical solution using the homogenization method are developed to determine the effective electromagnetic characteristics of honeycombs. Based on the proposed solution method, the electromagnetic properties are obtained by employing the multi-scale homogenization theory and periodical electric (magnetic) potential boundary conditions. Further, the effect of geometry of honeycomb's unit cell on effective electromagnetic properties is investigated with the use of the proposed method. The numerical results are compared with analytic results using the Smith-Scarpa's semi-empirical formula.

Mathematical modelling of the refractive index reconstruction through global optimization

We examine a mathematical model of non-destructive testing of planar waveguides, based on numerical solution of a nonlinear integral equation. Such problem is ill-posed, and the method of Tikhonov regularization is applied. To minimize Tikhonov functional, and find the parameters of the waveguide, we use two new optimization methods: the cutting angle method of global optimization, and the discrete gradient method of nonsmooth local optimization. We examine how the noise in the experimental data influences the solution, and how the regularization parameter has to be chosen. We show that even with significant noise in the data, the numerical solution is of high accuracy, and the method can be used to process real experimental da.ta..

A lower bound on effective performance testing for digital forensic tools

The increasing complexity and number of digital forensic tasks required in criminal investigations demand the development of an effective and efficient testing methodology, enabling tools of similar functionalities to be compared based on their performance. Assuming that the tool tester is familiar with the underlying testing platform and has the ability to use the tools correctly, we provide a numerical solution for the lower bound on the number of testing cases needed to determine comparative capabilities of any set of digital forensic tools. We also present a case study on the performance testing of password cracking tools, which allows us to confirm that the lower bound on the number of testing runs needed is closely related to the row size of certain orthogonal arrays. We show how to reduce the number of test runs by using knowledge of the underlying system

OWA operators in regression problems

We consider an application of fuzzy logic connectives to statistical regression. We replace the standard least squares, least absolute deviation, and maximum likelihood criteria with an ordered weighted averaging (OWA) function of the residuals. Depending on the choice of the weights, we obtain the standard regression problems, high-breakdown robust methods (least median, least trimmed squares, and trimmed likelihood methods), as well as new formulations. We present various approaches to numerical solution of such regression problems. OWA-based regression is particularly useful in the presence of outliers, and we illustrate the performance of the new methods on several instances of linear regression problems with multiple outliers.

Designing optimal fault tolerant Jacobian for robotic manipulators

Fault tolerance of robotic manipulators is determined based on the fault tolerance measures. In this study a Jacobian of a 7DOF optimal fault tolerant manipulator is designed based on optimality of worse case relative manipulability and worse case dexterity from geometric perspective instead of numerical solution of constrained optimisation problem or construction of optimal Jacobean through a desired null space. The proposed Jacobean matrix is optimal and equally fault tolerant for a single joint failure within any joint of the manipulators.

Kinematics analysis of 6-DOF parallel micro-manipulators with offset U-Joints : a case study

This paper analyses the kinematics of a special 6-DOF parallel micro-manipulator with offset RR-joint configuration. Kinematics equations are derived and numerical methodologies to solve the inverse and forward kinematics are presented. The inverse and forward kinematics of such robots compared with those of 6-UCU parallel robots are more complicated due to the existence of offsets between joints of RR-pairs. The characteristics of RR-pairs used in this manipulator are investigated and kinematics constraints of these offset U-joints are mathematically explained in order to find the best initial guesses for the numerical solution. Both inverse and forward kinematics of the case study 6-DOF parallel micro-manipulator are modelled and computational analyses are performed to numerically verify accuracy and effectiveness of the proposed methodologies.

Kinematics analysis of 6-DOF parallel micro-manipulators with offset u-joints : a case study

This paper analyses the kinematics of a special 6-DOF parallel micro-manipulator with offset RR-joint configuration. Kinematics equations are derived and numerical methodologies to solve the inverse and forward kinematics are presented. The inverse and forward kinematics of such robots compared with those of 6-UCU parallel robots are more complicated due to the existence of offsets between joints of RR-pairs. The characteristics of RR-pairs used in this manipulator are investigated and kinematics constraints of these offset U-joints are mathematically explained in order to find the best initial guesses for the numerical solution. Both inverse and forward kinematics of the case study 6-DOF parallel micro-manipulator are modelled and computational analyses are performed to numerically verify accuracy and effectiveness of the proposed methodologies.

Forward kinematics analysis of offset 6-RRCRR parallel manipulators

A methodology for the numerical solution of the forward kinematics problem of 6-RRCRR parallel manipulators with orthogonal non-intersecting RR-joint configuration is presented in this article. The inverse and forward kinematics solutions of such robots compared with that of parallel robots with orthogonal intersecting RR-joint or universal joint configurations are much more complicated due to the existence of dependent joint variables. The constraints of RR-joints are analysed and the numerical algorithm for the forward kinematics solution is assessed. Numerical results for the solution of the forward kinematics of 6-RRCRR parallel robot under study are provided to confirm the accuracy and efficiency of the procedure.

