  It is widely believed that the fermion determinant cannot be treated in
global acceptance-rejection steps of gauge link configurations that differ in a
large fraction of the links. However, for exact factorizations of the
determinant that separate the ultraviolet from the infrared modes of the Dirac
operator it is known that the latter show less variation under changes of the
gauge field compared to the former. Using a factorization based on recursive
domain decomposition allows for a hierarchical algorithm that starts with pure
gauge updates of the links within the domains and ends after a number of
filters with a global acceptance-rejection step. Ratios of determinants have to
be treated stochastically and we construct techniques to reduce the noise. We
find that the global acceptance rate is high on moderate lattice sizes and
demonstrate the effectiveness of the hierarchical filter.
