Some Comments on Multigrid Methods for Computing Propagators

I make three conceptual points regarding multigrid methods for computing propagators in lattice gauge theory: 1) The class of operators handled by the algorithm must be stable under coarsening. 2) Problems related by symmetry should have solution methods related by symmetry. 3) It is crucial to distinguish the vector space V from its dual space V . All the existing algorithms violate one or more of these principles. There has recently been much interest in developing multigrid methods for solving large linear systems with disordered coefficients [1, 2, 3], and in particular for computing the bosonic or fermionic propagator in a background gauge field. 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. Perhaps a brief note explaining these points is therefore warranted. First point. We are interested in solving linear systems of the form Ax = b for some class of linear operators A. So the first order of business must be to specify clearly the class C of operators to which our algorithm is intended to apply. This is not a completely trivial matter, because 1) For a multigrid (as opposed to two-grid) algorithm to be well-defined, the class C must be stable under coarsening. In other words, we start with some class C0 of operators on the finest grid, for which we want to solve Ax = b. But then our multigrid algorithm produces operators on the first coarse grid, which belong to some quite possibly larger class C1 — what this class is depends on our procedure for choosing interpolation, restriction and coarse-grid operators. If we are then to continue to the second coarse grid, this procedure must be defined for all the operators in this larger class. If we want to allow an arbitrary number of grids, then we cannot stop short of a class C ⊃ C0 that is stable under coarsening . For example, our initial interest might be in nearest-neighbor Laplace operators in an SU(N) gauge field, with x-independent values for the hopping parameter β and the mass m: C0 = {A ≡ −β∆U +m: Uxy ∈ SU(N), β ≥ 0, m ∈ R, A > 0} , (1)