Talk:Multigrid method

Merger

Merging looks like a good idea 153.90.192.27 18:06, 18 October 2005 (UTC)UncleKensson

I have performed the merger. I'd appreciate it if someone who is familiar with the subject could look at the post-merger cleanup [1], and check that I didn't foul anything up. --PeR 08:31, 10 March 2006 (UTC)

Stub

I think this article needs stub status due to very little useful information.

Convexification?

I'm at the edge of my understanding, but would it be accurate to say that multigrid smoothing is a form of convexification, that is, it makes the objective function convex in the search space? It seems like that's what's going on: that smoothing the objective function creates a single global optimum which can then be used as an initial guess for finer grids. Thoughts? Comments? —Ben FrantzDale (talk) 17:02, 8 May 2008 (UTC)

The basic multigrid idea is independent of such local/global considerations. In the symmetric positive definite case, which is the most common setting, the objective function is always convex anyway (being quadratic). However, for some nonlinear problems, using the coarse solution as initial guess for the finer grid tends to get a better solution when there are many solutions (or local optima). Jmath666 (talk) 05:52, 9 May 2008 (UTC)

Interesting. Do you have an intuitive sense of how multigrid helps in the positive-definite (convex) case? Is it just that it is less expensive to work with fewer degrees of freedom, so you can get close to the solution easier this way? —Ben FrantzDale (talk) 16:46, 10 May 2008 (UTC)

There is a little more to it. Smoothing makes the error, well, smooth, then the error can be well approximated by solving a coarse problem with fewer degrees of freedom, and subtracted out. The dual view is that the coarse grid correction (minimize in the direction of the coarse space) makes the error rough, so the smoother is effective. Repeating this process you get eventually a good solution of the original (fine) problem. The important thing is that the coarse space is made out of functions smooth in some sense (in the simplest case, piecewise linear) and the smoother makes the error closer to that space. This is well explained for example in the Multigrid tutorial by Bill Briggs. Jmath666 (talk) 06:56, 11 May 2008 (UTC)

Thanks. I'll look into that. —Ben FrantzDale (talk) 12:49, 11 May 2008 (UTC)

mesh

Is it possible to apply the multigrid method to an arbitrary mesh of triangles (an unstructured grid), or is it limited to rectilinear regular grids? --68.0.124.33 (talk) 14:06, 20 September 2008 (UTC)

Well, it has been a while, but just for the record: that's what algebraic multigrid does. Jmath666 (talk) 17:10, 5 December 2009 (UTC)

Sources added

I've added a number of sources to back up various statements in the Intro. These sources also provide a point of entry to more complete discussions of numerous points via the links to google books. The wording of the article has been somewhat amended to make more clear the bearing of the sources. Brews ohare (talk) 17:00, 5 December 2009 (UTC)

Plagiarism

Some of this article appears to be verbatim from What are multigrid methods?. Brews ohare (talk) 20:03, 5 December 2009 (UTC)

If so, by all means remove that text. —Ben FrantzDale (talk) 13:21, 7 December 2009 (UTC)

Translation

I'm a French PhD student currently working on multigrid technology to solve large systems of equations arising from the discretization of mechanical problems. Are you interested in letting me translate in French this article, or is anyone else already working on it ? —Preceding unsigned comment added by The Unforgiv3n (talk • contribs) 15:30, 22 December 2009 (UTC)

Algebraic Multigrid

This page could do with some explanation of how Algebraic multigrid works. In particular, how can we use the system matrix to find a new set of DoFs and an coarsening operation? As a simple example, I'm thinking of a Poisson equation in 1D, which has a tridiagonal matrix (at least if the DoFs are sorted). Knowing this, it is easy to say we might halve the number of DoFs and halve the Laplacian operator (I think). But an algebraic multigrid solver wouldn't know these things a priori. I can imagine that it could use the matrix to get a sense of the topology of the space (even if the DoFs aren't ordered nicely into the tridiagonal form), but how would it choose how many DoFs to use? —Ben FrantzDale (talk) 02:02, 19 April 2010 (UTC)

This page could do with much more than that... and algebraic multigrid warrants a page by itself. In the meantime, maybe try P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems, Computing 56 (1996) 179-196 and Algebraic multigrid (AMG) for sparse matrix equations, A. Brandt, S. McCormick, and J. Ruge, Sparsity and Its Applications (D.J. Evans, ed.), Cambridge Univ. Press (1984) for the basics... Jmath666 (talk) 02:37, 19 April 2010 (UTC)

Thanks, this looks interesting; I'll need to read it more carefully. —Ben FrantzDale (talk) 14:48, 19 April 2010 (UTC)

Introduction

I disagree with the following sentence in the introduction: "The main idea of multigrid is to accelerate the convergence of a basic iterative method by global correction from time to time, accomplished by solving a coarse problem." If this was the whole truth, there would be no point in having more than two grids. An at least as important idea of multigrid in my opinion is that error components with different frequencies can be best removed on different grids with standard smoothers. Gernegross (talk) 10:47, 11 March 2014 (UTC)

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