  We introduce a new and very convenient approach to multi-grid Monte Carlo
(MGMC) algorithms for general nonlinear $\sigma$-models: it is based on
embedding an $XY$ model into the given $\sigma$-model, and then updating the
induced $XY$ model using a standard $XY$-model MGMC code. We study the dynamic
critical behavior of this algorithm for the two-dimensional $O(N)$
$\sigma$-models with $N = 3,4,8$ and for the $SU(3)$ principal chiral model. We
find that the dynamic critical exponent $z$ varies systematically between these
different asymptotically free models: it is approximately 0.70 for $O(3)$, 0.60
for $O(4)$, 0.50 for $O(8)$, and 0.45 for $SU(3)$. It goes without saying that
we have no theoretical explanation of this behavior.
