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, and it is shown that more efficient computation can be obtained by relaxing the algebraic constraints on the factoriation. 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 Dissertation submitted to the School of Computational Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Summer Semester, 2008. / Date of Defense: May 22, 2008. / Riemannian Manifolds, Iterative Methods, Convergence Theory, Numerical Optimization, Eigenvalue Problems, Trust-Region Methods, Riemannian Optimization, Optimization on Manifolds / Includes bibliographical references. / Kyle Gallivan, Professor Co-Directing Dissertation; Pierre-Antoine Absil, Professor Co-Directing Dissertation; Anjaneyulu Krothapalli, Outside Committee Member; Gordon Erlebacher, Committee Member; Anuj Srivastava, Committee Member; Yousuff Hussaini, Committee Member.

This thesis presents and investigates ideas for improvement of the creation of quality centroidal voronoi tessellations on the sphere (SCVT). First, we discuss the theory of CVTs in general, and specifically on the sphere. Subsequently we consider the iterative processes, such as Lloyd's algorithm, which are used to construct them. Following this, we examine and introduce different schemes for creating their input values, known as generators, and compare the effects of these different initial points with respect to their ability to converge and the amount of work required to meet a given tolerance goal. In addition, we describe a method for density functions via images so that we can shape generator density in an intuitive manner and then implement this method with examples to demonstrate it's efficacy. / A Thesis submitted to the School of Computational Science in partial fulfillment of the requirements for the degree of Master of Science. / Degree Awarded: Summer Semester, 2008. / Date of Defense: June 18, 2008. / Cvt Scvt Centroidal Voronoi Tessellations Sphere D / Includes bibliographical references. / Max Gunzburger, Professor Co-Directing Thesis; Janet Peterson, Professor Co-Directing Thesis; Gordon Erlebacher, Committee Member.

In this work we apply quasirandom sequences to develop a derivative-free algorithm for approximating the global maximum of a given function. This work is based on previous results which used a single type of quasirandom sequence in a Brute Force approach and in an approach called Localization of Search. In this work we present several methods for computing quasirandom sequences as well as measures for determining their properties. We discuss the shortcomings of the Brute Force and Localization of Search methods and then present modifications which address these issues which culminate in a new algorithm which we call Modified Localization of Search. Our algorithm is applied to a test suite of problems and the results are discussed. Finally we present some comments on code development for our algorithm. / A Thesis submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Master of Science. / Degree Awarded: Summer Semester, 2011. / Date of Defense: April 25, 2011. / Search, Sequences, Numerical, Quasirandom, Optimization / Includes bibliographical references. / Janet Peterson, Professor Directing Thesis; Max Gunzburger, Committee Member; Gordon Erlebacher, Committee Member; John Burkardt, Committee Member.

Thesis (Ph. D.)--University of Chicago. / Vita. From the American journal of mathematics. Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--Oregon State University, 1977. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

As a part of the design process for hydro-electric generating stations, hydraulic engineers typically conduct some form of model testing. The desired outcome from the testing can vary considerably depending on the specific situation, but often characteristics such as velocity patterns, discharge rating curves, water surface profiles, and pressures at various locations are measured. Due to recent advances in computational power and numerical techniques, it is now possible to obtain much of this information through numerical modeling. Computational fluid dynamics (CFD) is a type of numerical modeling that is used to solve problems involving fluid flow. Since CFD can provide a faster and more economical solution than physical modeling, hydraulic engineers are interested in verifying the capability of CFD software. Although some literature shows successful comparisons between CFD and physical modeling, a more comprehensive study would provide the required confidence to use numerical modeling for design purposes. This study has examined the ability of the commercial CFD software Flow-3D to model a variety of spillway configurations by making data comparisons to both new and old physical model experimental data. In general, the two types of modeling have been in agreement with the provision that discharge comparisons appear to be dependent on a spillway’s height to design head ratio (P/Hd). Simulation times and required mesh resolution were also examined as part of this study. / February 2009

Thesis (M.S.)--Middle East Technical University, 2003. / Keywords: Rotary wing, thin₆layer Navier₆Stokes equations, finite volume method, structured grid, parallel processing, MPI, blade₆vortex interactions, flat plate.

Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1} Writing T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set {(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}. This leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by W(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}. The joint numerical range has been studied extensively in order to understand the joint behaviour of operators. Let K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by W_(e ) (T)=∩{W(T+K) :K∈K(H) }. The joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by W_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }. These notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given. In 2010, Müller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979. Let H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. The maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range. However the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong. / published_or_final_version / Mathematics / Master / Master of Philosophy

This work discusses isogeometric analysis as a promising alternative to standard finite element analysis. Isogeometric analysis has emerged from the idea that the act of modeling a geometry exactly at the coarsest levels of discretization greatly simplifies the refinement process by obviating the need for a link to an external representation of that geometry. The NURBS based implementation of the method is described in detail with particular emphasis given to the numerous refinement possibilities, including the use of functions of higher-continuity and a new technique for local refinement. Examples are shown that highlight each of the major features of the technology: geometric flexibility, functions of high continuity, and local refinement. New numerical approaches are introduced for modeling the fine scales within the variational multiscale method. First, a general framework is presented for seeking solutions to differential equations in a way that approximates optimality in certain norms. More importantly, it makes possible for the first time the approximation of the fine-scale Green's functions arising in the formulation, leading to a better understanding of machinery of the variational multiscale method and opening new avenues for research in the field. Second, a simplified version of the approach, dubbed the "parameter-free variational multiscale method," is proposed that constitutes an efficient stabilized method, grounded in the variational multiscale framework, that is free of the ad hoc stabilization parameter selection that has plagued classical stabilized methods. Examples demonstrate the efficacy of the method for both linear and nonlinear equations. / text

published_or_final_version / Mathematics / Master / Master of Philosophy