China
Africa
Explore chapters and articles related to this topic
Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
The multigrid method is one of the most efficient general iterative methods known today. The key word here is “general.” More efficient schemes can be found for certain problems or certain choices of grids, but it is difficult to find a method more efficient than multigrid for the general case. The multigrid technique can be applied using any of the iterative schemes discussed in this chapter as the “smoother,” although the Gauss–Seidel procedure will be used to illustrate the main points of this technique in the introductory material presented here. The objective of the multigrid technique is to accelerate the convergence of an iterative scheme.
Numerical Methods for ODEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Multigrid methods are iterative methods for solving systems of linear equations arising from differential equations. Generally, different grid sizes are used with only a few iterations per grid. The last approximation on one grid becomes the first approximation on the next grid.
Speckle Methods and Applications
Published in Toru Yoshizawa, Handbook of Optical Metrology, 2015
Unwrapping is the procedure which removes these 2π phase jumps and the result is converted into desired continuous phase function, thus phase unwrapping is an important final procedure for DSPI fringe analysis process. The process is carried out by adding or subtracting 2π each time the phase map presents a discontinuity. Even though these methods are well suited for unwrapping noise-free data, the process becomes more difficult when the absolute phase difference between the adjacent pixels with no discontinuities is greater than π. These spurious jumps may be caused by noise, discontinuous phase jumps, and regional under-sampling in the fringe pattern. If any of such local phase inconsistencies are present, an error appears which is propagated along the unwrapping path. To avoid this problem, sophisticated algorithms have been proposed [57]. These methods include unwrapping by the famous branch-cut technique [63] and with discrete cosine transform (DCT) [64]. The former method detects error inducing locations in the phase map and connects them to each other by a branch-cut, which must not be crossed by a subsequent unwrapping with a simple path dependent method. The DCT approaches the problem of unwrapping by solving the Poisson equation relating wrapped and unwrapped phases by 2D DCT. This way, any path dependency is evaded and error propagation does not appear “localized” as for scanning methods. The advantage of this technique is that noise in wrapped data has less influence, and that the whole unwrapping is performed in a single step (no step function is created), though the time consumed is rather high compared to path dependent methods. The multigrid algorithm is an iterative algorithm that adopts a recursive mode of operation and forms the basis for most multigrid algorithms [57,65]. Multigrid methods are a class of techniques for rapidly solving partial differential equations (PDEs) on large grids. These methods are based on the idea of applying Gauss–Seidel relaxation schemes on coarser, smaller grids. After application of any phase unwrapping algorithm, “true” phase jumps in unwrapped phase data higher than 2π cannot be identified correctly, as these are interpreted as changes of order. Figure 8.9 shows the speckle fringe analysis on a centrally loaded, rigidly clamped circular diaphragm using the TPS “difference-of-phases” method [37] and DCT unwrapping algorithm [57].
Newton-type multilevel optimization method
Published in Optimization Methods and Software, 2022
Chin Pang Ho, Michal Kočvara, Panos Parpas
This idea of multigrid was extended to optimization algorithms. Nash [19] proposed a multigrid framework for unconstrained infinite-dimensional convex optimization problems. Examples of such problems could be found in the area of optimal control. Following the idea of Nash, many multigrid optimization methods were further developed [10,16–20,25]. In particular, Wen and Goldfarb [25] provided a line search-based multigrid optimization algorithm under the framework in [19], and further extended the framework to nonconvex problems. Gratton et al. [10] provided a sophisticated trust-region version of multigrid optimization algorithms, which they called it multiscale algorithm. In this paper, we will consistently use the name multilevel algorithms for all these optimization algorithms, but we emphasize that the terms multilevel, multigrid and multiscale were used interchangeably in different papers. On the other hand, we keep the name multigrid methods for the conventional multigrid methods that solve linear or nonlinear equations that are discretizations arising from partial differential equations (PDEs).
A multidomain multigrid pseudospectral method for incompressible flows
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Wenqian Chen, Yaping Ju, Chuhua Zhang
To speed up the solution convergence of numerical methods, the multigrid method is proven to yield a significant speedup for the numerical solutions of both the linear and nonlinear partial differential equations, especially, for finite difference method and finite volume method. However, quite a few works were carried out for the multigrid pseudospectral solutions of Navier-Stokes equations, and most of the available works were devoted to simple geometry, e.g., lid-driven cavity problem [7,13] and thermally driven cavity problem [14]. The suitability and performance of the multigrid pseudospectral method for complex geometry remain unknown. For complex geometry, where the aforementioned multidomain method is applied, the unknowns on interior points are solved using the governing equations while the unknowns on interface points are solved using the interface transfer conditions. This renders the design of multigrid operators much more complex for the multidomain cases than the single domain cases. To this end, Haupt et al. [15] recently proposed an interface/boundary condensation method for the design and implementation of multigrid operators under the framework of spectral element method [15]. The basic idea behind their work is that the updating of unknowns on interface/boundary is adequate due to the elliptic character of the Helmholtz equation. Inspired by the work of Haupt et al. [15], we shall develop an interface/boundary condensation method for the multidomain multigrid operators under the framework of the pseudospectral method. Through condensing the interface/boundary equations on the finest grid, the unknowns at the interface/boundary are updated only on the finest grid while not on the coarser grids.
Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines
Published in Nuclear Science and Engineering, 2018
R. N. Slaybaugh, M. Ramirez-Zweiger, Tara Pandya, Steven Hamilton, T. M. Evans
The idea of multigrid methods is to take advantage of the smoothing effects of iterative methods by making smooth errors look oscillatory and thus easier to remove. Errors that are low-frequency on a fine grid can be mapped onto a coarser grid where they are high frequency. A relaxer is applied on the coarser grid to remove the now oscillatory error components. The remaining error is mapped to a still coarser grid and smoothed again. The problem is restricted to coarser and coarser grids until the coarsest grid is reached.