This thesis develops an early warning system framework for assessing systemic risks and for predicting systemic events, i.e. periods of extreme financial instability with potential real costs, over the short horizon of six quarters and the long horizon of twelve quarters on the panel of 14 countries both advanced and developing. Firstly, Financial Stress Index is built aggregating indicators from equity, foreign exchange, security and money markets in order to identify starting dates of systemic financial crises for each country in the panel. Secondly, the selection of early warning indicators for assessment and prediction of systemic risks is undertaken in a two- step approach; relevant prediction horizons for each indicator are found by means of a univariate logit model followed by the application of Bayesian model averaging method to identify the most useful indicators. Next, logit models containing useful indicators only are estimated on the panel while their in-sample and out-of-sample performance is assessed by a variety of measures. Finally, having applied the constructed EWS for both horizons to the Czech Republic it was found that even though models for both horizons perform very well in-sample, i.e. both predict 100% of crises, only the long model attains the maximum utility of 0,5 as...

I investigate the relationship between expected stock returns and trading volume. I collect together 522 estimates from 46 studies and conduct the first meta-analysis in this field. Use of Bayesian model averaging and Frequentist model averaging help me to discover the most influential factors that affect the return-volume relationship, since I control for more than 50 differences among primary articles such as midyear and type of data, length of the primary dataset, size of market, or model employed. In the end, I find out that the relation between expected stock returns and trading volume is rather negligible. On the other hand, the contemporaneous relation between returns and volume is positive. These two findings cut the mixed results from previously written studies. Moreover, the investigated relationship is influenced by the size of country of interest and the level of its development. Besides the primary studies that employ higher data frequency provide substantially larger estimates than the studies with data from longer time periods. On the contrary, there is no difference among different estimation methodologies used. Finally, I employ classical and modern techniques such as stem-based methodology for publication bias detection, and I find evidence for it in this field. 1

Este trabalho aborda o problema da inversão do vetor momento magnético, com ampla aplicação na Ressonância Nuclear Magnética (RNM). Em vez de uma sequência de impulsos e de abordarmos somente o problema de conduzir o vetor de -e3 para +e3, escolhemos uma lei de controle limitada e analisamos o processo de várias iterações (voltas completas). Através do método da medianização, obtemos uma solução explícita aproximada para o sistema e, através dela e de alguns teoremas auxiliares sobre rotações, discutimos a propagação do erro em módulo e fase cometido após a realização dessas iterações. / This dissertation considers the problem of inversion of the magnetic moment vector, with wide application in Nuclear Magnetic Resonance (NMR). Instead of a pulse sequence and only approach the problem of driving the vector from -e3 to +e3, we choose limited controls and we analyze several iterations of the process (laps). By the averaging method, we obtain an approximate explicit solution for the system and through this method, together with some auxiliary theorems on rotations, we discuss the propagation of error in magnitude and phase committed after performing these iterations.

In this doctoral thesis, we propose new iterative methods for solving a class of convex optimization problems. In general, we consider problems in which the objective function is composed of a finite sum of convex functions and the set of constraints is, at least, convex and closed. The iterative methods we propose are basically designed through the combination of incremental subgradient methods and string-averaging algorithms. Furthermore, in order to obtain methods able to solve optimization problems with many constraints (and possibly in high dimensions), generally given by convex functions, our analysis includes an operator that calculates approximate projections onto the feasible set, instead of the Euclidean projection. This feature is employed in the two methods we propose; one deterministic and the other stochastic. A convergence analysis is proposed for both methods and numerical experiments are performed in order to verify their applicability, especially in large scale problems. / Nesta tese de doutorado, propomos novos métodos iterativos para a solução de uma classe de problemas de otimização convexa. Em geral, consideramos problemas nos quais a função objetivo é composta por uma soma finita de funções convexas e o conjunto de restrições é, pelo menos, convexo e fechado. Os métodos iterativos que propomos são criados, basicamente, através da junção de métodos de subgradientes incrementais e do algoritmo de média das sequências. Além disso, visando obter métodos flexíveis para soluções de problemas de otimização com muitas restrições (e possivelmente em altas dimensões), dadas em geral por funções convexas, a nossa análise inclui um operador que calcula projeções aproximadas sobre o conjunto viável, no lugar da projeção Euclideana. Essa característica é empregada nos dois métodos que propomos; um determinístico e o outro estocástico. Uma análise de convergência é proposta para ambos os métodos e experimentos numéricos são realizados a fim de verificar a sua aplicabilidade, principalmente em problemas de grande escala.

