Monte Carlo Methods in Statistical Mechanics: Foundations and ...

solution is reached in a very few MG iterations (in particular, critical slowing-down is
completely eliminated). This viewpoint also explains why MG convergence is more delicate
for piecewise-constant interpolation than for piecewise-linear: the point isthat a
sine wave (orother slowly varying function) can be approximated to arbitrary accuracy
(in energy norm) by piecewise-linear functions but not by piecewise-constant functions.
We remark that McCormick andRuge [39] have advocated the \unigrid" idea not
just as an alternate point of view on the multi-grid algorithm, but asanalternate computational
procedure. To be sure, the unigrid method is somewhat simpler to program,
and this could have pedagogical advantages. But oneofthe key properties of the multigrid
method, namely the O(L d ) computational labor per iteration, is sacri ced in the
unigrid scheme. Instead of (5.33){(5.34) one has
and hence
work(UG) WM
l=M
Wl WM (5.36)
0X
M;l
MWM if =1
M WM if >1
Since M log2 L and WM Ld ,weobtain
(
d L log L if =1
work(UG)
Ld+log2 if >1
(5.37)
(5.38)
For a V-cycle the additional factor of log L is perhaps not terribly harmful, but for a
W-cycle the additional factor of L is a severe drawback (though not as severe as the
O(L 2 )critical slowing-down of the traditional algorithms). Thus, we donot advocate
the use of unigrid as a computational method if there is a viable multi-grid alternative.
Unigrid could, however, be of interest in cases where true multi-grid is unfeasible, as
may occur for non-Abelian lattice gauge theories.
Multi-grid algorithms can also be devised for some models in which state space is a
nonlinear manifold, such as nonlinear -models and lattice gauge theories [20, Sections
3{5]. The simplest case is the XY model: both the ne-grid and coarse-grid eld
variables are angles, andthe interpolation operator is piecewise-constant (withangles
added modulo 2 ). Thus, a coarse-grid variable y speci es the angle by whichthe
2 d spins in the block By are to besimultaneously rotated. A similar strategy can be
employed for nonlinear -models taking values in a group G (the so-called \principal
chiral models"): the coarse-grid variable y simultaneously left-multiplies the2 d spins in
the block By. For nonlinear -models taking values in a nonlinear manifold M on which
a group G acts [e.g. the n-vector model with M = Sn;1 and G = SO(n)], the coarsegrid-correction
moves are still simultaneous rotation this means that while the ne-grid
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