Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples

The Health Status of Southern Children: A Neglected Regional Disparity

Purpose: Great variations exist in child health outcomes among states in the United States, with southern states consistently ranked among the lowest in the country. Investigation of the geographical distribution of children’s health status and the regional factors contributing to these outcomes has been neglected. We attempted to identify the degree to which region of residence may be linked to health outcomes for children with the specific aim of determining whether living in the southern region of the United States is adversely associated with children’s health status. Methods: A child health index (CHI) that ranked each state in the United States was computed by using statespecific composite scores generated from outcome measures for a number of indicators of child health. Five indicators for physical health were chosen (percent low birth weight infants, infant mortality rate, child death rate, teen death rate, and teen birth rates) based on their historic and routine use to define health outcomes in children. Indicators were calculated as rates or percentages. Standard scores were calculated for each state for each health indicator by subtracting the mean of the measures for all states from the observed measure for each state. Indicators related to social and economic status were considered to be variables that impact physical health, as opposed to indicators of physical health, and therefore were not used to generate the composite child health score. These variables were subsequently examined in this study as potential confounding variables. Mapping was used to redefine regional groupings of states, and parametric tests (2-sample t test, analysis of means, and analysis-of-variance F tests) were used to compare the means of the CHI scores for the regional groupings and test for statistical significance. Multiple regression analysis computed the relationship of region, social and economic indicators, and race to the CHI. Simple linear-regression analyses were used to assess the individual effect of each indicator. Results: A geographic region of contiguous states, characterized by their poor child health outcomes relative to other states and regions of the United States, exists within the “Deep South” (Mississippi, Louisiana, Arkansas, Tennessee, Alabama, Georgia, North Carolina, South Carolina, and Florida). This Deep-South region is statistically different in CHI scores from the US Census Bureau– defined grouping of states in the South. The mean of CHI scores for the Deep-South region was \u3e1 SD below the mean of CHI scores for all states. In contrast, the CHI score means for each of the other 3 regions were all above the overall mean of CHI scores for all states. Regression analysis showed that living in the Deep- South region is a stronger predictor of poor child health outcomes than other consistently collected and reported variables commonly used to predict children’s health. Conclusions: The findings of this study indicate that region of residence in the United States is statistically related to important measures of children’s health and may be among the most powerful predictors of child health outcomes and disparities. This clarification of the poorer health status of children living in the Deep South through spatial analysis is an essential first step for developing a better understanding of variations in the health of children. Similar to early epidemiology work linking geographic boundaries to disease, discovering the mechanisms/pathways/causes by which region influences health outcomes is a critical step in addressing disparities and inequities in child health and one that is an important and fertile area for future research. The reasons for these disparities may be complex and synergistically related to various economic, political, social, cultural, and perhaps even environmental (physical) factors in the region. This research will require the use and development of new approaches and applications of spatial analysis to develop insights into the societal, environmental, and historical determinants of child health that have been neglected in previous child health outcomes and policy research. The public policy implications of the findings in this study are substantial. Few, if any, policies identify these children as a high-risk group on the basis of their region of residence. A better understanding of the depth and breadth of disparities in health, education, and other social outcomes among and within regions of the United States is necessary for the generation of policies that enable policy makers to address and mitigate the factors that influence these disparities. Defining and clarifying the regional boundaries is also necessary to better inform public policy decisions related to resource allocation and the prevention and/or mitigation of the effects of region on child health. The identification of the Deep South as a clearly defined sub-region of the Census Bureau’s regional definition of the South suggests the need to use more culturally and socially relevant boundaries than the Census Bureau regions when analyzing regional data for policy development

Analysis of Iterative Methods for the Linear Boltzmann Transport Equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the discretisation parameters. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport.Comment: 27 pages, 8 figure

Discontinuous Galerkin Methods for the Linear Boltzmann Transport Equation

Radiation transport is an area of applied physics that is concerned with the propagation and distribution of radiative particle species such as photons and electrons within a material medium. Deterministic models of radiation transport are used in a wide range of problems including radiotherapy treatment planning, nuclear reactor design and astrophysics. The central object in many such models is the (linear) Boltzmann transport equation, a high-dimensional partial integro-differential equation describing the absorption, scattering and emission of radiation. In this thesis, we present high-order discontinuous Galerkin finite element discretisations of the time-independent linear Boltzmann transport equation in the spatial, angular and energetic domains. Efficient implementations of the angular and energetic components of the scheme are derived, and the resulting method is shown to converge with optimal convergence rates through a number of numerical examples. The assembly of the spatial scheme on general polytopic meshes is discussed in more detail, and an assembly algorithm based on employing quadrature-free integration is introduced. The quadrature-free assembly algorithm is benchmarked against a standard quadrature-based approach, and an analysis of the algorithm applied to a more general class of discontinuous Galerkin discretisations is performed. In view of developing efficient linear solvers for the system of equations resulting from our discontinuous Galerkin discretisation, we exploit the variational structure of the scheme to prove convergence results and derive a posteriori solver error estimates for a family of iterative solvers. These a posteriori solver error estimators can be used alongside standard implementations of the generalised minimal residual method to guarantee that the linear solver error between the exact and approximate finite element solutions (measured in a problem-specific norm) is below a user-specified tolerance. We discuss a family of transport-based preconditioners, and our linear solver convergence results are benchmarked through a family of numerical examples

