  We study the dynamic critical behavior of the multi-grid Monte Carlo (MGMC)
algorithm with piecewise-constant interpolation and a W-cycle, applied to the
one-dimensional $O(4)$-symmetric nonlinear $\sigma$-model [= $SU(2)$ principal
chiral model], on lattices from $L=128$ to $L=16384$. Our data for the
integrated autocorrelation time $\tau_{int,{\cal M}^2}$ are well fit by a
logarithmic growth. We have no idea why the critical slowing-down is not
completely eliminated.
