Section: New Results

Tiling for Stencils

Participants : Tobias Grosser, Sven Verdoolaege, Albert Cohen.

This research project aims with optimizing time-iterated stencil operations.

Iterative stencil computations are important in scientific computing and more and more also in the embedded and mobile domain. Recent publications have shown that tiling schemes that ensure concurrent start provide efficient ways to execute these kernels. Diamond tiling and hybrid-hexagonal tiling are two tiling schemes that enable concurrent start. Both have different advantages: diamond tiling has been integrated in a general purpose optimization framework and uses a cost function to choose among tiling hyperplanes, whereas the greater flexibility with tile sizes for hybrid-hexagonal tiling has been exploited for effective generation of GPU code.

We undertook a comparative study of these two tiling approaches and proposed a hybrid approach that combines them. We analyzed the effects of tile size and wavefront choices on tile-level parallelism, and formulate constraints for optimal diamond tile shapes. We then extended, for the case of two dimensions, the diamond tiling formulation into a hexagonal tiling one, which offers both the flexibility of hexagonal tiling and the generality of the original diamond tiling implementation. We also showed how to compute tile sizes that maximize the compute-to-communication ratio, and apply this result to compare the best achievable ratio and the associated synchronization overhead for diamond and hexagonal tiling.

One particularly exciting result is the ability to apply tiling to periodic data domains. These computations are prevalent in computational sciences, particularly in partial differential equation solvers. We proposed 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 $5\times$ and $4.2\times$ 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 $1.5\times$ 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.

These works resulted in a number of high-profile publications, including a nommination for a best paper award, and culminated with the PhD thesis defense of Tobias Grosser.