A relatively new preconditioning technique called support graph preconditioning has many merits over the traditional incomplete factorization based methods. A major limitation of this technique is that it is applicable to symmetric diagonally dominant matrices only. This work presents a technique that can be used to transform the symmetric positive definite matrices arising from elliptic finite element problems into symmetric diagonally dominant M-matrices. The basic idea is to approximate the element gradient matrix by taking the gradients along chosen edges, whose unit vectors form a new coordinate system. For Lagrangian elements, the rows of the element gradient matrix in this new coordinate system are scaled edge vectors, thus a diagonally dominant symmetric semidefinite M-matrix can be generated to approximate the element stiffness matrix. Depending on the element type, one or more such coordinate systems are required to obtain a global nonsingular M-matrix. Since such approximation takes place at the element level, the degradation in the quality of the preconditioner is only a small constant factor independent of the size of the problem. This technique of element coordinate transformations applies to a variety of first order Lagrangian elements. Combination of this technique and other techniques enables us to construct an M-matrix preconditioner for a wide range of second order elliptic problems even with higher order elements. Another contribution of this work is the proposal of a new variant of Vaidya’s support graph preconditioning technique called modified domain partitioned support graph preconditioners. Numerical experiments are conducted for various second order elliptic finite element problems, along with performance comparison to the incomplete factorization based preconditioners. Results show that these support graph preconditioners are superior when solving ill-conditioned problems. In addition, the domain partition feature provides inherent parallelism, and initial experiments show a good potential of parallelization and scalability of these preconditioners.

The purpose of the inverse-transport problems that we address is to reconstruct the material distribution inside an unknown object undergoing a nondestructive evaluation. We assume that the object is subjected to incident beams of photons or particles and that the exiting radiation is measured with detectors around the periphery of the object. In the present work we focus on problems in which radiation can undergo significant scattering within the optically thick object. We develop a set of reconstruction strategies to infer the material distribution inside such objects. When we apply these strategies to a set of neutron-tomography test problems we find that the results are substantially superior to those obtained by previous methods. We first demonstrate that traditional analytic methods such as filtered back projection (FBP) methods do not work for very thick, highly scattering problems. Then we explore deterministic optimization processes, using the nonlinear conjugate gradient iterative updating scheme to minimize an objective functional that characterizes the misfits between forward predicted measurements and actual detector readings. We find that while these methods provide more information than the analytic methods such as FBP, they do not provide sufficiently accurate solutions of problems in which the radiation undergoes significant scattering. We proceed to present some advances in inverse transport methods. Our strategies offer several advantages over previous reconstruction methods. First, our optimization procedure involves the systematic use of both deterministic and stochastic methods, using the strengths of each to mitigate the weaknesses of the other. Another key feature is that we treat the material (a discrete quantity) as the unknown, as opposed to individual cross sections (continuous variables). This changes the mathematical nature of the problem and greatly reduces the dimension of the search space. In our hierarchical approach we begin by learning some characteristics of the object from relatively inexpensive calculations, and then use knowledge from such calculations to guide more sophisticated calculations. A key feature of our strategy is dimension-reduction schemes that we have designed to take advantage of known and postulated constraints. We illustrate our approach using some neutron-tomography model problems that are several mean-free paths thick and contain highly scattering materials. In these problems we impose reasonable constraints, similar to those that in practice would come from prior information or engineering judgment. Our results, which identify exactly the correct materials and provide very accurate estimates of their locations and masses, are substantially better than those of deterministic minimization methods and dramatically more efficient than those of typical stochastic methods.

Elliptic partial differential equations that are used to model physical phenomena give rise to large sparse linear systems. Such systems can be symmetric positive definite and can be solved by the preconditioned conjugate gradients method. In this thesis, we develop support graph preconditioners for symmetric positive definite matrices that arise from the finite element discretization of elliptic partial differential equations. An object oriented code is developed for the construction, integration and application of these preconditioners. Experimental results show that the advantages of support graph preconditioners are retained in the proposed extension to the finite element matrices.

