A multigrid perspective on the parallel full approximation scheme in space and time

Abstract

SUMMARYFor the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the "Parallel Full Approximation Scheme in Space and Time" (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigridin-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.

Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems

“…The algorithm in its original form is rather complex and not even straightforward to write down, posing a severe obstacle for any attempt to even formulate a conclusive theory. Yet, with the formal equivalence to multigrid methods, as shown in our preceding work, a mathematical framework now indeed exists that allows to use a broad range of established methods for the analysis of PFASST, at least for linear problems. While a detailed semialgebraic Fourier mode analysis revealed many interesting features and limitations of PFASST in our preceding work, a rigorous convergence proof has not been given so far.…”

“…Yet, with the formal equivalence to multigrid methods, as shown in our preceding work, a mathematical framework now indeed exists that allows to use a broad range of established methods for the analysis of PFASST, at least for linear problems. While a detailed semialgebraic Fourier mode analysis revealed many interesting features and limitations of PFASST in our preceding work, a rigorous convergence proof has not been given so far. In this paper, we used the iteration matrices of PFASST, its smoother, and the coarse‐grid correction to establish an asymptotic convergence theory for PFASST.…”

“…A natural question to ask is whether the smoother satisfies the smoothing property, which would make PFASST an actual multigrid algorithm in the classical sense. However, this is not the case, as we can also observe numerically . Still, we can bound the norm of the iteration matrix.…”

“…In addition, the composite collocation problem has to be modified on the coarse level. This is done by the τ ‐correction of the FAS scheme, and we refer to our preceding work and the work of Moser for details on this because the actual formulation does not matter here. We now have all the ingredients for one iteration of the two‐level version of PFASST using postsmoothing: restriction to the coarse level including the formulation of the τ ‐correction, serial approximative block Gauss‐Seidel iteration on the modified composite collocation problem on the coarse level, coarse‐grid correction of the fine‐level values, smoothing of the composite collocation problem on the fine level using parallel approximative block Jacobi iteration. …”

“…Proof This is taken from our preceding work and the work of Moser, and a much more detailed derivation and discussion can be found there.…”