Combining sparse approximate factorizations with mixed precision iterative refinement

Iterative refinement is seeing its popularity growing again with the new promises of accessible and efficient hardware support for half precision arithmetics. Novel variants of this methods were recently proposed that rely either or the LU factorization or a LU-preconditioned GMRES method for the solution of the correction equation and can employ up to three precisions. The effectiveness of these methods was extensively proved on dense linear systems but the case of sparse problems was not studied as much. Our work aims to fill this gap. First, we have extended the theoretical ground and proposed novel variants that can employ up to five precisions concurrently. Furthermore, we have studied the use of approximation techniques that are commonly used to improve the performance and scalability of sparse direct methods. In all cases we derived theoretical bounds for the convergence conditions and the associate solution accuracy. Second we have implemented these variants on top of a parallel sparse direct solver. We will present the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that the proposed approach can lead to considerable reductions of both the time and memory consumption.