The classic view of applying multigrid to parabolic (space-time) problems is based on a time-marching approach: discretization of the PDE leads to a discrete elliptic problem at each time step when an implicit scheme is used. Multigrid is then used as an iterative solver for these elliptic equations. Parallelization in this approach is limited to parallelization in the elliptic (spatial) solver, since the time-stepping procedure is sequential. However, with current trends in computer architectures leading towards systems with more, but not faster, processors, faster compute speeds must come from increased concurrency.
In this talk, we present a non-intrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). Being a non-intrusive approach which only uses an existing time propagator, this multigrid-reduction-in-time algorithm (MGRIT) easily allows one to exploit substantially more computational resources than standard sequential time stepping. We discuss progress to date in applying MGRIT to parabolic (space-time) problems.