Many applications in engineering practice can be described by thehyperbolic partial differential equations (PDEs). Numerical modeling of this typeof equations often involves large gradients or shocks, which makes it achallenging task for conventional numerical methods to accurately simulate suchsystems. Thus developing accurate and efficient shock capturing numericalschemes becomes important for the study of hyperbolic equations.In this dissertation, a detailed study of the numerical methods for linearand nonlinear unsteady hyperbolic equations was carried out. A new finitedifference shock capturing scheme of finite volume style was developed. Thisscheme is based on the high order Pad?? type compact central finite differencemethod with the weighted essentially non-oscillatory (WENO) reconstruction toeliminate non-physical oscillations near the discontinuities while maintain stablesolution in the smooth areas. The unconditionally stable semi-implicit Crank-Nicolson (CN) scheme is used for time integration.The theoretical development was conducted based on one-dimensionalhomogeneous scalar equation and system equations. Discussions were alsoextended to include source terms and to deal with problems of higher dimension.For the treatment of source terms, Strang splitting was used. For multidimensionalequations, the ?? -form Douglas-Gunn alternating direction implicit(ADI) method was employed. To compare the performance of the scheme withENO type interpolation, the current numerical framework was also applied usingENO reconstruction.The numerical schemes were tested on 1-D and 2-D benchmark problems,as well as published experimental results. The simulated results show thecapability of the proposed scheme to resolve discontinuities while maintainingaccuracy in smooth regions. Comparisons with the experimental results validatethe method for dam break problems. It is concluded that the proposed scheme isa useful tool for solving hyperbolic equations in general, and from engineeringapplication perspective it provides a new way of modeling open channel flows.

Shallow water equations are widely used to model water flows in the field of hydrodynamics and civil engineering. They are complex, and except for some simplified cases, no analytical solution exists for them. Therefore, the partial differential equations of the shallow water system have been the subject of various numerical analyses and studies in past decades. In this study, we construct a stable and robust finite volume scheme for the shallow water equations over quadtree grids. Quadtree grids are two-dimensional semi-structured Cartesian grids that have different applications in several fields of engineering, such as computational fluid dynamics. Quadtree grids refine or coarsen where it is required in the computational domain, which gives the advantage of reducing the computational cost in some problems. Numerical schemes on quadtree grids have different properties. An accurate and robust numerical scheme is able to provide a balance between the flux and source terms, preserve the positivity of the water height and water surface, and is capable of regenerating the grid with respect to different conditions of the problem and computed solution. The proposed scheme uses a piecewise constant approximation and employs a high-order Runge-Kutta method to be able to make the solution high-order in space and time. Hence, in this thesis, we develop an adaptive well-balanced positivity preserving scheme for the shallow water system over quadtree grids utilizing different techniques. We demonstrate the formulations of the proposed scheme over one of the different configurations of quadtree cells. Six numerical benchmark tests confirm the ability of the scheme to accurately solve the problems and to capture small perturbations. Furthermore, we extend the proposed scheme to the coupled variable density shallow water flows and establish an extended method where we focus on eliminating nonphysical oscillations, as well as well-balanced, positivity preserving, and adaptivity properties of the scheme. Four different numerical benchmark tests show that the proposed extension of the scheme is accurate, stable, and robust.

After presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order. / text

This thesis deals with the development of algorithms for recursive estimation within the errors-in-variables framework. Within this context attention is focused on two major threads of research: Recursive system identification based on the Frisch scheme and the extension and application of errors-in-variables Kalman filtering techniques. In the first thread, recursive algorithms for the approximate update of the estimates obtained via the Frisch scheme, which makes use of the Yule-Walker model selection criterion, are developed for the case of white measurement noise. Gradient-based techniques are utilised to update the Frisch scheme equations, which involve the minimisation of the model selection criterion as well as the solution of an eigenvalue problem, in a recursive manner. The computational complexity of the resulting algorithms is critically analysed and, by introducing additional approximations, fast recursive Frisch scheme algorithms are developed, which reduce the computational complexity from cubic to quadratic order. In addition, it is investigated how the singularity condition within the Frisch scheme is affected when the estimates are computed recursively. Whilst this first group of recursive Frisch scheme algorithms is developed directly from the offline Frisch scheme equations, it is also possible to interpret the Frisch scheme within an extended bias compensating least squares framework. Consequently, the development of recursive algorithms, which update the estimate obtained from the extended bias compensated least squares technique, is considered. These algorithms make use of the bilinear parametrisation principle or, alternatively, the variable projection method. Finally, two recursive Frisch scheme algorithms are developed for the case of coloured output noise. The second thread, which considers the theory of errors-in-variables filtering for linear systems, extends the approach to deal with a class of bilinear systems, a frequently used subset of nonlinear systems. The application of errors-in-variables filtering for the purpose of system identification is also considered. This leads to the development of a prediction error method based on symmetric innovations, which resembles the joint output method. Both the offline and online implementation of this novel identification technique are investigated.

