scholarly journals Evaluation of the Capability of the Multigrid Method in Speeding Up the Convergence of Iterative Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-5
Author(s):  
Iman Harimi ◽  
Mohsen Saghafian
Keyword(s):  
Convergence Rate ◽  
Boundary Conditions ◽  
Laplace Equation ◽  
Multigrid Method ◽  
Low Frequency ◽  
Dirichlet Boundary Conditions ◽  
Frequency Function ◽  
Frequency Error ◽  
Multigrid Algorithm ◽  
The Difference

The performance of the multigrid method and the effect of different grid levels on the convergence rate are evaluated. The two-, three-, and four-level V-cycle multigrid methods with the Gauss-Seidel iterative solver are employed for this purpose. The numerical solution of the one-dimensional Laplace equation with the Dirichlet boundary conditions is obtained using these methods. For the Laplace equation, a two-frequency function involving high- and low-frequency components is defined. It is observed that, however, the GS method can smooth out the high-frequency error components properly, but because the difference scheme for Laplace equation is remarkably concise, in the fine grids, a very large number of iterations are needed for extending the boundary conditions into the domain. Furthermore, the obtained results reveal that the number of necessary iterations for convergence is reduced considerably by employing the two-level multigrid algorithm. But increasing the number of levels of algorithm does not have any significant effect on the convergence rate in this study.

scholarly journals Another Look at Neural Multigrid

1997 ◽  
Vol 08 (02) ◽  
pp. 191-205
Author(s):  
Martin Bäker
Keyword(s):  
Laplace Equation ◽  
Multigrid Method ◽  
Main Idea ◽  
Neural Nets ◽  
Critical Slowing Down ◽  
Neural Net ◽  
Multigrid Algorithm ◽  
Slowing Down ◽  
Interpolation Operators ◽  
Conjugate Gradient Solver

We present a new multigrid method called neural multigrid which is based on joining multigrid ideas with concepts from neural nets. The main idea is to use the Greenbaum criterion as a cost functional for the neural net. The algorithm is able to learn efficient interpolation operators in the case of the ordered Laplace equation with only a very small critical slowing down and with a surprisingly small amount of work comparable to that of a Conjugate Gradient solver In the case of the two-dimensional Laplace equation with SU(2) gauge fields at β = 0 the learning exhibits critical slowing down with an exponent of about z≈0.4. The algorithm is able to find quite good interpolation operators in this case as well. Thereby it is proven that a practical true multigrid algorithm exists even for a gauge theory. An improved algorithm using dynamical blocks that will hopefully overcome the critical slowing down completely is sketched.

A NEWTON-MULTIGRID METHOD FOR NUMERICAL SIMULATION OF SLIDER AIR BEARING

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1647-1650
Author(s):  
HEJUN DU ◽  
QIANG LI ◽  
JOR HUAT ONG ◽  
CHANG SHU
Keyword(s):  
Numerical Simulation ◽  
Multigrid Method ◽  
Low Frequency ◽  
Least Square ◽  
Frequency Error ◽  
Air Bearing ◽  
Slider Air Bearing ◽  
Simulation Problem ◽  
Late Stages

In this paper, a Newton-Multigrid method is presented to solve the numerical simulation of the slider air bearing. For each fixed attitude in the specified grid, the Newton method is used to achieve the pressure distribution of the slider by solving the generalized Reynolds equations discretized by the least square finite difference (LSFD) method. Between the different sizes of grids, full approximation multigrid method is used to accelerate the convergence rate by eliminating the dominating low frequency error mode in the late stages of convergence. From the case study of the slider air bearing, it shows that the Newton-Multigrid method is accurate and efficient for the numerical simulation problem.

scholarly journals SELF-CORRECTING MULTIGRID SOLVER

2006 ◽  
Vol 03 (02) ◽  
pp. 137-151
Author(s):  
JEROME L. V. LEWANDOWSKI
Keyword(s):  
Convergence Rate ◽  
Elliptic Problem ◽  
Short Wavelength ◽  
Multigrid Method ◽  
Source Term ◽  
Elliptic Problems ◽  
The Self ◽  
Multigrid Algorithm ◽  
Algebraic Error ◽  
Computational Work

A new multigrid algorithm based on the method of self-correction for the solution of elliptic problems is described. The method exploits information contained in the residual to dynamically modify the source term (right-hand side) of the elliptic problem. It has shown that the self-correcting solver is more efficient at damping the short wavelength modes of the algebraic error than its standard equivalent. When used in conjunction with a multigrid method, the resulting solver displays an improved convergence rate with no additional computational work.

