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

minimizer we know, namely =0! Thegoalofthis coarse-grid correction step is to
reduce signi cantly the low-frequency components ofthe error in ' (hopefully without
creating large new high-frequency error components).
3) Post-smoothing. We apply, for good measure, a few more sweeps of the basic
smoother. (This would protect against any high-frequency error components which
may inadvertently have been created by the coarse-grid correction step.)
The foregoing constitutes, of course, a single step of the multi-grid algorithm. In
practice this step would be repeated several times, as in any other iteration, until the
error has been reduced to an acceptably small value. The advantage of multi-grid over
the traditional (e.g. damped Jacobi) iterative methods is that, with a suitable choice
of the ingredients pll;1, Sl and so on, only a few (maybe ve orten) iterations are
needed to reduce the errortoasmall value, independent of the lattice sizeL. This
contrasts favorably with the behavior (5.13) of the damped Jacobi method, in which
O(L 2 )iterations are needed.
The multi-grid algorithm is thus a general framework the userhas considerable
freedom in choosing the speci c ingredients, which must be adapted to the speci c
problem. We now discuss brie y each ofthese ingredients more details can be found
in Chapter 3 of the book of Hackbusch [32].
Coarse grids. Most commonly one usesauniform factor-of-2 coarsening between
each grid l and the next coarser grid l;1. The coarse-grid points could be either a
subset of the ne-grid points (Fig. 1) or a subset of the dual lattice (Fig. 2). These
schemes have obvious generalizations to higher-dimensional cubic lattices. In dimension
d =2,another possibility isauniform factor-of- p 2 coarsening (Fig. 3) note that the
coarse grid is again a square lattice, rotated by 45. Figs. 1{3 are often referred to as
\standard coarsening", \staggered coarsening", and \red-black (orcheckerboard) coarsening",
respectively. Coarsenings by a larger factor (e.g. 3) could also be considered,
but are generally disadvantageous. Note that eachofthe above schemes works also for
periodic boundary conditions provided that the linear size Ll of the grid l is even. For
this reason it is most convenient totake the linear size L LM of the original ( nest)
grid M to beapower of 2, or at least a power of 2 times a small integer. Other
de nitions of coarse grids (e.g. anisotropic coarsening) are occasionally appropriate.
Prolongation operators. For a coarse grid as in Fig. 2, a natural choice of prolongation
operator is piecewise-constant injection:
(pll;1'l;1) x1
1
2 x2
1
2
= ('l;1)x1x2 for all x 2 l;1 (5.15)
(illustrated here for d = 2). It can be represented in an obvious shorthand notation by
the stencil "
1
1
#
1
:
1
(5.16)
For a coarse grid as in Fig. 1, a natural choice is piecewise-linear interpolation, one
29