Hybrid Lanczos-switching models for solving large linear systems and their parallel versions

Abstract

Lanczos iterative methods for solving a large sparse linear systems typically face the latent breakdown which strikes every time these methods are deployed. A number of approaches to deal with this issue have been investigated. One of them is by switching between the solvers preemptively breakdown. However, the problem is not fully solved yet. Here, we propose switching models combined with particular Lanczos iterative methods. The first model is by using the last iterate as the switching point with an unlimited number of iterations, the second model is by using the iterate with the minimum residual norm as the initial point and the third model is by using the iterate with the minimum of minimum residual norms as the switching point. These three models lead algorithms of SLULast, SLUMinRes, and SLUMoM, respectively. The parallel version of the proposed algorithms is also provided to speed up their convergence. In this case, we constructed the parallel of SLUMoM and we call it pSLUMoM. The numerical results showed that our switching models performed better than the existing switching strategy in terms of robustness and efficiency. In fact, under a parallel framework, pSLUMoM showed a performance gain of up to 50% in our experiments.