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. Classical tomography fails to reconstruct the optical properties of thick scattering objects because it does not adequately account for the scattering component of the neutron beam intensity exiting the sample. We proposed a new method of computed tomography which employs an inverse problem analysis of both the transmitted and scattered images generated from a beam passing through an optically thick object. This inverse problem makes use of a computationally efficient, two-dimensional forward problem based on neutron transport theory that effectively calculates the detector readings around the edges of an object. The forward problem solution uses a Step-Characteristic (SC) code with known uncollided source per cell, zero boundary flux condition and Sn discretization for the angular dependence. The calculation of the uncollided sources is performed by using an accurate discretization scheme given properties and position of the incoming beam and beam collimator. The detector predictions are obtained considering both the collided and uncollided components of the incoming radiation. The inverse problem is referred as an optimization problem. The function to be minimized, called an objective function, is calculated as the normalized-squared error between predicted and measured data. The predicted data are calculated by assuming a uniform distribution for the optical properties of the object. The objective function depends directly on the optical properties of the object; therefore, by minimizing it, the correct property distribution can be found. The minimization of this multidimensional function is performed with the Polack Ribiere conjugate-gradient technique that makes use of the gradient of the function with respect to the cross sections of the internal cells of the domain. The forward and inverse models have been successfully tested against numerical results obtained with MCNP (Monte Carlo Neutral Particles) showing excellent agreements. The reconstructions of several objects were successful. In the case of a single intrusion, TNTs (Tomography Neutron Transport using Scattering) was always able to detect the intrusion. In the case of the double body object, TNTs was able to reconstruct partially the optical distribution. The most important defect, in terms of gradient, was correctly located and reconstructed. Difficulties were discovered in the location and reconstruction of the second defect. Nevertheless, the results are exceptional considering they were obtained by lightening the object from only one side. The use of multiple beams around the object will significantly improve the capability of TNTs since it increases the number of constraints for the minimization problem.

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [52]-58)

In each iteration of the interior point method (IPM) at least one linear system has to be solved. The main computational effort of IPMs consists in the computation of these linear systems. Solving the corresponding linear systems with a direct method becomes very expensive for large scale problems. In this thesis, we have been concerned with using an iterative method for solving the reduced KKT systems arising in IPMs for linear programming. The augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the normal equations form. We design a block triangular preconditioner for this system which is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. We use the preconditioned conjugate gradients (PCG) method to solve the augmented system. Although the augmented system is indefinite, short recurrence iterative methods such as PCG can be applied to indefinite system in certain situations. This approach has been implemented within the HOPDM interior point solver. The KKT system is solved approximately. Therefore, it becomes necessary to study the convergence of IPM for this inexact case. We present the convergence analysis of the inexact infeasible path-following algorithm, prove the global convergence of this method and provide complexity analysis.

