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

(plus a little auxiliary computation) on the nestgridalone|irrespective of the total
number of levels. The most common choices are = 1 (which is called the V-cycle) and
=2(the W-cycle).
Convergence proofs. For certain classes of Hamiltonians H |primarily quadratic
ones|andsuitable choices of the coarse grids, prolongations, smoothing iterations
and cycle control parameters, it can be proven rigorously 18 that the multi-grid iteration
matrices Ml satisfy a uniform bound
jjMljj C < 1 (5.35)
valid irrespective of the total number of levels. Thus, a xed number of multi-grid
iterations (maybe ve orten) are su cient toreduce the error to asmall value, independent
of the lattice sizeL. Inother words, critical slowing-down has been completely
eliminated.
The rigorous convergence proofs are somewhat arcane, so we cannot describe them
here in anydetail, but certain general features are worth noting. The convergence proofs
are most straightforward when linear or higher-order interpolation and restriction are
used, and >1 (e.g. the W-cycle). When either low-order interpolation (e.g. piecewiseconstant)
or =1(the V-cycle) is used, the convergence proofs become much more
delicate. Indeed, if both piecewise-constant interpolation and a V-cycle are used, then
the uniform bound (5.35) has not yet been proven, and itismostlikely false! To some
extent these features may be artifacts ofthe current methods of proof, but we suspect
that they do also re ect real properties of the multi-grid method, and sotheconvergence
proofs may serve asguidance for practice. For example, in our work wehave used
piecewise-constantinterpolation (so as to preserve the simple nearest-neighbor coupling
on the coarse grids), and thus for safety westick tothe W-cycle. Thereisinany case
much room for further research, both theoretical and experimental.
To recapitulate, the extraordinary e ciency of the multi-grid method arises from the
combination of two key features:
1) The convergence estimate (5.35). This means that onlyO(1) iterations are needed,
independent ofthe lattice size L.
2) The work estimate (5.34). This means that eachiteration requires only a computational
labor of order L d (the ne-grid lattice volume).
It follows that the complete solution of the minimization problem, to any speci ed
accuracy ", requires a computational labor of order L d .
Unigrid point of view. Let us look again at the multi-grid algorithm from the variational
standpoint. One natural class of iterative algorithms for minimizing H are the
18 For a detailed exposition of multi-grid convergence proofs, see [32, Chapters 6{8, 10, 11], [40] and
the references cited therein. The additional work needed to handle the piecewise-constant interpolation
can be found in[41].
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