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On Uniform Convergence

Description: In this paper, we will be concerned primarily with series of functions and a particular type of convergence which will be described. The purpose of this paper is to familiarize the reader with the concept of uniform convergence. In the main it is a compilation of material found in various references and revised to conform to standard notation.
Date: 1951
Creator: Drew, Dan Dale
Partner: UNT Libraries

Convergence Tests for Infinite Series

Description: The field of infinite series is so large that any investigation into that field must necessarily be limited to a particular phase. An attempt has been made to develop a number of tests having a wide range of applications. Particular emphasis has been placed on tests for series of positive terms.
Date: 1950
Creator: Latimer, Philip W.
Partner: UNT Libraries

Convergence of Infinite Series

Description: The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.
Date: August 1983
Creator: Abbott, Catherine Ann
Partner: UNT Libraries

Weak and Norm Convergence of Sequences in Banach Spaces

Description: We study weak convergence of sequences in Banach spaces. In particular, we compare the notions of weak and norm convergence. Although these modes of convergence usually differ, we show that in โ„“ยน they coincide. We then show a theorem of Rosenthal's which states that if {๐“โ‚™} is a bounded sequence in a Banach space, then {๐“โ‚™} has a subsequence {๐“'โ‚™} satisfying one of the following two mutually exclusive alternatives; (i) {๐“'โ‚™} is weakly Cauchy, or (ii) {๐“'โ‚™} is equivalent to the unit vector basis of โ„“ยน.
Date: December 1993
Creator: Hymel, Arthur J. (Arthur Joseph)
Partner: UNT Libraries

Towards bulk based preconditioning for quantum dotcomputations

Description: This article describes how to accelerate the convergence of Preconditioned Conjugate Gradient (PCG) type eigensolvers for the computation of several states around the band gap of colloidal quantum dots. Our new approach uses the Hamiltonian from the bulk materials constituent for the quantum dot to design an efficient preconditioner for the folded spectrum PCG method. The technique described shows promising results when applied to CdSe quantum dot model problems. We show a decrease in the number of iteration steps by at least a factor of 4 compared to the previously used diagonal preconditioner.
Date: May 25, 2006
Creator: Dongarra, Jack; Langou, Julien; Tomov, Stanimire; Channing,Andrew; Marques, Osni; Vomel, Christof et al.
Partner: UNT Libraries Government Documents Department

MILLIMETER-SCALE GENETIC GRADIENTS AND COMMUNITY-LEVEL MOLECULAR CONVERGENCE IN A HYPERSALINE MICROBIAL MAT

Description: To investigate the extent of genetic stratification in structured microbial communities, we compared the metagenomes of 10 successive layers of a phylogenetically complex hypersaline mat from Guerrero Negro, Mexico. We found pronounced millimeter-scale genetic gradients that are consistent with the physicochemical profile of the mat. Despite these gradients, all layers displayed near identical and acid-shifted isoelectric point profiles due to a molecular convergence of amino acid usage indicating that hypersalinity enforces an overriding selective pressure on the mat community.
Date: April 30, 2008
Creator: Fenner, Marsha W; Kunin, Victor; Raes, Jeroen; Harris, J. Kirk; Spear, John R.; Walker, Jeffrey J. et al.
Partner: UNT Libraries Government Documents Department

Efficient computation of matched solutions of the KV envelopeequation for periodic focusing lattices

Description: A new iterative method is developed to numerically calculate the periodic, matched beam envelope solution of the coupled Kapchinskij-Vladimirskij (KV) equations describing the transverse evolution of a beam in a periodic, linear focusing lattice of arbitrary complexity. Implementation of the method is straightforward. It is highly convergent and can be applied to all usual parameterizations of the matched envelope solutions. The method is applicable to all classes of linear focusing lattices without skew couplings, and also applies to parameters where the matched beam envelope is strongly unstable. Example applications are presented for periodic solenoidal and quadrupole focusing lattices. Convergence properties are summarized over a wide range of system parameters.
Date: January 3, 2006
Creator: Lund, Steven M.; Chilton, Sven H. & Lee, Edward P.
Partner: UNT Libraries Government Documents Department

