  We study the relation between the dynamical critical behavior and the
kinematics of stochastic multigrid algorithms. The scale dependence of
acceptance rates for nonlocal Metropolis updates is analyzed with the help of
an approximation formula. A quantitative study of the kinematics of multigrid
algorithms in several interacting models is performed. We find that for a
critical model with Hamiltonian H(phi) absence of critical slowing down can
only be expected if the expansion of <H(phi+psi)> in terms of the shift psi
contains no relevant term (mass term). The predictions of this rule are
verified in a multigrid Monte Carlo simulation of the Sine Gordon model in two
dimensions. Our analysis can serve as a guideline for the development of new
algorithms: We propose a new multigrid method for nonabelian lattice gauge
theory, the time slice blocking. For SU(2) gauge fields in two dimensions,
critical slowing down is almost completely eliminated by this method, in
accordance with the theoretical prediction. The generalization of the time
slice blocking to SU(2) in four dimensions is investigated analytically and by
numerical simulations. Compared to two dimensions, the local disorder in the
four dimensional gauge field leads to kinematical problems.
