In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.

Thesis (M.S.)--University of Missouri--Rolla, 2007. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed March 18, 2008) Includes bibliographical references (p. 89-90).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. 55).

Thesis--Bonn. / Extra t.p. with thesis statement inserted. Includes bibliographical references (p. 55).

A common approach for evaluating complex multiattributed choice alternatives is judgment decomposition: the alternatives are decomposed into a number of value-relevant attributes, the decision maker evaluates each alternative with respect to each attribute, and those single-attribute evaluations are aggregated across the attributes by a formal composition rule. One primary assumption behind decomposition is that it would produce a more reliable outcome than direct holistic evaluations. Although there is some empirical evidence that decomposed procedures can improve the reliability of evaluations, the extent of decomposition can have a considerable effect on the resulting evaluations. This research investigated, theoretically and experimentally, the effects of decomposition level on intrarater reliability in multiattribute alternative evaluation. In a theoretical study, using an additive value composition model with random variables, the composite variance of alternative evaluation was analyzed with respect to the level of decomposition. The composite variance of decomposed evaluation was derived from the variances in the components recomposed using a Statistical method of error propagation. By analyzing the composite variance as a function of the number of attributes used, possible effects of decomposition level were predicted and explained. The analysis showed that the variance of an alternative evaluation is a decreasing function with respect to the level of decomposition, in most cases, and that the marginal reduction of variance diminishes as decomposition level increases. In an experimental study, intrarater test-retest Convergence was examined for a job evaluation with different levels of decomposition. Subjects evaluated six hypothetical job alternatives using four levels of decomposition that ranged from a single overall evaluation to evaluations on twelve highly specific attributes. Intrarater convergence was measured by mean absolute deviations and Pearson correlations between the evaluation scores in two identical sessions separated by two weeks. The mean absolute deviations decreased significantly with respect to the decomposition levels while the Pearson correlations were not significant. Further analyses indicated that the mean absolute deviations decreased with a diminishing rate of reduction, as the decomposition level increased. The research results suggest that decomposition reduces the variability of each alternative evaluation, in most situations. The results, however, also suggest that decomposition may not improve the consistency of preference order of the alternatives that is often important in practical choice decisions. / Ph. D.

Large scale optimization problems are tractable only if they are somehow decomposed. Hierarchical decompositions are inappropriate for some types of problems and do not parallelize well. Sobieszczanski-Sobieski has proposed a nonhierarchical decomposition strategy for nonlinear constrained optimization that is naturally parallel. Despite some successes on engineering problems, the algorithm as originally proposed fails on simple two dimensional quadratic programs. Here, the algorithm is carefully analyzed by testing it on simple quadratic programs, thereby recognizing the problems with the algorithm. Different modifications are made to improve its robustness and the best version is tested on a larger dimensional example. Some of the changes made are very fundamental, affecting the updating of the various tuning parameters present in the original algorithm. The algorithm involves solving a given problem by dividing it into subproblems and a final coordination phase. The results indicate good success with small problems. On testing it with a larger dimensional example, it was discovered that there is a basic flaw in the coordination phase which needs to be rectified. / Master of Science

Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers. Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems. The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiprocessor. / Ph. D.

In this study, a heterogeneous flow model is proposed based on a non-overlapping domain decomposition method. The model combines potential flow and incompressible viscous flow. Both flow domains contain a free surface boundary. The heterogeneous domain decomposition method is formulated following the Dirichlet-Neumann method. Both an implicit scheme and an explicit scheme are proposed. The algebraic form of the implicit scheme is of the same form of the Dirichlet--Neumann method, whereas the explicit scheme can be interpreted as the classical staggered scheme using the splitting of the Dirichlet-Neumann method. The explicit scheme is implemented based on two numerical solvers, a Boundary element method (BEM) solver for the potential flow model, and a finite element method (FEM) solver for the Navier-Stokes equations (NSE). The implementation based on the two solvers is validated using numerical examples. / Graduation date: 2013

Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 127-134).

Thesis (M.S.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009. / Includes bibliographical references.