Quantification of the finger functional range via explicit descriptions of reachable subspaces

In recent times, the finger flexibility assessment by means of reachable space is considered as an effective tool to describe the range of motion of the hand. Existing approaches numerically compute the reachable space using forward kinematics such as exhaustive scanning or Monte Carlo methods. In this paper, we provide explicit formulas mathematically determining the reachable space boundary. Green's theorem is used to deduce the corresponding capacity formula for the size of the reachable space as opposed to an implicit numerical solution. Using this new mechanism, we accurately quantify and compare the reachable space of different subjects in order to effectively compare the functionality of the fingers. We evaluate the performance of our proposed method against the kinematic feed-forward (KFF) approach in calculating the reachable space. The execution time to capture the reachable space is significantly less than that for the standard KFF method. The computational cost for quantifying the reachable space capacity is significantly improved due to explicit capacity formulas resulting from the abstract form of boundary descriptions of the reachable space, unique to the proposed approach.

Integral approach and numerical improvement to calculate the carbon concentration profiles in carburising process

In the present work, the carbon diffusion in steel, where the carbon diffusivity varies with the carbon content, was solved with the integral methods under the third boundary condition. The variation of carbon diffusivity in steel with the carbon content was described with two different functions ie. linear dependence and exponential dependence. The integral approximation for both cases was improved with the numerical computation to more accurately predict the carbon profiles. The integral solution is more accurate than the formulation based on the assumption of a constant diffusivity or those based on the assumption of a constant diffusivity and/or constant carbon content at part surface. It is also more easily used in practice than the numerical method to describe the carburising process and predict the carbon content at steel surface and carbon profiles in treated layer.

Evaluation of the thermal behaviour of bevelled cutting inserts using a numerical approach

The nose geometry of a hard and brittle metal cutting tool is generally modified in order to avoid the premature failure due to fracture under tensile stresses. While most research findings point to a favourable mechanical load pattern, the possible influence of the shape of the geometry on the thermal fields and the consequent changes in the stressed state of the tool seem to have attained less attention. The present work aims at establishing the thermal behaviour of bevelled tools under varying geometrical and process parameters. Data generated from statistically designed experiments and quick-stop chip samples are coupled to conduct numerical investigations using a mixed finite and boundary element solution to obtain the temperature distribution in bevelled carbide inserts. Due consideration is given to the presence of the stagnation zone and its size and shape. While the cutting forces and temperatures increased owing to the blunt shape of the tool, the possible absence of tensile stresses was found to be the likely effect of a more uniform temperature distribution resulting from a significant plastic contact on the principal flank and the consequent flank heat source. The characteristic low-temperature zones close to the nose of the conventional tool are taken over by the stagnation zone in bevelled tools. © IMechE 2007.

Numerical investigation of natural convection inside complex enclosures

In this article, the analyses of heat transfer and free convective motion have been carried out numerically for various structures. The solution is based on a finite element method with the frontal solver to examine the flow parameters and heat transfer characteristics. Several dome configurations--such as flat, inclined, and dome shapes--are considered for the top of the enclosure. A general conic equation is considered to represent the dome as circular, elliptical, parabolic, or hyperbolic shape. The findings from this study indicate that the convective phenomenon is greatly influenced by the shape of the top cover dome and tends to form a secondary core even at a moderate Rayleigh number when compared with an equivalent rectangular enclosure. In addition, the circular and elliptical shapes of the dome give higher heat transfer rate. The effect of various "offset" of the dome and inclined roof on convective heat transfer is also found to be quite significant. However, beyond 0.3 of offset of the top cover for the dome and inclined roof, the change in overall heat transfer rate is minimal. The heat transfer coefficients of dome shaped and inclined roof enclosures are given and discussed.

Integral approach and numerical improvement to calculate carbon concentration profiles in carburising

The carbon diffusion in steel, where the carbon diffusivity varies with the carbon content, was solved with the integral methods under the third boundary condition. The variation of carbon diffusivity in steel with the carbon content was described with two different functions, linear dependence and exponential dependence. The integral approximation for both cases was improved with the numerical computation to more accurately predict the carbon profiles. The integral solution is more accurate than the formulation based on the assumption of a constant diffusivity or those based on the assumption of a constant diffusivity and/or constant carbon content at part surface. It is also more easily used in practice than the numerical method to describe the carburising process and predict the carbon content at steel surface and carbon profiles in treated layer.