Neste trabalho, nós consideramos o seguinte problema de valor inicial para uma equação diferencial funcional retardada com impulsos { \'x PONTO\' = \'varepsilon\' f (t, \'x IND.t\'), t \' DIFERENTE\' \'t IND. k\', \'DELTA\' x(\'t IND. k\') = \'varepsilon\' \' I IND. k\' (x ( \'t IND.k\')), k = 0, 1, 2, ... \'x IND. t IND.0\' = \' phi\', onde f está definida em um aberto \' OMEGA\' de R x \' G POT. -\' ([- r, 0], \' R POT. n\') e assume valores em \'R POT. n\', \' \'varepsilon\' \'G POT. - ([ - r, 0], \'R POT.n\'), r .0, onde \' G POT -\' ([ - r, 0], \' R POT. n\') denota o espaço das funções de [ - r, 0] em \' R POT. n\' que estão regradas e contínuas à esquerda. Além disso, \' t IND.0 < \' t IND. 1\'< ... \'t IND. k\' < ... são momentos pré determinados de impulsos tais que \'lim SOBRE k SETA + \' INFINITO\' \'t IND. k = + \' INFINITO\' e \'DELTA\'x (\' t IND.k\') = x ( \'t POT. + IND > k) - x (\'t IND. k). Os operadores de impulso \' I IND. k\', k = 0, 1, ... são funções contínuas de \'R POT. n\' em \' R POT. n\'. Consideramos, também, que para cada x \'varepsilon\' \' G POT. -\' ([- r, \' INFINITO\'), \'R POT. n\'), t \'SETA\' f (t, \'x IND. t\') é uma função localmente Lebesgue integrável e sua integral indefinida satisfaz uma condição do tipo Carathéodory. Além disso, f é Lipschitziana na segunda variável. Definimos \' f IND. 0\' ( \'phi\') = \' lim SOBRE T \' SETA\' \' INFINITO\' \'1 SUP. T \' INT. SUP. T INF. \' T IND.0\' f (t, \' PSI\') dt e \' I IND. 0(x) = \' lim SOBRE T \'SETA\' \' INFINITO\' \' 1 SUP. T\' \' SIGMA\' IND. 0 < ou = \' t IND. i\' < T onde \' psi\' \'varepsilon\' \' G POT. -\' ([ - r, 0], \' R POT. n\', e consideremos a seguinte equação diferencial funcioonal autônoma \" média\" y PONTO = \' varepsilon\' [ \' f IND. 0\' (\' y IND. t\' + \' I IND> 0\' (y (t))], \'y IND. t IND. 0 = \' phi\'. Então provamos que, sob certas condições, a solução x(t) de (1) se aproxima da solução y(t) de (2) em tempo assintoticamente grande / In this present work, we condider the following initial value problem for a retarded functional differential equation with impulses { \'x POINT\' = \'varepsilon\' f (t, \'x IND.t\'), t \' DIFFERENT\' \'t IND. k\', \'DELTA\' x(\'t IND. k\') = \'varepsilon\' \' I IND. k\' (x ( \'t IND.k\')), k = 0, 1, 2, ... \'x IND. t IND.0\' = \' phi\', where f está defined in a open set \' OMEGA\' de R x \' G POT. -\' ([- r, 0], \' R POT. n\'), r >0, and takes values in \'R POT. n\', \' \'varepsilon\' \'G POT. - ([ - r, 0], \'R POT.n\'), r .0, where \' G POT -\' ([ - r, 0], \' R POT. n\') denotes the space of regulated functions from [ - r, 0] to \' R POT. n\' which are left continuous. Furthermore, \' t IND.0 < \' t IND. 1\'< ... \'t IND. k\' < ... are pre-assigned moments of impulse effects such that \'lim ON k ARROW + \' THE INFINITE\' \'t IND. k = + \' THE INFINITE\' e \'DELTA\'x (\' t IND.k\') = x ( \'t POT. + IND>k) - x (\'t IND. k). The impulse operators \' I IND. k\', k = 0, 1, ... are continuous mappings from \'R POT. n\' to \' R POT. n\'. For each x \'varepsilon\' \' G POT. -\' ([- r, \' THE INFINITE\'), \'R POT. n\'), t \'ARROW\' f (t, \'x IND. t\') is locally Lebesgue integrable and its indefinite integral satisfies a Carathéodory. Moreover, f é Lipschitzian with respect to the second variable. We define \' f IND. 0\' ( \'phi\') = \' lim ON T \' ARROW\' \' THE INFINITE\' \'1 SUP. T \' INT. SUP. T INF. \' T IND.0\' f (t, \' PSI\') dt and \' I IND. 0(x) = \' lim ON T \'ARROW\' \' THE INFINITE\' \' 1 SUP. T\' \' SIGMA\' IND. 0 < or = \' t IND. i\' < T where \' psi\' \'varepsilon\' \' G POT. -\' ([ - r, 0], \' R POT. n\', and consider the \"averaged\" autonomous functional differential equation \'y PONTO = \' varepsilon\' [ \' f IND. 0\' (\' y IND. t\' + \' I IND> 0\' (y (t))], \'y IND. t IND. 0 = \' phi\'. Then we prove that, under certain conditions, the solution x(t) of (1) in aproximates the solution y(t) de (2) in an asymptotically large time interval

