Mixed Precision Iterative Refinement with Sparse Approximate Inverse Preconditioning

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 all based on using a full LU factorization 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 within GMRES-based iterative refinement. We analyze the computation of sparse approximate inverses in finite precision and derive constraints under which the user-specified stopping criteria will be satisfied. We then analyze the behavior of and convergence constraints for a GMRES-based iterative refinement scheme that uses sparse approximate inverse preconditioning, which we call SPAI-GMRES-IR. Our numerical experiments confirm that in some cases, sparse approximate inverse preconditioning can have an advantage over using a full LU factorization.