Mixed Precision GMRES‐Based Iterative Refinement With ILU Preconditioning

Abstract

State‐of‐the‐art iterative refinement (IR)‐based methods mainly employ LU factorization to construct preconditioners, which severely limits their applicability to sparse linear systems due to excessive memory overhead. In order to address such issues, we exploit incomplete LU ( ILU ) factorization as a preconditioning technique for the mixed‐precision GMRES‐IR to develop a new method named as ILU ‐GMRES‐IR. A theoretical error bound of the residual matrix associated with ILU factors is established, which, combined with experimental observations, provides a reference for selecting appropriate ILU parameters to balance memory overhead and convergence rate. In addition, a bound analysis is made for the condition number of the preconditioned matrix to characterize the effectiveness of ILU preconditioning. In comparison with the popular LU ‐GMRES‐IR method, it is verified from numerical experiments that our ILU ‐GMRES‐IR provides a tunable trade‐off between memory consumption and convergence performance, allowing substantial memory savings while retaining satisfactory convergence.