La contamination des ressources en eaux souterraines par une phase organique non miscible à l'eau couramment appelée NAPL (Non Aqueous Phase Liquid) constitue aujourd'hui un défi scientifique majeur compte tenu de la durée de vie d'un tel polluant. Bien que l'activité bactérienne (généralement présente sous forme de biofilm) joue un rôle crucial dans le devenir à long terme de ces effluents, peu d'études existent à l'heure actuelle sur son impact dans des conditions multiphasiques (i.e., à proximité de la source). En effet, dans la zone saturée, sous l'action des forces capillaires, le NAPL se retrouve souvent piégé, en effet, sous forme de «gouttelettes» au niveau des pores. Ce comportement spécifique au polluant modifie la dynamique du système biofilm/milieu poreux saturé et d'importantes questions restent encore ouvertes : accessibilité du polluant, modification de la tension interfaciale, production de biosurfactant, effet de toxicité (inhibition de la croissance bactérienne). Pour tenter de répondre à ces questions, nous avons adopté une approche aussi bien théorique qu'expérimentale. L'approche théorique porte sur le développement d'un modèle macroscopique décrivant le transport multiphasique en milieu poreux pour un système eau/NAPL/biofilm. Elle repose sur la méthode de prise de moyenne volumique, appliqué aux équations décrivant le couplage écoulement/transport à l'échelle du pore, permettant d'effectuer le changement d'échelle et dériver un modèle à deux équations. Le modèle est établit sous les hypothèses d'équilibre de masse local à l'interface fluide/biofilm et les contraintes associées à ces hypothèses ont étés définies. L'influence des caractéristiques microscopiques (arrangement des grains, fraction volumique du biofilm, distribution des blobs de NAPL, mouillabilité) sur les propriétés effectives du milieu (coefficient de dispersion, coefficient d'échange de masse) est discutée au travers des résultats issus des simulations. Ensuite, le modèle macroscopique a été comparé avec succès à la simulation numérique direct à l'échelle du pore pour la géométrie 2D complexe considérée. Quant à l'approche expérimentale, elle consiste à étudier le transport et la biodégradation du toluène en présence des bactéries Pseudomonas Putida F1 à l'aide d'un milieu poreux transparent 2D (micromodèle). Premièrement, nous avons étudié la dissolution du toluène résiduel sans bactéries et des courbes de dissolution du toluène ont été obtenues. Les résultats de dissolution du toluène en condition abiotique ont été comparés avec succès aux résultats du modèle théorique. Ensuite, l'étude expérimentale en micromodèle a porté sur la dissolution du toluène en condition biotique. Les résultats de ces études (courbes de dissolution et évolution de la saturation résiduelle) ont montré un impact significatif de la présence des bactéries sur les processus de dissolution par comparaison au cas abiotique. / Contamination of groundwater resources by an immiscible organic phase commonly called NAPL (Non Aqueous Phase Liquid) represents a major scientific challenge considering the residence time of such a pollutant. Although bacterial activity (usually in the form of biofilm) plays a crucial role in the long term fate of these effluents, very few works are focused on the study of such processes in multiphase conditions (oil/water/biofilm systems). The NAPL often gets trapped, in fact, under the action of capillary forces in the saturated zone in the form of «droplets» within the pores. This specific pollutant behavior changes the dynamics of biofilm /saturated porous medium system where important questions remain open: accessibility of the pollutant, changes in interfacial tension, biosurfactant production, toxicity effect (inhibition of bacterial growth). Modeling the transport of chemical species in the presence of bacteria is an extremely complex issue in terms of scale. We will use an experimental and theoretical approach to address these questions. In this thesis, we developed a macroscopic model describing multiphase transport in porous media for a water/NAPL/biofilm system. A volume averaging method has been applied here to the equations at the pore scale to make the upscaling and derive