Highly optimized fourth-order short-time approximation for pathintegrals

Description: We derive a fourth-order short-time approximation for use in imaginary-time path-integral simulations. The short-time approximation converges for all continuous and bounded from below potentials, attains quartic order of convergence for sufficiently smooth potentials, and utilizes statistically independent random variables for its construction. These properties recommend the approximation as a natural replacement of the trapezoidal Trotter-Suzuki approximation for physical systems with continuous distributions.
Date: October 1, 2006
Creator: Predescu, Cristian
Partner: UNT Libraries Government Documents Department

A three-level BDDC algorithm for Mortar discretizations

Description: In this paper, a three-level BDDC algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically non-conforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work, the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method and numerical experiments are also discussed.
Date: December 9, 2007
Creator: Kim, H. & Tu, X.
Partner: UNT Libraries Government Documents Department

Convergence analysis of a balalncing domain decomposition method for solving interior Helmholtz equations

Description: A variant of balancing domain decomposition method by constraints (BDDC) is proposed for solving a class of indefinite system of linear equations, which arises from the finite element discretization of the Helmholtz equation of time-harmonic wave propagation in a bounded interior domain. The proposed BDDC algorithm is closely related to the dual-primal finite element tearing and interconnecting algorithm for solving Helmholtz equations (FETI-DPH). Under the condition that the diameters of the subdomains are small enough, the rate of convergence is established which depends polylogarithmically on the dimension of the individual subdomain problems and which improves with the decrease of the subdomain diameters. These results are supported by numerical experiments of solving a Helmholtz equation on a two-dimensional square domain.
Date: December 10, 2008
Creator: Li,Jing & Tu, Xuemin
Partner: UNT Libraries Government Documents Department

BDDC for nonsymmetric positive definite and symmetric indefinite problems

Description: The balancing domain decomposition methods by constraints are extended to solving both nonsymmetric, positive definite and symmetric, indefinite linear systems. In both cases, certain nonstandard primal constraints are included in the coarse problems of BDDC algorithms to accelerate the convergence. Under the assumption that the subdomain size is small enough, a convergence rate estimate for the GMRES iteration is established that the rate is independent of the number of subdomains and depends only slightly on the subdomain problem size. Numerical experiments for several two-dimensional examples illustrate the fast convergence of the proposed algorithms.
Date: December 10, 2008
Creator: Tu, Xuemin & Li, Jing
Partner: UNT Libraries Government Documents Department

A Three-level BDDC algorithm for saddle point problems

Description: BDDC algorithms have previously been extended to the saddle point problems arising from mixed formulations of elliptic and incompressible Stokes problems. In these two-level BDDC algorithms, all iterates are required to be in a benign space, a subspace in which the preconditioned operators are positive definite. This requirement can lead to large coarse problems, which have to be generated and factored by a direct solver at the beginning of the computation and they can ultimately become a bottleneck. An additional level is introduced in this paper to solve the coarse problem approximately and to remove this difficulty. This three-level BDDC algorithm keeps all iterates in the benign space and the conjugate gradient methods can therefore be used to accelerate the convergence. This work is an extension of the three-level BDDC methods for standard finite element discretization of elliptic problems and the same rate of convergence is obtained for the mixed formulation of the same problems. Estimate of the condition number for this three-level BDDC methods is provided and numerical experiments are discussed.
Date: December 10, 2008
Creator: Tu, X.
Partner: UNT Libraries Government Documents Department

Scalable parallel Newton-Krylov solvers for discontinuous Galerkin discretizations

