Improving Multigrid and Conventional Relaxation Algorithms for Propagators

Abstract

Practical modifications of deterministic multigrid and conventional relaxation algorithms are discussed. New parameters need not be tuned but are determined by the algorithms themselves. One modification can be thought of as "updating on a last layer consisting of a single site". It eliminates critical slowing down in computations of bosonic and fermionic propagators in a fixed volume. Here critical slowing down means divergence of asymptotic relaxation times as the propagators approach criticality. A remaining volume dependence is weak enough in case of bosons so that conjugate gradient can be outperformed. However, no answer can be given yet if the same is true for staggered fermions on lattices of realizable sizes. Numerical results are presented for propagators of bosons and of staggered fermions in 4-dimensional SU (2) gauge fields.

Progress in lattice field theory algorithms

“…Multigrid methods are also under active study as a means of inverting the lattice Dirac operator more efficiently [57,58,59,60,61]. Another method used for this purpose has been proposed by Vyas [62,63].…”

Idealized multigrid algorithm for staggered fermions

“…In Monte Carlo simulations of lattice gauge theories with fermions the most time-consuming part is the computation of the gauge field dependent fermion propagators. Great hopes to compute propagators without any critical slowing down (CSD) are attached to multigrid (MG) methods [1,2,3,4,5,6,7,8,9,10,11]. However, up to now no practical MG algorithm has been found for fermions.…”

Some comments on multigrid methods for computing propagators

“…S 0 is A 0 -symmetric). 20 A similar use of B 0 -or more precisely, of the diagonal part of A 0 -can be found in the norms defined by Ruge and Stüben [46, p. 78]. 21 Such a basis can always be achieved by a GL(N, C) gauge transformation A xy → U x A xy U * y .…”

“…1 Many interesting ideas in this direction have been proposed by the Amsterdam [4,5,6,7,8,9,10], Boston [11,12,13,14,15], Hamburg [16,17,18,19,20,21,22,23,24,25] and Israeli [26,27,28,29,30,31,32,33,34,35] groups. However, all these discussions have missed what I consider to be some key conceptual points.…”

“…3 If this occurs, we must then go back to the beginning, and define our coarsening procedure for all the operators in this larger class -and so on successively until the class stabilizes. 4 This comment is relevant to the ground-state-projection (GSP) multigrid algorithms proposed by the Boston [11,12,13,14,15] and Hamburg [17,18,19,20,21,24,25] groups, in which the range of the interpolation operator is spanned by the lowest eigenvectors of the operator A modified to impose Neumann boundary conditions on 2 × 2 or 3 × 3 blocks. The trouble here is that the concept of "Neumann b.c.…”