the model. This two-equation model is established under the assumptions of local mass equilibrium at the fluid/biofilm interface and the constraints associated with these assumptions were defined. The effect of microscopic features (arrangement of grains, volume fraction of the biofilm, distribution of NAPL blobs, wettability) on the effective properties of the media (dispersion coefficient, mass exchange coefficient) is discussed through some results from simulations. Subsequently, the macroscopic model has been successfully compared with the direct numerical simulation at pore scale for a 2D complex geometry. The experimental approach consists of studying transport and biodegradation of toluene in the presence of bacteria Pseudomonas Putida F1 using a flowcell. First, we studied the dissolution of toluene in abiotic conditions and toluene dissolution curves were obtained. The results of toluene dissolution in abiotic conditions were compared with success the results of the theoretical model. Finally, an experimental study in flowcell on the dissolution of toluene under biotic conditions was performed. The results of these studies (dissolution curve and evolution of toluene saturation) showed a significant impact of the presence of bacteria on the dissolution process compared to the abiotic case.

Neste trabalho foi proposta uma estratégia de modelagem hidrológica para a transformação chuva vazão da Bacia do Ribeirão da Onça (B.R.O) utilizando-se técnicas de auto calibração com análise de incertezas e de ponderação de modelos. Foi utilizado o modelo hidrológico Soil and Water Assessment Tool (SWAT), por ser um modelo que possui uma descrição física e de maneira distribuída dos processos hidrológicos da bacia. Foram propostas cinco diferentes séries de precipitação e esquemas de interpolação espacial a serem utilizados como dados de entrada para o modelo SWAT. Em seguida, utilizou-se o método semiautomático Sequential Uncertainty Fitting ver.-2 (SUFI-2) para a auto calibração e análise de incertezas dos parâmetros do modelo e produção de respostas com intervalos de incerteza para cada uma das séries de precipitação utilizadas. Por fim, foi utilizado o método de ponderação bayesiana de modelos (BMA) para o pós-processamento estocástico das respostas. Os resultados da análise de incerteza dos parâmetros do modelo SWAT indicam uma não adequação do método Soil Conservation Service (SCS) para simulação da geração do escoamento superficial, juntamente com uma necessidade de maior investigação das propriedades físicas do solo da bacia. A análise da precisão e acurácia dos resultados das séries de precipitação em comparação com a resposta combinada pelo método BMA sugerem a última como a mais adequada para a simulação chuva-vazão na B.R.O. / This study proposed an approach to the hydrological modeling of the Ribeirão da Onças Basin (B.R.O) based on automatic calibration and uncertainty analysis methods, together with model averaging. The Soil and Water Assessment Tool (SWAT) was used due to its distributed nature and physical description of hydrologic processes. An ensemble, composed by five different precipitation schemes, based on different sources and spatial interpolation methods was used. The Sequential Uncertainty Fitting ver-2 (SUFI-2) procedure was used for automatic calibration and uncertainty analysis of the SWAT model parameters, together with generation of streamflow simulations with uncertainty intervals. Following, the Bayesian Model Averaging (BMA) was used to merge the different responses into a single probabilistic forecast. The results of the uncertainty analysis for the SWAT parameters show that the Soil Conservation Service (SCS) model for surface runoff prediction may not be suitable for the B.R.O, and that more investigations about the soil physical properties at the Basin are recommended. An analysis of the accuracy and precision of the simulations produced by the precipitation ensemble members against the BMA simulation supports the use of the latter as a suitable framework for streamflow simulations at the B.R.O.

