Parallel-Transported Multigrid for Inverting the Dirac-Operator: Variants of the Method and Their Efficiency.

摘要

Using U(1) lattice gauge theory in two dimensions as a test case, we discuss in detail several variants of the Parallel-Transported Multigrid (PTMG) with different restriction and interpolation procedures, resulting in radically different efficiencies. In particular, it is optimal to stop the coarsening once the coarse mesh size is of the order of the largest correlation length xi. A dependence of the efficiency on the topological charge Q of the two-dimensional U(1) configurations was not detected, at least not for the most efficient PTMG variant in the physically relevant range of the parameters. In order to investigate this phenomenon in a reliable way, we developed and applied a Monte-Carlo simulation algorithm using global update steps, which ensures ergodic covering of all topological sectors of the theory. For large enough systems and correlation lengths, an optimally tuned PTMG algorithm yields a very high convergence rate (one order of magnitude per cycle) and it beats by far the Conjugate-Gradient method in CPU-time. Similar conclusions hold for other relevant lattice gauge theories. (Copyright (c) GMD 1992.)