A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

The resistance of PRESENT-80 against related-key differential attacks

We examine the security of the 64-bit lightweight block cipher PRESENT-80 against related-key differential attacks. With a computer search we are able to prove that for any related-key differential characteristic on full-round PRESENT-80, the probability of the characteristic only in the 64-bit state is not higher than 2−64. To overcome the exponential (in the state and key sizes) computational complexity of the search we use truncated differences, however as the key schedule is not nibble oriented, we switch to actual differences and apply early abort techniques to prune the tree-based search. With a new method called extended split approach we are able to make the whole search feasible and we implement and run it in real time. Our approach targets the PRESENT-80 cipher however,with small modifications can be reused for other lightweight ciphers as well.

Dissection of class III major histocompatibility complex haplotypes associated with rheumatoid arthritis

Objective. We have previously identified a single-nucleotide polymorphism (SNP) haplotype involving the lymphotoxin α (LTA) and tumor necrosis factor (TNF) loci (termed haplotype LTA-TNF2) on chromosome 6 that shows differential association with rheumatoid arthritis (RA) on HLA-DRB1*0404 and *0401 haplotypes, suggesting the presence of additional non-HLA-DRB1 RA susceptibility genes on these haplotypes. To refine this association, we performed a case-control association study using both SNPs and microsatellite markers in haplotypes matched either for HLA-DRB1*0404 or for HLA-DRB1*0401. Methods. Fourteen SNPs lying between HLA-DRB1 and LTA were genotyped in 87 DRB1*04-positive families. High-density microsatellite typing was performed using 24 markers spanning 2,500 kb centered around the TNF gene in 305 DRB1*0401 or *0404 cases and 400 DRB1*0401 or *0404 controls. Single-marker, 2-marker, and 3-marker minihaplotypes were constructed and their frequencies compared between the DRB1*0401 and DRB1*0404 matched case and control haplotypes. Results. Marked preservation of major histocompatibility complex haplotypes was seen, with chromosomes carrying LTA-TNF2 and either DRB1*0401 or DRB1*0404 both carrying an identical SNP haplotype across the 1-Mb region between TNF and HLA-DRB1. Using microsatellite markers, we observed two 3-marker minihaplotypes that were significantly overrepresented in the DRB1*0404 case haplotypes (P = 0.00024 and P = 0.00097). Conclusion. The presence of a single extended SNP haplotype between LTA-TNF2 and both DRB1*0401 and DRB1*0404 is evidence against this region harboring the genetic effects in linkage disequillbrium with LTA-TNF2. Two RA-associated haplotypes on the background of DRB1*0404 were identified in a 126-kb region surrounding and centromeric to the TNF locus.

QoS-aware web service composition using genetic algorithms

Web service technology is increasingly being used to build various e-Applications, in domains such as e-Business and e-Science. Characteristic benefits of web service technology are its inter-operability, decoupling and just-in-time integration. Using web service technology, an e-Application can be implemented by web service composition — by composing existing individual web services in accordance with the business process of the application. This means the application is provided to customers in the form of a value-added composite web service. An important and challenging issue of web service composition, is how to meet Quality-of-Service (QoS) requirements. This includes customer focused elements such as response time, price, throughput and reliability as well as how to best provide QoS results for the composites. This in turn best fulfils customers’ expectations and achieves their satisfaction. Fulfilling these QoS requirements or addressing the QoS-aware web service composition problem is the focus of this project. From a computational point of view, QoS-aware web service composition can be transformed into diverse optimisation problems. These problems are characterised as complex, large-scale, highly constrained and multi-objective problems. We therefore use genetic algorithms (GAs) to address QoS-based service composition problems. More precisely, this study addresses three important subproblems of QoS-aware web service composition; QoS-based web service selection for a composite web service accommodating constraints on inter-service dependence and conflict, QoS-based resource allocation and scheduling for multiple composite services on hybrid clouds, and performance-driven composite service partitioning for decentralised execution. Based on operations research theory, we model the three problems as a constrained optimisation problem, a resource allocation and scheduling problem, and a graph partitioning problem, respectively. Then, we present novel GAs to address these problems. We also conduct experiments to evaluate the performance of the new GAs. Finally, verification experiments are performed to show the correctness of the GAs. The major outcomes from the first problem are three novel GAs: a penaltybased GA, a min-conflict hill-climbing repairing GA, and a hybrid GA. These GAs adopt different constraint handling strategies to handle constraints on interservice dependence and conflict. This is an important factor that has been largely ignored by existing algorithms that might lead to the generation of infeasible composite services. Experimental results demonstrate the effectiveness of our GAs for handling the QoS-based web service selection problem with constraints on inter-service dependence and conflict, as well as their better scalability than the existing integer programming-based method for large scale web service selection problems. The major outcomes from the second problem has resulted in two GAs; a random-key GA and a cooperative coevolutionary GA (CCGA). Experiments demonstrate the good scalability of the two algorithms. In particular, the CCGA scales well as the number of composite services involved in a problem increases, while no other algorithms demonstrate this ability. The findings from the third problem result in a novel GA for composite service partitioning for decentralised execution. Compared with existing heuristic algorithms, the new GA is more suitable for a large-scale composite web service program partitioning problems. In addition, the GA outperforms existing heuristic algorithms, generating a better deployment topology for a composite web service for decentralised execution. These effective and scalable GAs can be integrated into QoS-based management tools to facilitate the delivery of feasible, reliable and high quality composite web services.

