Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning

TL;DR

This paper explores the use of sparse approximate inverse preconditioners in mixed precision GMRES iterative refinement, analyzing convergence and performance tradeoffs for practical sparse linear system solutions.

Contribution

It introduces a novel analysis of sparse approximate inverse preconditioning within mixed precision iterative refinement, including convergence constraints and practical implementation insights.

Findings

Sparse approximate inverse preconditioners are effective within GMRES iterative refinement.

Convergence depends on preconditioner sparsity and precision levels.

Numerical experiments validate theoretical convergence and performance tradeoffs.

Abstract

With the commercial availability of mixed precision hardware, mixed precision GMRES-based iterative refinement schemes have emerged as popular approaches for solving sparse linear systems. Existing analyses of these approaches, however, are based on using full LU factorizations to construct preconditioners for use within GMRES in each refinement step. In practical applications, inexact preconditioning techniques, such as incomplete LU or sparse approximate inverses, are often used for performance reasons. In this work, we investigate the use of sparse approximate inverse preconditioners based on Frobenius norm minimization within GMRES-based iterative refinement. We analyze the computation of sparse approximate inverses in finite precision and derive constraints under which user-specified stopping criteria will be satisfied. We then analyze the behavior of and convergence constraints for a five-precision GMRES-based iterative refinement scheme that uses sparse approximate inverse preconditioning, which we call SPAI-GMRES-IR. Our numerical experiments confirm the theoretical analysis and illustrate the resulting tradeoffs between preconditioner sparsity and GMRES-IR convergence rate.