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

Note rst that the only slow modes in the damped Jacobi iteration are the longwavelengthmodes
(provided that ! is not near 1): as long as, say, max(p1p2) 2 ,we
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have 0 p 4 (for ! = 2 ), independent ofL. Itfollows that the short-wavelength
components oftheerror e (n) ' (n) ; ' can be e ectively killed by a few (say, ve
or ten) damped Jacobi iterations. The remaining error has primarily long-wavelength
components, and so is slowly varying inx-space. But a slowly varying function can
be well represented on a coarser grid: if, for example, we were told e (n)
x only at even
values of x, we could nevertheless reconstruct with highaccuracythe function e (n)
x at all
x by, say, linear interpolation. This suggests an improved algorithm for solving (5.1):
perform a few damped Jacobi iterations on the original grid, until the (unknown) error
is smooth inx-space then set up an auxiliary coarse-grid problem whose solution will
be approximately this error (this problem will turn out tobeaPoisson equation on the
coarser grid) perform a few damped Jacobi iterations on the coarser grid and then
transfer (interpolate) the resultbacktothe original ( ne) grid and add it in to the
current approximate solution.
There are two advantages to performing the damped Jacobi iterations on the coarse
grid. Firstly, the iterations take lesswork, because there are fewer lattice points onthe
coarse grid (2 ;d times as many for a factor-of-2 coarsening ind dimensions). Secondly,
with respect to the coarse grid the long-wavelength modes no longer have such long
wavelength: their wavelength has been halved (i.e. their wavenumber has been doubled).
This suggests that those modes with, say, max(p1p2) 4 can be e ectively killed
by a few damped Jacobi iterations on the coarse grid. And then we can transfer the
remaining (smooth) error to ayet coarser grid, and so on recursively. Theseare the
essential ideas of the multi-grid method.
Let us now give aprecisede nition of the multi-grid algorithm. For simplicity
we shall restrict attention to problems de ned in variational form16 : thus, the goal is
to minimize a real-valued function (\Hamiltonian") H('), where ' runs over some
N-dimensional real vector space U. Weshall treat quadratic and non-quadratic Hamiltonians
on an equal footing. In order to specify the algorithm we must specify the
following ingredients:
1) A sequence of coarse-grid spaces UM U UM;1 UM;2 ::: U0. Here dim Ul
Nl and N = NM >NM;1 >NM;2 > >N0.
2) Prolongation (or \interpolation") operators pll;1: Ul;1 ! Ul for 1 l M.
3) Basic (or \smoothing") iterations Sl: Ul Hl ! Ul for 0 l M. HereHl is
a space of \possible Hamiltonians" de ned on Ul we discuss this in more detail
below. Therole of Sl is to takeanapproximate minimizer ' 0 l of the Hamiltonian Hl
andcomputeanew (hopefully better) approximate minimizer ' 00
l = Sl(' 0 l Hl). [For
16 In fact, the multi-grid method can be applied to the solution of linear or nonlinear systems of
equations, whether or not these equations come from a variational principle. See, for example, [32] and
[20, Section 2].
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