Scalability of encryption techniques for electronic health record (EHR) retrieval

Electronic Health Record (EHR) retrieval processes are complex demanding Information Technology (IT) resources exponentially in particular memory usage. Database-as-a-service (DAS) model approach is proposed to meet the scalability factor of EHR retrieval processes. A simulation study using ranged of EHR records with DAS model was presented. The bucket-indexing model incorporated partitioning fields and bloom filters in a Singleton design pattern were used to implement custom database encryption system. It effectively provided faster responses in the range query compared to different types of queries used such as aggregation queries among the DAS, built-in encryption and the plain-text DBMS. The study also presented with constraints around the approach should consider for other practical applications.

Analytical solutions of the multi-term modified power law wave equations in a finite domain

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n) (n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko’s Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi term time-space fractional models including fractional Laplacian.

Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain

Multi-term time-fractional differential equations have been used for describing important physical phenomena. However, studies of the multi-term time-fractional partial differential equations with three kinds of nonhomogeneous boundary conditions are still limited. In this paper, a method of separating variables is used to solve the multi-term time-fractional diffusion-wave equation and the multi-term time-fractional diffusion equation in a finite domain. In the two equations, the time-fractional derivative is defined in the Caputo sense. We discuss and derive the analytical solutions of the two equations with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin conditions, respectively.

Mixed convection along a vertical flat plate in a non-absorbing medium

Here mixed convection boundary layer flow of a viscous fluid along a heated vertical semi-infinite plate is investigated in a non-absorbing medium. The relationship between convection and thermal radiation is established via boundary condition of second kind on the thermally radiating vertical surface. The governing boundary layer equations are transformed into dimensionless parabolic partial differential equations with the help of appropriate transformations and the resultant system is solved numerically by applying straightforward finite difference method along with Gaussian elimination technique. It is worthy to note that Prandlt number, Pr, is taken to be small (<< 1) which is appropriate for liquid metals. Moreover, the numerical results are demonstrated graphically by showing the effects of important physical parameters, namely, the modified Richardson number (or mixed convection parameter), Ri*, and surface radiation parameter, R, in terms of local skin friction and local Nusselt number coefficients.

A stochastic exponential Euler scheme for simulation of stiff biochemical reaction systems

In order to simulate stiff biochemical reaction systems, an explicit exponential Euler scheme is derived for multidimensional, non-commutative stochastic differential equations with a semilinear drift term. The scheme is of strong order one half and A-stable in mean square. The combination with this and the projection method shows good performance in numerical experiments dealing with an alternative formulation of the chemical Langevin equation for a human ether a-go-go related gene ion channel mode

The tectonic significance of lower Permain successions in the Texas Orocline (Eastern Australia)

The Texas Orocline is a prominent orogenic curvature that developed during the early Permian in the southern New England Orogen. Outliers preserving lower Permian sedimentary successions (Bondonga, Silver Spur, Pikedale, Terrica, Alum Rock and Ashford beds) approximately outline the oroclinal structure, but the tectonic processes responsible for the development of these basinal successions, and their relationships to the Texas Orocline, are unclear. Here we address this shortcoming by providing new U–Pb detrital and primary zircon ages from these successions, as well as detailed stratigraphic and structural data from the largest exposed succession (Bondonga beds). Field observations and U–Pb geochronological data suggest that the lower Permian successions in the Texas Orocline are remnants of a single, formerly larger basin that was deposited after ca 302 Ma. Time constraints for formation of this basin are correlative with constraints from the lower Permian Nambucca Block, which was likely deposited in response to regional back-arc extension during and/or after the development of the Texas Orocline. The conclusion that the lower Permian sedimentary basins in the Texas Orocline belong to this back-arc extensional system supports the suggestion that oroclinal bending in the New England Orogen was primarily controlled by trench retreat and associated overriding-plate extension.