Residual Replacement in Mixed-Precision Iterative Refinement for Sparse Linear Systems

Evaluating Performance of Mixed Precision Linear Solvers with Iterative Refinement

“…The problem of residuals deviation for the Krylov subspace iterative methods has been analyzed in the literature, e.g., in [3,13,16]. The observed convergence issue in some cases can be resolved by the correction of the recurrent residuals, which can be done in several ways.…”

“…In [13,16] the authors propose the reliable updates to correct the recurrent residual with the true one and several criteria when these updates must be performed. Alternatively, in [3] the inner solver restarts are considered for the mixed precision iterative refinement algorithm. Following [3], the proper choice of the conditions to perform the restarts allows this approach to outperform the reliable updates strategy.…”

“…An alternative way of utilizing the mixed precision calculations is the use of the iterative refinement procedure [17]. Initially developed in the 1940s, this method receives great attention in the light of recent activity with introducing the mixed precision calculations in iterative methods for solving SLAEs [3,5]. These publications show promising results with attractive speedup numbers.…”

“…The proposed methods combination allowed to speed up the corresponding SLAE solver for NVIDIA Volta GPUs compared to the basic solver performed in double precision by a factor of 4-6. In [3] the authors investigated the combination of CG solver with a lightweight plain Jacobi (diagonal) preconditioner and the top-level iterative refinement procedure. The outer solver calculations were performed in double precision, and the preconditioned CG solver was performed in the single precision.…”

“…While the results mentioned above show good potential in using mixed precision calculations, the numerical methods and test matrices considered have their specific, which hardens the conclusion on the applicability of the methods proposed to the wider range of applications. Additionally, the energy consumption results presented in [3] can only give a rough estimate to the real calculation time improvement by using the mixed precision iterative refinement procedure. The current paper focuses on the applicability of mixed precision calculations to solve the elliptic equations, and, specifically, the pressure Poisson equation occurring in incompressible flow simulations.…”