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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 83 (1999), S. 231-257 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N22
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In this work we calculate the eigenvalues obtained by preconditioning the discrete Helmholtz operator with Sommerfeld-like boundary conditions on a rectilinear domain, by a related operator with boundary conditions that permit the use of fast solvers. The main innovation is that the eigenvalues for two and three-dimensional domains can be calculated exactly by solving a set of one-dimensional eigenvalue problems. This permits analysis of quite large problems. For grids fine enough to resolve the solution for a given wave number, preconditioning using Neumann boundary conditions yields eigenvalues that are uniformly bounded, located in the first quadrant, and outside the unit circle. In contrast, Dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased. These eigenvalue properties yield the first insight into the behavior of iterative methods such as GMRES applied to these preconditioned problems.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 23 (1982), S. 20-33 
    ISSN: 1436-4646
    Keywords: Discrete Newton Algorithm ; Function Minimization ; Conjugate Gradient
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A Newton-like method is presented for minimizing a function ofn variables. It uses only function and gradient values and is a variant of the discrete Newton algorithm. This variant requires fewer operations than the standard method whenn 〉 39, and storage is proportional ton rather thann 2.
    Type of Medium: Electronic Resource
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  • 3
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir besprechen mehrere Varianten des konjugierten Gradienten-Algorithmus unter Hervorhebung der Parameterbestimmung ohne Minimierung entlang von Linien und der Konvergenzbeschleunigung durch Zerlegung. Die hier verwendeten Zerlegungen sind dem nichtlinearen SSOR-Algorithmus verwandt. Das Verhalten der Methoden wird illustriert an der Diskretisierung eines nichtlinearen elliptischen partiellen Randwertproblems, nämlich der Minimalflächen-Gleichung. Wir entwickeln auch einen konjugierten Gradienten-Algorithmus mit Zerlegungen für Minimierung mit von oben und unten beschränkten Variabeln; ferner zeigen wir eine Anwendung der Methode auf das Hindernisproblem bei der Minimalflächen-Gleichung.
    Notes: Abstract Several variants of the conjugate gradient algorithm are discussed with emphasis on determining the parameters without performing line searches and on using splitting techniques to accelerate convergence. The splittings used here are related to the nonlinear SSOR algorithm. The behavior of the methods is illustrated on a discretization of a nonlinear elliptic partial differential boundary value problem, the minimal surface equation. A conjugate gradient algorithm with splittings is also developed for constrained minimization with upper and lower bounds on the variables, and the method is applied to the obstacle problem for the minimal surface equation.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Numerical algorithms 25 (2000), S. 387-406 
    ISSN: 1572-9265
    Keywords: interior point methods ; linear programming ; iterative methods for linear systems ; adaptive algorithms ; self-timing algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this work we devise efficient algorithms for finding the search directions for interior point methods applied to linear programming problems. There are two innovations. The first is the use of updating of preconditioners computed for previous barrier parameters. The second is an adaptive automated procedure for determining whether to use a direct or iterative solver, whether to reinitialize or update the preconditioner, and how many updates to apply. These decisions are based on predictions of the cost of using the different solvers to determine the next search direction, given costs in determining earlier directions. We summarize earlier results using a modified version of the OB1-R code of Lustig, Marsten, and Shanno, and we present results from a predictor–corrector code PCx modified to use adaptive iteration. If a direct method is appropriate for the problem, then our procedure chooses it, but when an iterative procedure is helpful, substantial gains in efficiency can be obtained.
    Type of Medium: Electronic Resource
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