Mixed Precision Iterative Refinement with Adaptive Precision Sparse Approximate Inverse Preconditioning

TL;DR

This paper introduces an adaptive precision sparse approximate inverse preconditioner integrated with a five-precision GMRES iterative refinement method, aiming to reduce computational costs while maintaining accuracy in solving sparse linear systems.

Contribution

It develops a novel adaptive precision preconditioner and analyzes its convergence, demonstrating potential cost savings with trade-offs in iteration count.

Findings

Potential reduction in storage and application costs for preconditioners

Trade-off between cost savings and increased GMRES iterations

Convergence conditions established for the adaptive method

Abstract

Hardware trends have motivated the development of mixed precision algo-rithms in numerical linear algebra, which aim to decrease runtime while maintaining acceptable accuracy. One recent development is the development of an adaptive precision sparse matrix-vector produce routine, which may be used to accelerate the solution of sparse linear systems by iterative methods. This approach is also applicable to the application of inexact preconditioners, such as sparse approximate inverse preconditioners used in Krylov subspace methods. In this work, we develop an adaptive precision sparse approximate inverse preconditioner and demonstrate its use within a five-precision GMRES-based iterative refinement method. We call this algorithm variant BSPAI-GMRES-IR. We then analyze the conditions for the convergence of BSPAI-GMRES-IR, and determine settings under which BSPAI-GMRES-IR will produce similar backward and forward errors as the existing SPAI-GMRES-IR method, the latter of which does not use adaptive precision in preconditioning. Our numerical experiments show that this approach can potentially lead to a reduction in the cost of storing and applying sparse approximate inverse preconditioners, although a significant reduction in cost may comes at the expense of increasing the number of GMRES iterations required for convergence.