Mixed precision multigrid with smoothing based on incomplete Cholesky factorization

TL;DR

This paper develops a mixed precision multigrid method with incomplete Cholesky smoothing, demonstrating that lower precision can be used in certain components to achieve speedups and energy savings without sacrificing accuracy.

Contribution

It introduces a theoretical framework for mixed precision multigrid methods with incomplete Cholesky smoothing and validates it through numerical experiments showing practical efficiency gains.

Findings

Lower precisions in IC smoothing are feasible in certain settings.

Significant speedups up to 1.43x achieved with mixed precision.

Energy savings of up to 71% demonstrated in experiments.

Abstract

Multigrid methods are popular iterative methods for solving large-scale sparse systems of linear equations. We present a mixed precision formulation of the multigrid V-cycle with general assumptions on the finite precision errors coming from the application of coarsest-level solver and smoothing. Inspired by existing analysis, we derive a bound on the relative finite precision error of the V-cycle which gives insight into how the finite precision errors from the individual components of the method may affect the overall finite precision error. We use the result to study V-cycle methods with smoothing based on incomplete Cholesky factorization. The results imply that in certain settings the precisions used for applying the IC smoothing can be significantly lower than the precision used for computing the residual, restriction, prolongation and correction on the concrete level. We perform numerical experiments using simulated floating point arithmetic with the MATLAB Advanpix toolbox as well as experiments computed on GPUs using the Ginkgo library. The experiments illustrate the theoretical findings and show that in the considered settings the IC smoothing can be applied in relatively low precisions, resulting in significant speedups (up to 1.43x) and energy savings (down to 71%) in comparison with the uniform double precision variant.