Segmental Refinement: A Multigrid Technique for Data Locality

TL;DR

This paper analyzes segmental refinement, a multigrid method, demonstrating its potential to eliminate communication bottlenecks in parallel computations for nonlinear elliptic problems, with theoretical and experimental validation on large-scale systems.

Contribution

It provides a detailed complexity analysis and performance evaluation of segmental refinement, extending prior work by confirming communication reduction and exploring parameter dependencies.

Findings

Communication can be eliminated on fine grids with modest extra work.

Segmental refinement maintains asymptotic exactness of multigrid.

Performance results show scalability up to 64K cores.

Abstract

We investigate a domain decomposed multigrid technique, segmental refinement, for solving general nonlinear elliptic boundary value problems. Brandt and Diskin first proposed this method in 1994; we continue this work by analytically and experimentally investigating its complexity. We confirm that communication of traditional parallel multigrid can be eliminated on fine grids with modest amounts of extra work and storage while maintaining the asymptotic exactness of full multigrid, although we observe a dependence on an additional parameter not considered in the original analysis. We present a communication complexity analysis that quantifies the communication costs ameliorated by segmental refinement and report performance results with up to 64K cores of a Cray XC30.