Com o avanço tecnológico e econômico, que facilitaram o processo de comunicação e aumento do poder de compra, transações com cartão de crédito tornaram-se o principal meio de pagamento no varejo nacional e internacional (Bolton e Hand , 2002). Neste aspecto, o aumento do número de transações com cartão de crédito é crucial para a geração de mais oportunidades para fraudadores produzirem novas formas de fraudes, o que resulta em grandes perdas para o sistema financeiro (Chan et al. , 1999). Os índices de fraudes têm mostrado que transações no comércio eletrônico (e-commerce) são mais arriscadas do que transações presencias em terminais, pois aquelas não fazem uso de processos seguros e eficientes de autenticação do portador do cartão, como utilização de senha eletrônica. Como os fraudadores se adaptam rapidamente às medidas de prevenção, os modelos estatísticos para detecção de fraudes precisam ser adaptáveis e flexíveis para evoluir ao longo do tempo de maneira dinâmica. Raftery et al. (2010) desenvolveram um método chamado Dynamic Model Averaging (DMA), ou Ponderação Dinâmica de Modelos, que implementa um processo de atualização contínuo ao longo do tempo. Nesta dissertação, desenvolvemos modelos DMA no espaço de transações eletrônicas oriundas do comércio eletrônico que incorporem as tendências e características de fraudes em cada período de análise. Também desenvolvemos modelos de regressão logística clássica com o objetivo de comparar as performances no processo de detecção de fraude. Os dados utilizados para tal são provenientes de uma empresa de meios de pagamentos eletrônico. O experimento desenvolvido mostra que os modelos DMA apresentaram resultados melhores que os modelos de regressão logística clássica quando analisamos a medida F e a área sob a curva ROC (AUC). A medida F para o modelo DMA ficou em 58% ao passo que o modelo de regressão logística clássica ficou em 29%. Já para a AUC, o modelo DMA alcançou 93% e o modelo de regressão logística clássica 84%. Considerando os resultados encontrados para os modelos DMA, podemos concluir que sua característica de atualização ao longo do tempo se mostra um grande diferencial em dados como os de fraude, que sofrem mudanças de comportamento a todo momento. Deste modo, sua aplicação se mostra adequada no processo de detecção de transações fraudulentas no ambiente de comércio eletrônico. / Regarding technological and economic development, which made communication process easier and increased purchasing power, credit card transactions have become the primary payment method in national and international retailers (Bolton e Hand , 2002). In this scenario, as the number of transactions by credit card grows, more opportunities are created for fraudsters to produce new ways of fraud, resulting in large losses for the financial system (Chan et al. , 1999). Fraud indexes have shown which e-commerce transactions are riskier than card present transactions, since those do not use secure and efficient processes to authenticate the cardholder, such as using personal identification number (PIN). Due to fraudsters adapt quickly to fraud prevention measures, statistical models for fraud detection need to be adaptable and flexible to change over time in a dynamic way. Raftery et al. (2010) developed a method called Dynamic Model Averaging (DMA), which implements a process of continuous updating over time. In this thesis, we develop DMA models within electronic transactions coming from ecommerce environment, which incorporate the trends and characteristics of fraud in each period of analysis. We have also developed classic logistic regression models in order to compare their performances in the fraud detection processes. The database used for the experiment was provided by a electronic payment service company. The experiment shows that DMA models present better results than classic logistic regression models in respect to the analysis of the area under the ROC curve (AUC) and F measure. The F measure for the DMA was 58% while the classic logistic regression model was 29%. For the AUC, the DMA model reached 93% and the classical model reached 84%. Considering the results for DMA models, we can conclude that its update over time characteristic makes a large difference when it comes to the analysis of fraud data, which undergo behavioral changes continuously. Thus, its application has proved to be appropriate for the detection process of fraudulent transactions in the e-commerce environment.

