Fermions as global correction: The QCD case

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.