Thesis (Ph. D.)--University of Wisconsin, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Meshless methods have been developed to avoid the numerical burden imposed by meshing in the Finite Element Method. Such methods are especially attrac- tive in problems that require repeated updates to the mesh, such as problems with discontinuities or large geometrical deformations. Although meshing is not required for solving problems with meshless methods, the use of meshless methods gives rise to different challenges. One of the main challenges associated with meshless methods is imposition of essential boundary conditions. If exact interpolants are used as shape functions in a meshless method, imposing essen- tial boundary conditions can be done in the same way as the Finite Element Method. Another attractive feature of meshless methods is that their use involves compu- tations that are largely independent from one another. This makes them suitable for implementation to run on highly parallel computing systems. Highly par- allel computing has become widely available with the introduction of software development tools that enable developing general-purpose programs that run on Graphics Processing Units. In the current work, the Moving Regularized Interpolation method has been de- veloped, which is a novel method of constructing meshless shape functions that achieve exact interpolation. The method is demonstrated in data interpolation and in partial differential equations. In addition, an implementation of the Element-Free Galerkin method has been written to run on a Graphics Processing Unit. The implementation is described and its performance is compared to that of a similar implementation that does not make use of the Graphics Processing Unit.

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This thesis presents and evaluates a generic algorithm for incrementally computing the dominant singular subspaces of a matrix. The relationship between the generality of the results and the necessary computation is explored. The performance of this method, both numerical and computational, is discussed in terms of the algorithmic parameters, such as block size and acceptance threshhold. Bounds on the error are presented along with a posteriori approximations of these bounds. Finally, a group of methods are proposed which iteratively improve the accuracy of computed results and the quality of the bounds. / A Thesis submitted to the Department of Computer Science in partial fulfillment of the requirements for the degree of Master of Science. / Degree Awarded: Summer Semester, 2004. / Date of Defense: April 19, 2004. / Updating, Numerical Linear Algebra, Singular Value Decomposition, URV Factorization, Subspace Tracking / Includes bibliographical references. / Kyle Gallivan, Professor Directing Thesis; Anuj Srivastava, Committee Member; Robert van Engelen, Committee Member.

The presence of data dependences between statements in a loop iteration space imposes strict constraints on statement order and loop restructuring when preserving program semantics. A compiler determines the safe partial ordering of statements that enhance performance by explicitly disproving the presence of dependences. As a result, the false positive rate of a dependence analysis technique is a crucial factor in the effectiveness of a restructuring compiler's ability to optimize the execution of performance-critical code fragments. This dissertation investigates reducing the false positive rate by improving the accuracy of analysis methods for dependence problems and increasing the total number of problems analyzed. Fundamental to these improvements is the rephrasing of the dependence problem in terms of Chains of Recurrences (CR), a formalism that has been shown to be conducive to efficient loop induction variable analysis. An infrastructure utilizing CR-analysis methods and enhanced dependence testing techniques is developed and tested. Experimental results indicate capabilities of dependence analysis methods can be improved without a reduction in efficiency. This results in a reduction in the false positive rate and an increase in the number of optimized and parallelized code fragments. / A Dissertation submitted to the Department of Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester, 2007. / July 2, 2007. / Chains of Recurrences, Depedence Testing, Loop Analysis, Induction Variable, Loop Analysis, CR / Includes bibliographical references. / Robert Van Engelen, Professor Directing Dissertation; Paul Ruscher, Outside Committee Member; Kyle Gallivan, Committee Member; David Whalley, Committee Member; Xin Yuan, Committee Member.

Discusses types of mathematical errors that may occur in scientific work. / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: H. C. Griffith, Professor Directing Paper. / "August, 1958." / Includes bibliographical references (leaf 26).

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In this thesis, we describe Schensted's algorithm for finding the length of a longest increasing subsequence of a finite sequence. Schensted's algorithm also constructs a bijection between permutations of the first N natural numbers and standard bitableaux of size N. We also describe the Hillman-Grassl algorithm which constructs a bijection between reverse plane partitions and the solutions in natural numbers of a linear equation involving hook lengths. Pascal programs and sample output for both algorithms appear in the appendix. In addition, we describe the operations on standard bitableaux corresponding to the operations of inverting and reversing permutations. Finally, we show that these operations generate the dihedral group D_4

Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and specification of the model error covariance. Up to now, the increased computational cost associated with w4D-Var has prevented its practical implementation. Various simplifications to reduce the computational burden have been considered, including writing the model error covariance as a scalar multiple of the background error covariance and modeling the model error. In this thesis, the main objective is the development of computationally feasible techniques for the improved representation of the model error statistics in a data assimilation system. Three new approaches are considered. A Monte Carlo method that uses an ensemble of w4D-Var systems to obtain flow-dependent estimates to the model error statistics. The evaluation of statistical diagnostic equations involving observation residuals to estimate the model error covariance matrix. An adaptive tuning procedure based on the sensitivity of a short-range forecast error measure to the model error DAS parametrization. The validity and benefits of these approaches are shown in two stages of numerical experiments. A proof-of-concept is shown using the Lorenz multi-scale model and the shallow water equations for a one-dimensional domain. The results show the potential of these methodologies to produce improved state estimates, as compared to other approaches in data assimilation. It is expected that the techniques presented will find an extended range of applications to assess and improve the performance of a w4D-Var system.

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