Compiler generation and autotuning of communication-avoiding operators for geometric multigrid

Abstract

This paper describes a compiler approach to communicationavoiding optimizations in geometric multigrid (GMG), one of the most popular methods for solving partial differential equations. Communication-avoiding optimizations reduce vertical communication through the memory hierarchy and horizontal communication across processes or threads, usually at the expense of introducing redundant computation. We focus on applying these optimizations to the smooth operator, which successively reduces the error and accounts for the largest fraction of the GMG execution time. Our compiler technology applies both novel and known transformations to derive an implementation comparable to manually-tuned code. To make the approach portable, an underlying autotuning system explores the tradeoff between reduced communication and increased computation, as well as tradeoffs in threading schemes, to automatically identify the best implementation for a particular architecture and at each computation phase. Results show that we are able to quadruple the performance of the smooth operation on the finest grids while attaining similar or better performance than manually-tuned code.

Compiler-Directed Transformation for Higher-Order Stencils

“…The partial sum optimization reduces floating-point operations in compute-bound kernels to make them more memory bound. Once memory bound, communication-avoiding optimizations can be used to further improve performance, such as overlapped tiling (via larger ghost zones), loop fusion and wavefronts [8,16,17,26]. Overlapped tiling reduces inter-processor communication.…”

“…The partial sum transformation described in the previous section has been implemented in the CHiLL polyhedral transformation and code generation framework [18] and extends previous communication-avoiding optimizations in CHiLL targeting GMG [16,17]. Building new transformations into a polyhedral framework easily allows for composition of transformations as long as data dependences are not violated.…”

“…A new script command for the partial sum transformation has been implemented in CHiLL, which identifies the stencil statement to which this transformation should be applied. In this paper, which explores the impact of different optimization strategies, the transformation recipes were manually written, extending the recipes for communication-avoiding optimization described in [16] to also invoke the partial sum transformation. This section describes the abstractions used by the compiler in the automation of the partial sum transformation.…”

“…Previous work studied some of the performance and productivity aspects of miniGMG [16,26]. The benchmark creates a block structured 3D grid partitioned into subdomains (boxes) which are distributed among At each level, we apply four weighted Jacobi smooths using the relevant stencil shown in Figure 1.…”

“…We evaluate our approach in the context both of applying the operators in a stand-alone fashion, and within a Geometric Multigrid (GMG) method for solving Poisson's equation using these discretizations. We combine partial sums with the communication-avoiding and thread scheduling optimizations of prior work [16,17]. Our compiler's ability to compose transformations allows it to take a higher-order smoother, remove its computation bottleneck and make it bandwidthlimited, enabaling it to then further apply bandwidth-reducing optimizations.…”