  We examine what is an efficient and scalable nonlinear solver, with low work
and memory complexity, for many classes of discretized partial differential
equations (PDEs) - matrix-free Full multigrid (FMG) with a Full Approximation
Storage (FAS) - in the context of current trends in computer architectures.
Brandt proposed an extremely low memory FMG-FAS algorithm over 25 years ago
that has several attractive properties for reducing costs on modern - memory
centric -- machines and has not been developed to our knowledge. This method,
segmental refinement (SR), has very low memory requirements because the finest
grids need not be held in memory at any one time but can be "swept" through,
computing coarse grid correction and any quantities of interest, allowing for
orders of magnitude reduction in memory usage. This algorithm has two useful
ideas for effectively exploiting future architectures: improved data locality
and reuse via "vertical" processing of the multigrid algorithms and the method
of $\tau$-corrections, which allows for not storing the entire fine grids at
any one time. This report develops this algorithm for a model problem and a
parallel generalization of the original sweeping technique. We show that
FMG-FAS-SR can work as originally predicted, solving systems accurately enough
to maintain the convergence rate of the discretization with one FMG iteration,
and that the parallel algorithm provides a natural approach to fully exploiting
the available parallelism of FMG.