Les systèmes de climatisation et de ventilation sont souvent composés de configurations type jets impactant, sur leur partie terminale. Ainsi, les flux d’air soufflés viennent impacter des obstacles munis de fentes, de différentes formes, afin d’améliorer le mélange. Les conditions de confinement et de soufflage provoquent parfois un inconfort au niveau acoustique. Les nuisances acoustiques générées sont dues à un phénomène de boucles de rétroaction se traduisant par l’apparition des sons auto-entretenus. La production du son par un écoulement fluide en champ libre ou en interaction avec une structure a fait l’objet de nombreuses études. Dans le cas d’un champ acoustique et pour un écoulement à faible nombre de Mach la résolution du corollaire énergétique de Howe permet d’évaluer la puissance acoustique générée ou absorbée par les interactions entre le champ acoustique et l’écoulement. Le calcul de cette puissance nécessite la connaissance de trois paramètres : la vorticité, la vitesse et la vitesse acoustique par des méthodes analytiques ou en utilisant des données expérimentales. Expérimentalement, la mesure du champ cinématique, pour en déduire la vorticité, nécessite une technique de mesure tridimensionnelle. Pour cela une plate-forme expérimentale, utilisant de la vélocimétrie Laser, a été développée, et équipée pour générer les écoulements d’un jet plan. Les champs cinématiques de ces écoulements ont été mesurés en utilisant la technique PIV, avec un dispositif de PIV stéréoscopique. Les champs cinématiques de trente plans parallèles ont été mesurés afin d’étudier les champs de vitesses correspondants. Deux méthodes de reconstruction ont été appliquées à ces plans : la POD et la moyenne de phase. Le volume est obtenu par une interpolation des plans reconstruits donnant accès aux trois composantes de la vitesse. Pour valider ces méthodes de reconstruction en 3D à faible coût, elles étaient comparées à des mesures expérimentales réalisées par le même dispositif expérimental, dans les mêmes conditions, par la PIV tomographique donnant accès aux champs cinématiques tridimensionnels. / Air conditioning and ventilation systems are often composed of jets having a configuration of an impinging jet, on their end part. Thus, the blown airflows impact slotted obstacles of different shapes to improve mixing. The conditions of confinement and blowing sometimes cause acoustic incompatibility. The acoustic noises generated are due to a phenomenon of feedback loops resulting of the appearance of self-sustained sounds. The production of sound by a free flow or in interaction with a structure has been the subject of many studies. In the case of an acoustic field and for a flow of low Mach number, Howe's energetic correlation is used to evaluate the acoustic power generated or absorbed by the interactions between the acoustic field and the flow. The calculation of this power requires the knowledge of three parameters : vorticity, velocity and acoustic velocity by analytical methods or by using experimental data. Experimentally, the measurement of the kinematic field, to deduce the vorticity, requires a three-dimensional measurement technique. For this purpose, an experimental platform, using laser velocimetry, has been developed and equipped to generate flows of a plane jet. The kinematic fields of these flows were measured using the PIV technique, with a stereoscopic PIV device. The kinematic fields of thirty parallel planes were measured to study the corresponding velocity fields. Two reconstruction methods have been applied to these plans : the POD and the phase average. The volume is obtained by an interpolation of the reconstructed planes giving access to the three components of the velocity. To validate these low-cost 3D reconstruction methods, they were compared to experimental measurements made by the same experimental setup, under the same conditions, by using the tomographic PIV giving access to the three-dimensional kinematic fields.

Proton computed tomography (pCT) is an image modality that will improve treatment planning for patients receiving proton radiation therapy compared with the current techniques, which are based on X-ray CT. Images are reconstructed in pCT by solving a large and sparse system of linear equations. The size of the system necessitates matrix-partitioning and parallel reconstruction algorithms to be implemented across some sort of cluster computing architecture. The prototypical algorithm to solve the pCT system is the algebraic reconstruction technique (ART) that has been modified into parallel versions called block-iterative-projection (BIP) methods and string-averaging-projection (SAP) methods. General purpose graphics processing units (GPGPUs) have hundreds of stream processors for massively parallel calculations. A GPGPU cluster is a set of nodes, with each node containing a set of GPGPUs. This thesis describes a proton simulator that was developed to generate realistic pCT data sets. Simulated data sets were used to compare the performance of a BIP implementation against a SAP implementation on a single GPGPU with the data stored in a sparse matrix structure called the compressed sparse row (CSR) format. Both BIP and SAP algorithms allow for parallel computation by creating row partitions of the pCT linear system. The difference between these two general classes of algorithms is that BIP permits parallel computations within the row partitions yet sequential computations between the row partitions, whereas SAP permits parallel computations between the row partitions yet sequential computations within the row partitions. This thesis also introduces a general partitioning scheme to be applied to a GPGPU cluster to achieve a pure parallel ART algorithm while providing a framework for column partitioning to the pCT system, as well as show sparse visualization patterns that can be found via specified ordering of the equations within the matrix.