Talk:Multigrid method

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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)

[edit] Stub

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

[edit] 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)