Tiling and optimizing time-iterated computations over periodic domains

This paper deals with optimizing time-iterated computations on periodic data domains. These computations are prevalent in computational sciences, particularly in partial differential equation solvers. We propose a fully automatic technique suitable for implementation in a compiler or in a domain-specific code generator for such computations. Dependence patterns on periodic data domains prevent existing algorithms from finding tiling opportunities. Our approach augments a state-of-the-art parallelization and locality-enhancing algorithm from the polyhedral framework to allow time-tiling of stencil computations on periodic domains. Experimental results on the swim SPEC CPU2000fp benchmark show a speedup of 5× and 4.2× over the highest SPEC performance achieved by native compilers on Intel Xeon and AMD Opteron multicore SMP systems, respectively. On other representative stencil computations, our scheme provides performance similar to that achieved with no periodicity, and a very high speedup is obtained over the native compiler. We also report a mean speedup of about 1.5 χ over a domain-specific stencil compiler supporting limited cases of periodic boundary conditions. To the best of our knowledge, it has been infeasible to manually reproduce such optimizations on swim or any other periodic stencil, especially on a data grid of two-dimensions or higher.