Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

Railway junction traffic control by heuristic methods

Traffic conflicts at railway junctions are very conmon, particularly on congested rail lines. While safe passage through the junction is well maintained by the signalling and interlocking systems, minimising the delays imposed on the trains by assigning the right-of-way sequence sensibly is a bonus to the quality of service. A deterministic method has been adopted to resolve the conflict, with the objective of minimising the total weighted delay. However, the computational demand remains significant. The applications of different heuristic methods to tackle this problem are reviewed and explored, elaborating their feasibility in various aspects and comparing their relative merits for further studies. As most heuristic methods do not guarantee a global optimum, this study focuses on the trade-off between computation time and optimality of the resolution.

Wavelet based approaches and high resolution methods for complex process models

Many industrial processes and systems can be modelled mathematically by a set of Partial Differential Equations (PDEs). Finding a solution to such a PDF model is essential for system design, simulation, and process control purpose. However, major difficulties appear when solving PDEs with singularity. Traditional numerical methods, such as finite difference, finite element, and polynomial based orthogonal collocation, not only have limitations to fully capture the process dynamics but also demand enormous computation power due to the large number of elements or mesh points for accommodation of sharp variations. To tackle this challenging problem, wavelet based approaches and high resolution methods have been recently developed with successful applications to a fixedbed adsorption column model. Our investigation has shown that recent advances in wavelet based approaches and high resolution methods have the potential to be adopted for solving more complicated dynamic system models. This chapter will highlight the successful applications of these new methods in solving complex models of simulated-moving-bed (SMB) chromatographic processes. A SMB process is a distributed parameter system and can be mathematically described by a set of partial/ordinary differential equations and algebraic equations. These equations are highly coupled; experience wave propagations with steep front, and require significant numerical effort to solve. To demonstrate the numerical computing power of the wavelet based approaches and high resolution methods, a single column chromatographic process modelled by a Transport-Dispersive-Equilibrium linear model is investigated first. Numerical solutions from the upwind-1 finite difference, wavelet-collocation, and high resolution methods are evaluated by quantitative comparisons with the analytical solution for a range of Peclet numbers. After that, the advantages of the wavelet based approaches and high resolution methods are further demonstrated through applications to a dynamic SMB model for an enantiomers separation process. This research has revealed that for a PDE system with a low Peclet number, all existing numerical methods work well, but the upwind finite difference method consumes the most time for the same degree of accuracy of the numerical solution. The high resolution method provides an accurate numerical solution for a PDE system with a medium Peclet number. The wavelet collocation method is capable of catching up steep changes in the solution, and thus can be used for solving PDE models with high singularity. For the complex SMB system models under consideration, both the wavelet based approaches and high resolution methods are good candidates in terms of computation demand and prediction accuracy on the steep front. The high resolution methods have shown better stability in achieving steady state in the specific case studied in this Chapter.

A Taxonomy of image matching techniques for stereo vision

Stereo vision is a method of depth perception, in which depth information is inferred from two (or more) images of a scene, taken from different perspectives. Applications of stereo vision include aerial photogrammetry, autonomous vehicle guidance, robotics, industrial automation and stereomicroscopy. A key issue in stereo vision is that of image matching, or identifying corresponding points in a stereo pair. The difference in the positions of corresponding points in image coordinates is termed the parallax or disparity. When the orientation of the two cameras is known, corresponding points may be projected back to find the location of the original object point in world coordinates. Matching techniques are typically categorised according to the nature of the matching primitives they use and the matching strategy they employ. This report provides a detailed taxonomy of image matching techniques, including area based, transform based, feature based, phase based, hybrid, relaxation based, dynamic programming and object space methods. A number of area based matching metrics as well as the rank and census transforms were implemented, in order to investigate their suitability for a real-time stereo sensor for mining automation applications. The requirements of this sensor were speed, robustness, and the ability to produce a dense depth map. The Sum of Absolute Differences matching metric was the least computationally expensive; however, this metric was the most sensitive to radiometric distortion. Metrics such as the Zero Mean Sum of Absolute Differences and Normalised Cross Correlation were the most robust to this type of distortion but introduced additional computational complexity. The rank and census transforms were found to be robust to radiometric distortion, in addition to having low computational complexity. They are therefore prime candidates for a matching algorithm for a stereo sensor for real-time mining applications. A number of issues came to light during this investigation which may merit further work. These include devising a means to evaluate and compare disparity results of different matching algorithms, and finding a method of assigning a level of confidence to a match. Another issue of interest is the possibility of statistically combining the results of different matching algorithms, in order to improve robustness.

Patient-specific computational biomechanics for simulating adolescent scoliosis surgery : predicted vs clinical correction for a preliminary series of six patients

Scoliosis is a spinal deformity that requires surgical correction in progressive cases. In order to optimize surgical outcomes, patient-specific finite element models are being developed by our group. In this paper, a single rod anterior correction procedure is simulated for a group of six scoliosis patients. For each patient, personalised model geometry was derived from low-dose CT scans, and clinically measured intra-operative corrective forces were applied. However, tissue material properties were not patient-specific, being derived from existing literature. Clinically, the patient group had a mean initial Cobb angle of 47.3 degrees, which was corrected to 17.5 degrees after surgery. The mean simulated post-operative Cobb angle for the group was 18.1 degrees. Although this represents good agreement between clinical and simulated corrections, the discrepancy between clinical and simulated Cobb angle for individual patients varied between -10.3 and +8.6 degrees, with only three of the six patients matching the clinical result to within accepted Cobb measurement error of +-5 degrees. The results of this study suggest that spinal tissue material properties play an important role in governing the correction obtained during surgery, and that patient-specific modelling approaches must address the question of how to prescribe patient-specific soft tissue properties for spine surgery simulation.

Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

Asphalt concrete surfaces macrotexture determination from still images

Road surface macrotexture is identified as one of the factors contributing to the surface's skid resistance. Existing methods of quantifying the surface macrotexture, such as the sand patch test and the laser profilometer test, are either expensive or intrusive, requiring traffic control. High-resolution cameras have made it possible to acquire good quality images from roads for the automated analysis of texture depth. In this paper, a granulometric method based on image processing is proposed to estimate road surface texture coarseness distribution from their edge profiles. More than 1300 images were acquired from two different sites, extending to a total of 2.96 km. The images were acquired using camera orientations of 60 and 90 degrees. The road surface is modeled as a texture of particles, and the size distribution of these particles is obtained from chord lengths across edge boundaries. The mean size from each distribution is compared with the sensor measured texture depth obtained using a laser profilometer. By tuning the edge detector parameters, a coefficient of determination of up to R2 = 0.94 between the proposed method and the laser profilometer method was obtained. The high correlation is also confirmed by robust calibration parameters that enable the method to be used for unseen data after the method has been calibrated over road surface data with similar surface characteristics and under similar imaging conditions.

Accelerated leap methods for simulating discrete stochastic chemical kinetics

Biologists are increasingly conscious of the critical role that noise plays in cellular functions such as genetic regulation, often in connection with fluctuations in small numbers of key regulatory molecules. This has inspired the development of models that capture this fundamentally discrete and stochastic nature of cellular biology - most notably the Gillespie stochastic simulation algorithm (SSA). The SSA simulates a temporally homogeneous, discrete-state, continuous-time Markov process, and of course the corresponding probabilities and numbers of each molecular species must all remain positive. While accurately serving this purpose, the SSA can be computationally inefficient due to very small time stepping so faster approximations such as the Poisson and Binomial τ-leap methods have been suggested. This work places these leap methods in the context of numerical methods for the solution of stochastic differential equations (SDEs) driven by Poisson noise. This allows analogues of Euler-Maruyuma, Milstein and even higher order methods to be developed through the Itô-Taylor expansions as well as similar derivative-free Runge-Kutta approaches. Numerical results demonstrate that these novel methods compare favourably with existing techniques for simulating biochemical reactions by more accurately capturing crucial properties such as the mean and variance than existing methods.

Supplement : Efficient weak second-order stochastic Runge-Kutta methods for non-commutative Stratonvich stochastic differential equations

This paper gives a modification of a class of stochastic Runge–Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.

Computational approaches to the visual validation of 3D virtual environments

Virtual environments can provide, through digital games and online social interfaces, extremely exciting forms of interactive entertainment. Because of their capability in displaying and manipulating information in natural and intuitive ways, such environments have found extensive applications in decision support, education and training in the health and science domains amongst others. Currently, the burden of validating both the interactive functionality and visual consistency of a virtual environment content is entirely carried out by developers and play-testers. While considerable research has been conducted in assisting the design of virtual world content and mechanics, to date, only limited contributions have been made regarding the automatic testing of the underpinning graphics software and hardware. The aim of this thesis is to determine whether the correctness of the images generated by a virtual environment can be quantitatively defined, and automatically measured, in order to facilitate the validation of the content. In an attempt to provide an environment-independent definition of visual consistency, a number of classification approaches were developed. First, a novel model-based object description was proposed in order to enable reasoning about the color and geometry change of virtual entities during a play-session. From such an analysis, two view-based connectionist approaches were developed to map from geometry and color spaces to a single, environment-independent, geometric transformation space; we used such a mapping to predict the correct visualization of the scene. Finally, an appearance-based aliasing detector was developed to show how incorrectness too, can be quantified for debugging purposes. Since computer games heavily rely on the use of highly complex and interactive virtual worlds, they provide an excellent test bed against which to develop, calibrate and validate our techniques. Experiments were conducted on a game engine and other virtual worlds prototypes to determine the applicability and effectiveness of our algorithms. The results show that quantifying visual correctness in virtual scenes is a feasible enterprise, and that effective automatic bug detection can be performed through the techniques we have developed. We expect these techniques to find application in large 3D games and virtual world studios that require a scalable solution to testing their virtual world software and digital content.

Application of a new discreet form of Gauss’ theorem for measuring volume

Volume measurements are useful in many branches of science and medicine. They are usually accomplished by acquiring a sequence of cross sectional images through the object using an appropriate scanning modality, for example x-ray computed tomography (CT), magnetic resonance (MR) or ultrasound (US). In the cases of CT and MR, a dividing cubes algorithm can be used to describe the surface as a triangle mesh. However, such algorithms are not suitable for US data, especially when the image sequence is multiplanar (as it usually is). This problem may be overcome by manually tracing regions of interest (ROIs) on the registered multiplanar images and connecting the points into a triangular mesh. In this paper we describe and evaluate a new discreet form of Gauss’ theorem which enables the calculation of the volume of any enclosed surface described by a triangular mesh. The volume is calculated by summing the vector product of the centroid, area and normal of each surface triangle. The algorithm was tested on computer-generated objects, US-scanned balloons, livers and kidneys and CT-scanned clay rocks. The results, expressed as the mean percentage difference ± one standard deviation were 1.2 ± 2.3, 5.5 ± 4.7, 3.0 ± 3.2 and −1.2 ± 3.2% for balloons, livers, kidneys and rocks respectively. The results compare favourably with other volume estimation methods such as planimetry and tetrahedral decomposition.