It would appear at first that iteration with one of
the above relaxation schemes should result in rapid location of
the minimum of 
,
since this function
is so ``well behaved." However, we have discussed above another problem
associated with the length scales in the solution. Namely, it
takes longer and longer times to converge the long wavelength
modes as one goes to finer grids. Put another way,
if one plots S vs. iteration number, it decreases rapidly
at first (when the short wavelength modes are damped) and then
exhibits a long, slowly decaying tail due to the difficulty of
moving the solution `globally' (CSD). Therefore, no matter how efficient
the iterative method on a given scale,
approach to the converged solution in
a prescribed number of iterations is not guaranteed.
Furthermore, since the
long wavelength modes get longer as the system size is increased,
this problem leads to a method which does not scale
linearly with system size. This stimulated the development of modern
multiscale or multigrid methods in the 1960's
and 1970's, principally
by Brandt [21,22,23],
with later
refinements carried out by Hackbusch [24].
The initial multigrid techniques
were developed for linear elliptic partial differential equations
like the Poisson Eq. Later, nonlinear techniques were developed
which can handle problems like the PB Eq.