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 Title
 Distance Preserving Graphs
 Creator
 Zahedi, Emad
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

The computational complexity of exploring distance properties of large graphs such as realworld social networks which consist of millions of nodes is extremely expensive. Recomputing distances in subgraphs of the original graph will add to the cost. One way to avoid this is to use subgraphs where the distance between any pair of vertices is the same as in the original graph. Such a subgraph is called {\em isometric}. A connected graph is {\em distance preserving}, for which we use the...
Show moreThe computational complexity of exploring distance properties of large graphs such as realworld social networks which consist of millions of nodes is extremely expensive. Recomputing distances in subgraphs of the original graph will add to the cost. One way to avoid this is to use subgraphs where the distance between any pair of vertices is the same as in the original graph. Such a subgraph is called {\em isometric}. A connected graph is {\em distance preserving}, for which we use the abbreviation dp, if it has an isometric subgraph of every order. In this framework we study dp graphs from both the structural and algorithmic perspectives. First, we study the structural nature of dp graphs. This involves classifying graphs based on the dp property and the relation between dp graphs to other graph classes. Second, we study the recognition problem of dp graphs. We intend to develop efficient algorithms for finding isometric subgraphs as well as deciding whether a graph is dp or not.
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 Title
 HYBRID METHODS FOR RADIATION TRANSPORT USING INTEGRAL DEFERRED CORRECTION
 Creator
 Crockatt, Michael M.
 Date
 2018
 Collection
 Electronic Theses & Dissertations
 Description

This thesis provides roughly three contributions to the study of defect correction methods and hybrid methods for radiation transport. First, a modification of traditional integral deferred correction (IDC) timeintegration schemes is proposed that significantly reduces the storage requirements of the methods. These methods, which we call lowstorage IDC or LSIDC methods, require storing only one copy of each stage vector throughout the iteration process, whereas traditional IDC methods...
Show moreThis thesis provides roughly three contributions to the study of defect correction methods and hybrid methods for radiation transport. First, a modification of traditional integral deferred correction (IDC) timeintegration schemes is proposed that significantly reduces the storage requirements of the methods. These methods, which we call lowstorage IDC or LSIDC methods, require storing only one copy of each stage vector throughout the iteration process, whereas traditional IDC methods require two copies of each vector. The convergence and stability properties of the methods are examined in a variety of settings, and both analytical and experimental results are provided. A nonlinear ODE and a linear transport equation are used to compare the accuracy and storage requirements of LSIDC integrators with other fully implicit schemes including diagonally implicit RungeKutta (DIRK) and spacetime discontinuous Galerkin (STDG) methods. The results demonstrate that LSIDC methods have similar accuracy but significantly reduced memory requirements compared to other fully implicit methods.Second, extensions of a collisionbased hybrid method for timedependent radiation transport simulations are discussed. The hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low and highresolution discrete ordinates approximations are applied, respectively. The use of arbitrarily highorder numerical approximations is emphasized, with particular attention paid to highorder timeintegration schemes. A range of timeintegrators are considered including DIRK, STDG, IDC, and LSIDC methods of up to fifthorder accuracy. Convergence results in one spatial dimension are provided, and it is found that while the hybrid methods exhibit convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximations is comparable to standard discrete ordinates approximations in many cases. A test problem in twodimensional xygeometry consisting of a mockup of a standard hohlraum configuration is used to compare the computational efficiency of the hybrid methods with standard discrete ordinates methods. It is observed that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using angular quadratures of order 2N and N/2 for the uncollided and collided fluxes, respectively, usually reduces solve time by a factor of 2 or more and in many cases also yields a reduction in solution error by a factor of up to 2. However, it is noted that the specified hybrid methods require approximately 4.25 times the storage of the corresponding standard discrete ordinates methods.Finally, two mechanisms for increasing the effectiveness of the hybrid methods are presented. The first is a reconstruction procedure for mapping between arbitrary discrete ordinates quadratures within the context of these hybrid methods. This approach, called Nyström reconstruction, is shown to be significantly more accurate than previous reconstruction methods. When applied to the twodimensional hohlraum problem, it is observed that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using a Nyström reconstruction procedure and angular quadratures of order N and N/4 for the uncollided and collided fluxes, respectively, consistently reduces solve time by a factor of between 4 and 8 while increasing memory requirements by only 6% and producing little to no increase in solution error. Lastly, variations of hybrid methods using IDC integrators are described in which the hybrid approach is written as a twogrid iterative method in angle that is combined with an IDC timeintegration scheme. It is demonstrated that the resulting methods are able to iteratively reduce the error due to the application of discrete ordinates quadratures of different resolutions to the collided and uncollided components.
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 Title
 Estimating covariance structure in high dimensions
 Creator
 Maurya, Ashwini
 Date
 2016
 Collection
 Electronic Theses & Dissertations
 Description

Many of scientific domains rely on extracting knowledge from highdimensional data set to provide insights into complex mechanisms underlying these data. Statistical modeling hasbecome ubiquitous in the analysis of high dimensional data for exploring the largescale gen regulatory networks in hope of developing better treatments for deadly diseases, in search o better understanding of cognitive systems, and in prediction of volatility in stock market in the hope of averting the potential risk...
Show moreMany of scientific domains rely on extracting knowledge from highdimensional data set to provide insights into complex mechanisms underlying these data. Statistical modeling hasbecome ubiquitous in the analysis of high dimensional data for exploring the largescale gen regulatory networks in hope of developing better treatments for deadly diseases, in search o better understanding of cognitive systems, and in prediction of volatility in stock market in the hope of averting the potential risk. Statistical analysis in these highdimensional data set yields better results only if an estimation procedure exploits hidden structures underlying the data. This thesis develops flexible estimation procedures with provable theoretical guarantees for estimating the unknown covariance structures underlying data generating process. Of particular interest are procedures that can be used on high dimensional data sets where the number of samples n is much smaller than the ambient dimension p. Due to the importance of structure estimation, the methodology is developed for the estimation of both covariance and its inverse in parametric and as well in nonparametric framework.
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