The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple timesteps in parallel. In the first papers on PFASST, the numerical examples involve the approximation of PDEs on regular Cartesian-grid spatial discretizations and utilize a hierarchy of space-time grid resolutions. Explicit, fully implicit, and semi-implicit (IMEX) variations of PFASST are easily constructed and have been tested on a wide class of model PDEs.
However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any implicit spatial systems arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each substep and each iteration by some unspecified procedure. This is, in a sense, philosophically orthogonal to the vast majority of research on multigrid or iterative methods for time dependent PDEs that focus on the iterative solution of spatial equations arising from implicit (non-iterative) temporal integration methods. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach.
This talk will present some initial analysis on the relative performance of different strategies for coupling the PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each substep with a small fixed number of V-cycles or relaxation sweeps. This reduces the cost of each PFASST iteration at the expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency is compared to recent results coupling PFASST and full solves done with the parallel multigrid code PMG, which achieved space-time parallelism for the three-dimensional heat equation on up to 448K cores.