Optimizing Geometric Multigrid Method Computation using a DSL Approach

Abstract

The Geometric Multigrid (GMG) method is widely used in numerical analysis to accelerate the convergence of partial differential equations solvers using a hierarchy of grid discretizations. Multiple grid sizes and recursive expression of multigrid cycles make the task of program optimization tedious. A high-level language that aids domain experts for GMG with effective optimization and parallelization support is thus valuable. We demonstrate how high performance can be achieved along with enhanced programmability for GMG, with new language/optimization support in the PolyMage DSL framework. We compare our approach with (a) hand-optimized code, (b) hand-optimized code in conjunction with polyhedral optimization techniques, and (c) the existing PolyMage optimizer adapted to multigrid. We use benchmarks varying in multigrid cycle structure and smoothing steps for evaluation. On a 24-core Intel Xeon Haswell multicore system, our automatically optimized codes achieve a mean improvement of 3. 2x over straightforward parallelization, and 1. 31x over the PolyMage optimizer.CCS CONCEPTS• Software and its engineering →Compilers;