On the recursive structure of multigrid cycles

A new fixed (nonadaptive) recursive scheme for multigrid algorithms is introduced. Governed by a positive parameter kappa called the cycle counter, this scheme generates a family of multigrid cycles dubbed kappa-cycles. The well-known V-cycle, F-cycle, and W-cycle are shown to be particular members of this rich kappa-cycle family, which satisfies the property that the total number of recursive calls in a single cycle is a polynomial of degree kappa in the number of levels of the cycle. This broadening of the scope of fixed multigrid cycles is shown to be potentially significant for the solution of some large problems on platforms, such as graphics processing units, where the overhead induced by numerous sequential calls to the coarser levels may be relatively significant. In cases of problems for which the convergence of standard V-cycles or F-cycles (corresponding to kappa = 1 and kappa = 2, respectively) is particularly slow and yet the cost of W-cycles is very high due to the large number of coarse level calls (which is exponential in the number of levels), intermediate values of kappa may prove to yield significantly faster run-times. This is demonstrated in examples where kappa-cycles are used for the solution of rotated anisotropic diffusion problems, both as a stand-alone solver and as a preconditioner. Moreover, a simple model is presented for predicting the approximate run-time of the kappa-cycle, which is useful in preselecting an appropriate cycle counter for a given problem on a given platform. Implementing the kappa-cycle requires making just a small change in the classical multigrid cycle.