Rounding-Error Analysis of Multigrid V-Cycles

TL;DR

This paper analyzes the effects of rounding errors in multigrid V-cycle methods, providing a clear and extendable theoretical framework based on floating point arithmetic for various multigrid schemes.

Contribution

It offers a simplified, transparent analysis of rounding errors in two-grid and multigrid methods, including extensions to mixed-precision iterative refinement.

Findings

Provides a theoretical foundation for rounding-error behavior in multigrid cycles

Extends analysis to general V-cycle schemes and full multigrid methods

Facilitates understanding of error propagation in iterative refinement

Abstract

This paper provides a rounding-error analysis for two-grid methods that use one relaxation step both before and after coarsening. The analysis is based on floating point arithmetic and focuses on a two-grid scheme that is perturbed on the coarse grid to allow for an approximate coarse-grid solve. Leveraging previously published results, this two-grid theory can then be extended to general $V(\mu,\nu)$-cycles, as well as full multigrid (FMG). It can also be extended to mixed-precision iterative refinement (IR) based on these cycles. An added benefit of the theory here over previous work is that it is obtained in a more organized, transparent, and simpler way.