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an $hp$-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and $hp$-version a priori error analysis of the proposed method, by deriving suitable $hp$-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.Comment: 27 pages, 2 figure

Preservation of whole antibodies within ancient teeth

Archaeological remains can preserve some proteins into deep time, offering remarkable opportunities for probing past events in human history. Recovering functional proteins from skeletal tissues could uncover a molecular memory related to the life-history of the associated remains. We demonstrate affinity purification of whole antibody molecules from medieval human teeth, dating to the 13th–15th centuries, from skeletons with different putative pathologies. Purified antibodies are intact retaining disulphide-linkages, are amenable to primary sequences analysis, and demonstrate apparent immunoreactivity against contemporary EBV antigen on western blot. Our observations highlight the potential of ancient antibodies to provide insights into the long-term association between host immune factors and ancient microbes, and more broadly retain a molecular memory related to the natural history of human health and immunity

Conversion of colonoscopy to flexible sigmoidoscopy: an unintended consequence of quality measurement in endoscopy

OBJECTIVE: To quantify the proportion of requests for colonoscopy that are performed as flexible sigmoidoscopy and documented reasons for this in ordinary UK hospital practice. To determine the effect these requests have on colonoscopy completion rate if they are included in the denominator of the calculated rate by individual endoscopist. DESIGN: Retrospective study of 22 months flexible sigmoidoscopy practice at a major UK teaching hospital. All flexible sigmoidoscopies performed had their associated request form examined. SETTING: UK NHS University Hospital. PATIENTS: All patients receiving outpatient flexible sigmoidoscopy from January 2013 to October 2014 with no exclusions. INTERVENTION: Conversion of colonoscopy to flexible sigmoidoscopy. MAIN OUTCOME MEASURES: Conversion of colonoscopy to flexible sigmoidoscopy, reason for conversion and adjusted colonoscopy completion rate. RESULTS: 71 of the 3526 flexible sigmoidoscopies performed (2.0%), representing 71 of 5905 colonoscopy requests (1.2%). Conversion reason was noted only in 26 (37%) of converted cases. Adjustment of colonoscopy completion rate to include conversions pushed four of our unit's 22 endoscopists below the UK national 90% standard. CONCLUSIONS: Conversion to flexible sigmoidoscopy occurs in 1.2% of patients originally booked for colonoscopy. The reason for this conversion is often unqualified and may be inappropriate. Conversion can affect the colonoscopy completion rate, and therefore, should be included in endoscopists’ overall performance statistics

On emotions and salsa : some thoughts on dancing to rethink consumers

Dance forms are a big business, highly marketable commoditized cultural universes, with a plethora of markets constructed around their spirit, vitality and possibilities. In this paper, we explore one particular dance form, that of Salsa, arguing that as consumer researchers we look for a more vibrant vocabulary and mindset with which to capture the experiential and transcendental nature of such social associations. We demonstrate that the metaphor of dancing is useful to revitalize our notions of consumer actions; taking them out of the grey mundane of calculative and rational action into the possibilities of emotional economies constructed around the effervescence and vitality of the social (cf. Maffesoli, [1996])

Preservation of whole antibodies within ancient teeth

Archaeological remains can preserve some proteins into deep time, offering remarkable opportunities for probing past events in human history. Recovering functional proteins from skeletal tissues could uncover a molecular memory related to the life-history of the associated remains. We demonstrate affinity purification of whole antibody molecules from medieval human teeth, dating to the 13th – 15th centuries, from skeletons with different putative pathologies. Purified antibodies are intact retaining disulphide-linkages, are amenable to primary sequences analysis, and demonstrate apparent immunoreactivity against contemporary EBV antigen on western blot. Our observations highlight the potential of ancient antibodies to provide insights into the long-term association between host immune factors and ancient microbes, and more broadly retain a molecular memory related to the natural history of human health and immunity