Description: We present techniques for implicit solution of discontinuous Galerkin discretizations of the Navier-Stokes equations on parallel computers. While a block-Jacobi method is simple and straight-forward to parallelize, its convergence properties are poor except for simple problems. Therefore, we consider Newton-GMRES methods preconditioned with block-incomplete LU factorizations, with optimized element orderings based on a minimum discarded fill (MDF) approach. We discuss the difficulties with the parallelization of these methods, but also show that with a simple domain decomposition approach, most of the advantages of the block-ILU over the block-Jacobi preconditioner are still retained. The convergence is further improved by incorporating the matrix connectivities into the mesh partitioning process, which aims at minimizing the errors introduced from separating the partitions. We demonstrate the performance of the schemes for realistic two- and three-dimensional flow problems.
Date: December 31, 2008
Creator: Persson, P.-O.
Partner: UNT Libraries Government Documents Department

nu-TRLan User Guide Version 1.0: A High-Performance Software Package for Large-Scale Harmitian Eigenvalue Problems

Description: The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs, users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 ...
Date: October 27, 2008
Creator: Yamazaki, Ichitaro; Wu, Kesheng & Simon, Horst
Partner: UNT Libraries Government Documents Department

Removal of Singularities from Taylor Series

Description: A mathematical procedure is described whereby the radius of convergence of a Taylor series can be increased through the inclusion of complex poles in a rational approximation. Computer results show that this technique is quite independent of the asymptotic limit of the power series and only depends on the positions of the singularities. Aside from the applications in one variable, this method vastly improves perturbative solutions to symplectic, dynamical mappings in many dimensions by removing resonances in the complex plane.
Date: August 1, 1989
Creator: La Mon, K.
Partner: UNT Libraries Government Documents Department

Fourth-Order Method for Numerical Integration of Age- and Size-Structured Population Models

Description: In many applications of age- and size-structured population models, there is an interest in obtaining good approximations of total population numbers rather than of their densities. Therefore, it is reasonable in such cases to solve numerically not the PDE model equations themselves, but rather their integral equivalents. For this purpose quadrature formulae are used in place of the integrals. Because quadratures can be designed with any order of accuracy, one can obtain numerical approximations of the solutions with very fast convergence. In this article, we present a general framework and a specific example of a fourth-order method based on composite Newton-Cotes quadratures for a size-structured population model.
Date: January 8, 2008
Creator: Iannelli, M; Kostova, T & Milner, F A
Partner: UNT Libraries Government Documents Department

Training SVMs without offset

Description: We develop, analyze, and test a training algorithm for support vector machine cla.'>sifiers without offset. Key features of this algorithm are a new stopping criterion and a set of working set selection strategies that, although inexpensive, do not lead to substantially more iterations than the optimal working set selection strategy. For these working set strategies, we establish convergence rates that coincide with the best known rates for SYMs with offset. We further conduct various experiments that investigate both the run time behavior and the performed iterations of the new training algorithm. It turns out, that the new algorithm needs less iterations and run-time than standard training algorithms for SYMs with offset.
Date: January 1, 2009
Creator: Steinwart, Ingo; Hush, Don & Scovel, Clint
Partner: UNT Libraries Government Documents Department

The use of bulk states to accelerate the band edge statecalculation of a semiconductor quantum dot

Description: We present a new technique to accelerate the convergence of the folded spectrum method in empirical pseudopotential band edge state calculations for colloidal quantum dots. We use bulk band states of the materials constituent of the quantum dot to construct initial vectors and a preconditioner. We apply these to accelerate the convergence of the folded spectrum method for the interior states at the top of the valence and the bottom of the conduction band. For large CdSe quantum dots, the number of iteration steps until convergence decreases by about a factor of 4 compared to previous calculations.
Date: May 10, 2006
Creator: Vomel, Christof; Tomov, Stanimire Z.; Wang, Lin-Wang; Marques,Osni A. & Dongarra, Jack J.
Partner: UNT Libraries Government Documents Department