The Stationary Action Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Boundary Conditions ◽  
Lagrange Equation ◽  
Variational Calculus ◽  
Action Principle ◽  
Dirichlet Boundary Conditions ◽  
Lagrange Formulation ◽  
Helmholtz Conditions ◽  
Stationary Action ◽  
The Difference ◽  
Corner Conditions

This crucial chapter focuses on the stationary action principle. It introduces Lagrangian mechanics, using first-order variational calculus to derive the Euler–Lagrange equation, and the inverse problem is described. The chapter then considers the Ostrogradsky equation and discusses the properties of the extrema using the second-order variation to the action. It then discusses the difference between action functions (of Dirichlet boundary conditions) and action functionals of the extremal path. The different types of boundary conditions (Dirichlet vs Neumann) are elucidated. Topics discussed include Hessian conditions, Douglas’s theorem, the Jacobi last multiplier, Helmholtz conditions, Noether-type variation and Frenet–Serret frames, as well as concepts such as on shell and off shell. Actions of non-continuous extremals are examined using Weierstrass–Erdmann corner conditions, and the action principle is written in the most general form as the Hamilton–Suslov principle. Important applications of the Euler–Lagrange formulation are highlighted, including protein folding.

scholarly journals Low Frequency Estimates for Long Range Perturbations in Divergence Form

2011 ◽  
Vol 63 (5) ◽  
pp. 961-991 ◽  
Author(s):  
Jean-Marc Bouclet
Keyword(s):  
Boundary Conditions ◽  
Long Range ◽  
Dirichlet Boundary ◽  
Low Frequency ◽  
Divergence Form ◽  
Riesz Transforms ◽  
Dirichlet Boundary Conditions ◽  
Commutator Estimates ◽  
Positive Commutator

Abstract We prove a uniformcontrol as z → 0 for the resolvent (P−z)−1 of long range perturbations P of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension d ≥ 3 when P is defined on ℝd and in dimension d ≥ 2 when P is defined outside a compact obstacle with Dirichlet boundary conditions.

The low-frequency limiting behavior of ambipolar diffusive models of impedance spectroscopy

2021 ◽  
Vol 2021 (12) ◽  
pp. 123206
Author(s):  
G Barbero ◽  
L R Evangelista ◽  
P Tilli
Keyword(s):  
Boundary Conditions ◽  
Electrochemical Impedance ◽  
Low Frequency ◽  
Frequency Region ◽  
Interfacial Property ◽  
Limiting Behavior ◽  
The Difference ◽  
The One ◽  
Diffusional Model ◽  
Adsorption Desorption

Abstract The Poisson–Nernst–Planck (PNP) diffusional model is a successful theoretical framework to investigate the electrochemical impedance response of insulators containing ionic impurities to an external ac stimulus. Apparent deviations of the experimental spectra from the predictions of the PNP model in the low frequency region are usually interpreted as an interfacial property. Here, we provide a rigorous mathematical analysis of the low-frequency limiting behavior of the model, analyzing the possible origin of these deviation related to bulk properties. The analysis points toward the necessity to consider a bulk effect connected with the difference in the diffusion coefficients of cations and anions (ambipolar diffusion). The ambipolar model does not continuously reach the behavior of the one mobile ion diffusion model when the difference in the mobility of the species vanishes, for a fixed frequency, in the cases of ohmic and adsorption–desorption boundary conditions. The analysis is devoted to the low frequency region, where the electrodes play a fundamental role in the response of the cell; thus, different boundary conditions, charged to mimic the non-blocking character of the electrodes, are considered. The new version of the boundary conditions in the limit in which one of the mobility is tending to zero is deduced. According to the analysis in the dc limit, the phenomenological parameters related to the electrodes are frequency dependent, indicating that the exchange of electric charge from the bulk to the external circuit, in the ohmic model, is related to a surface impedance, and not simply to an electric resistance.

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