Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner is challenging. A resurgence of interest in using low precision arithmetic makes the search for robustness more urgent and tougher. In this paper, we focus on symmetric positive definite problems and explore a number of approaches: a look-ahead strategy to anticipate break down as early as possible, the use of global shifts, and a modification of an idea developed in the field of numerical optimization for the complete Cholesky factorization of dense matrices. Our numerical simulations target highly ill-conditioned sparse linear systems with the goal of computing the factors in half precision arithmetic and then achieving double precision accuracy using mixed precision refinement.

翻译：不完全分解长期以来一直是求解大规模稀疏线性方程组的通用代数预条件子。在计算高质量预条件子的同时确保分解不发生分解崩溃是一项具有挑战性的任务。低精度算术重新引起的研究兴趣使得对鲁棒性的探索更加紧迫和艰难。本文针对对称正定问题，探讨了多种方法：一种尽可能早地预测分解崩溃的前瞻策略、全局位移的使用，以及对数值优化领域用于稠密矩阵完全Cholesky分解的改进思路。我们的数值模拟以高度病态的稀疏线性系统为目标，旨在半精度算术下计算分解因子，并利用混合精度精化实现双精度精度。