The Alexander terrane is an allochthonous terrane within the North American Cordillera. New structural mapping and geochronology within the southern Alexander terrane in the vicinity of Porcher Island provides evidence for two major tectonic events. The oldest is Late Silurian to Early Devonian magmatism and deformation assigned to the Klakas orogeny. The area has subsequently been affected by mid-Cretaceous conjugate shear zones potentially associated with tectonic escape. Northwest-striking sinistral shear zones characterize mid-Cretaceous deformation in the western Coast Belt south of Prince Rupert in north coastal British Columbia. Structurally focused mapping and geochronology has revealed a component of lateral extension to this deformation. General flow characteristics of the shear zones are identified by comparison of fabric patterns to published models for fabric development in shear zones. U-Pb ages from synkinematic dykes constrain motion on northwest-striking sinistral transpressional shear zones, including the Useless, Barrett and Salt Lagoon shear zones, to ca. 104 – 96 Ma, and dextral transpression on the north-striking Telegraph Passage shear zone to ca. 97.6 ± 0.2 Ma. The geometry, kinematics, and coeval nature of these shear zones suggests that they formed in part as a ductile conjugate set. The presence of similarly-oriented conjugate shear bands in the apex zone between sinistral and dextral shear zones further reinforces this interpretation. The orientation of these conjugate sets indicates a component of north-northwest east-southeast extension. The conjugate shear zones merge together into the Grenville Channel shear zone, a sinistral transpressional shear zone with high strike-parallel stretch. A U-Pb age of 103 ± 32 Ma from a synkinematic dyke in the Grenville Channel shear zone coincides with a previously published Lu-Hf age of 102.6 ± 3.7 Ma on synkinematic garnet. Overall, structural and geochronological data from Porcher Island and surrounding area in north coastal British Columbia indicate that mid-Cretaceous deformation was characterized by ENE-WSW (orogen normal) shortening and NNW-SSE (orogen parallel) extension. This local strain regime is consistent with large-scale mid-Cretaceous tectonic escape as proposed for the northern Cordillera at that time, expressed in coeval sinistral faulting in the Coast Belt and dextral faulting in the northern Omineca belt. The Late Silurian to Early Devonian Ogden Channel complex is a mafic to intermediate metaplutonic-metamorphic complex within the southern Alexander terrane on southern Porcher Island and adjacent Pitt Island in north coastal British Columbia. Lithological characteristics of the complex suggest that it represents the mid-crustal roots of a volcanic arc. An age of 413.3 ± 2.5 Ma from a comparatively weakly deformed quartz diorite dyke indicates that the synkinematic Ogden Channel complex is at least in part Early Devonian in age, corresponding to the Klakas orogeny that affected the Alexander terrane in southeast Alaska. Crosscutting relationships indicate that individual intrusions within the Ogden Channel complex were emplaced syn- to post-kinematically with respect to southwest-vergent sinistral reverse deformation (present coordinates). The structural and lithological characteristics of the Ogden Channel complex are consistent with the interpretation that this part of the Alexander terrane was located in the upper plate of a northeast-dipping subduction zone, which culminated in the Klakas orogeny.

Thesis (Ph. D.)--University of Hong Kong, 1993.

A thesis by Marc Ashley Bayliss entitled ‘Virus-like particles as a novel platform for delivery of protective Burkholderia antigens’ and submitted to the University of Exeter for the degree of Doctor of Philosophy. There is currently no licensed vaccine available for the global tropical pathogen Burkholderia pseudomallei which is the causative agent of melioidosis and a potential bio-threat agent. The capsule polysaccharide (CPS) expressed by B. pseudomallei has been shown to offer some protection against bacterial challenge. Polysaccharide immunogenicity can be enhanced by conjugation to a carrier protein and several licensed vaccines utilise this technology. Virus-like particles (VLPs) are non-infectious, non-replicating, viral proteins that self-assemble into viral structures and are in several licensed vaccines as primary antigens. VLPs are also effective delivery platforms for foreign antigens by genetic insertion or chemical conjugation. iQur, a collaborator on this project, has developed Tandem CoreTM that consists of two genetically linked hepatitis B core proteins that allow insertion of large proteins into each core whilst remaining assembly competent. The aim of this thesis was to assess the protective efficacy of Tandem CoreTM VLPs chemically conjugated to CPS and Tandem CoreTM Burkholderia protein fusion constructs. This involved three objectives; reduce the cost of CPS extraction; identify immunogenic Burkholderia proteins; and test candidate vaccine efficacy in an animal model of acute melioidosis against B. pseudomallei challenge. To reduce the cost of extraction, CPS was purified from B. thailandensis strain E555 and bacterial culture CPS concentration optimised which first required development of a quantitative ELISA. Immunogenic Burkholderia proteins were identified from the literature but Tandem CoreTM fusion constructs containing these proteins were not assembly competent. The Burkholderia proteins were added as co-antigens to the VLP CPS conjugate vaccine but did not improve efficacy. Tandem CoreTM VLPs conjugated to CPS were protective against B. pseudomallei challenge and were compared to CPS conjugated to Crm197: a commercially available carrier protein used in several licensed vaccines. At lower challenge doses, survival was greater in mice vaccinated with the VLP-CPS conjugate although at higher doses, Crm197-CPS efficacy was greater.

Two alternative technologies of communication inside a parallelized Conjugate-Gradient algorithm are presented and compared to the well known hypercubecommunication. The amount of communication is diskussed in detail. A large range of numerical results corroborate the theoretical investigations.