Most stencil computations allow tile-wise concurrent start, i.e., there always exists a face of the iteration space and a set of tiling hyperplanes such that all tiles along that face can be started concurrently. This provides load balance and maximizes parallelism. However, existing automatic tiling frameworks often choose hyperplanes that lead to pipelined start-up and load imbalance. We address this issue with a new tiling technique that ensures concurrent start-up as well as perfect load-balance whenever possible. We first provide necessary and sufficient conditions on tiling hyperplanes to enable concurrent start for programs with affine data accesses. We then provide an approach to find such hyperplanes. Experimental evaluation on a 12-core Intel Westmere shows that our code is able to outperform a tuned domain-specific stencil code generator by 4% to 27%, and previous compiler techniques by a factor of 2× to 10.14×.

No abstract

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 localityenhancing algorithm from the polyhedral framework to allow timetiling 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× 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 twodimensions or higher.