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

Hamiltonian Hl;1 has the same functional form as the \ ne-grid" Hamiltonian Hl: itis
speci ed by the coe cients 0 , 0 , 0 y, A 0 y and h 0 y. Thestep \compute Hl;1" therefore
means to compute these coe cients. Note alsothe importance of allowing in (5.23)
for ' 3 and ' terms and for site-dependent coe cients: even if these are not present in
the original Hamiltonian H HM, they will be generated on coarser grids. Finally,
we emphasize that the coarse-grid Hamiltonian Hl;1 depends implicitly on the current
value of the ne-lattice eld ' 2 Ul although our notation suppresses this dependence,
it should be kept in mind.
Basic (smoothing) iterations. We have already discussed the damped Jacobi iteration
as one possible smoother. Note that inthis method only the \old" values ' (n) are
used on the right-hand side of (5.9)/(5.10), even though for someofthe terms the \new"
value ' (n+1) may already have been computed. An alternative algorithm is to useat
each stage on the right-hand sidethe \newest" available value. This algorithm is called
the Gauss-Seidel iteration. 17 Note that the Gauss-Seidel algorithm, unlike the Jacobi
algorithm, depends on the ordering of the grid points. For example, if a 2-dimensional
grid is swept in lexicographic order (1 1), (2 1), :::,(L 1), (1 2), (2 2), :::,(L 2), :::,
(1L), (2L), :::,(L L), then the Gauss-Seidel iteration becomes
' (n+1)
x1x2
1
=
4 ['(n) x1+1x2 + '(n+1) x1;1x2 + '(n) x1x2+1 + ' (n+1)
x1x2;1 + fx1x2]: (5.32)
Another convenient ordering isthe red-black (or checkerboard) ordering, inwhichthe
\red" sublattice r = fx 2 : x1 + + xd is eveng is swept rst, followed by the
\black" sublattice b = fx 2 : x1 + + xd is oddg. Notethat the ordering ofthe
grid points within each sublattice is irrelevant [forthe usual nearest-neighbor Laplacian
(5.1)], since the matrix A does not couple sites of the same color. This means that redblack
Gauss-Seidel is particularly well suited to vector or parallel computation. Note
that the red-black ordering makes sense with periodic boundary conditions only if the
linear size Ll of the grid l is even.
It turns out that Gauss-Seidel is a better smoother than damped Jacobi (even if the
latter is given its optimal !). Moreover, Gauss-Seidel is easier to program and requires
only half the storage space (no need for separate storage of \old" and \new" values).
The only reason we introduced damped Jacobi at allisthat itiseasiertounderstand
and toanalyze.
Many other smoothing iterations can be considered, and can be advantageous in
anisotropic or otherwise singular problems [32, Section 3.3 and Chapters 10{11]. But
we shall stick toordinary Gauss-Seidel, usually with red-black ordering.
will consist, respectively,ofm1 and m2 iterations of the Gauss-
Thus, S pre
l
and S post
l
Seidel algorithm. The balance between pre-smoothing and post-smoothing is usually
not very crucial only the total m1 + m2 seems to matter much. Indeed, one (butnot
17 It is amusing to note that \Gauss did not use a cyclic order of relaxation, and :::Seidel speci cally
recommended against using it" [36, p.44n]. See also [37].
33