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Frequently, testing of Single Photon Emission Computed Tomography (SPECT) motion correction algorithms is done either by using simplistic deformations that do not accurately simulate true patient motion or by applying the algorithms directly to data acquired from a real patient, where the true internal motion is unknown. In this work, we describe a way to combine these two approaches by using imaging data acquired from real volunteers to simulate the data that the motion correction algorithms would normally observe. The goal is to provide an assessment framework which can both: simulate realistic SPECT acquisitions that incorporate realistic body deformations and provide a ground truth volume to compare against. Every part of the motion correction algorithm needs to be exercised: from parameter estimation of the motion model, to the final reconstruction results. In order to build the ground truth anthropomorphic numerical phantoms, we acquire high resolution MRI scans and motion observation data of a volunteer in multiple different configurations. We then extract the organ boundaries using thresholding, active contours, and morphology. Phantoms of radioactivity uptake and density inside the body can be generated from these boundaries to be used to simulate SPECT acquisitions. We present results on extraction of the ribs, lungs, heart, spine, and the rest of the soft tissue in the thorax using our segmentation approach. In general, extracting the lungs, heart, and ribs in images that do not contain the spine works well, but the spine could be better extracted using other methods that we discuss. We also go in depth into the software development component of this work, describing the C++ coding framework we used and the High Level Interactive GUI Language (HLING). HLING solved a lot of problems but introduced a fair bit of its own. We include a set of requirements to provide a foundation for the next attempt at developing a declarative and minimally restrictive methodology for writing interactive image processing applications in C++ based on lessons learned during the development of HLING.

Background: Access to &quot high-cost medicines&quot under Australia's Pharmaceutical Benefits Scheme (PBS) is characterised by strict eligibility criteria. The PBS access scheme for the anti-rheumatic biologicals (etanercept, infliximab, and adalimumab) was examined for concordance with Australia?s National Medicines Policy. Methods: Semi-structured interviews with a range of stakeholders were conducted. National, aggregated prescription and expenditure data from Medicare Australia and dispensing data from the Drug Utilisation Sub-Committee were analysed. Access to biologicals was also examined from an ethical perspective. Results: Interviewees agreed that controlled access to high-cost medicines was broadly equitable and practical but specific concerns included: timeliness of access; bureaucracy of the process; contentious cases of individual patients being denied access; insufficient patient information; the quantum of resources required to administer the access scheme; inadequate stakeholder consultation. The access requirement of a history of failure of conventional anti-rheumatic drugs was supported. Recommendations included proactive review of the access criteria and outcomes; greater transparency and formal stakeholder involvement to increase public confidence in the definition of &quot target patient population" and a formal appeal mechanism to increase the fairness and accountability of the PBS. Establishment of an appeal mechanism is supported by &quot accountability for reasonableness&quot framework grounded in procedural justice. Data needed to examine the health outcomes associated with the use of biologicals on a national level was not easily available. This shortcoming is discordant with National Medicines Policy. Utilisation of biologicals over the first two years of PBS-subsidy was conservative but with considerable variability across States and Territories (an 8-fold difference between the jurisdictions), usage roughly correlating with access to rheumatologists. Introduction of PBS-subsidised biologicals did not alter the trends in utilisation of non-biological anti-rheumatic drugs. Conclusions: This research suggests that policy-makers focus upon: explicitly considering ethical principles and formally involving stakeholders when developing policies on access to high-cost medicines; improving communication and providing information based on increased transparency; and establishing formal mechanisms for review of and appeals against PBS decisions. The comprehensive evaluation of medicine use and outcomes post-subsidy is critical for the future of the PBS. The National Medicines Policy has proved a useful framework for evaluating this access scheme.