Converting Stencils to Accumulations for Communication-Avoiding Optimization in Geometric Multigrid

This paper describes a compiler transformation on stencil operators that automatically converts a standard stencil representation into an accumulation. We use this as an enabling transformation to optimize the stencil operators in the context of Geometric Multigrid (GMG), a widely used method to solve partial differential equations. GMG has four stencil operators, the smoother, residual, restriction, and interpolation some of which require inter-process and inter-thread communication. This new optimization allows us, at each level of a GMG V-Cycle, to fuse all operators when recursing down the V-Cycle, and all smooth operations when returning up the V-Cycle. In turn, this fusion allows us to create a parallel wavefront across the fused operators that reduces communication. Thus, these combined optimizations reduce vertical (through the memory hierarchy) data movement and horizontal (inter-thread and inter-process) messages and synchronization.