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型稀疏线性方程组时流行的通用代数预条件子。在保证分解无中断的同时计算高质量预条件子极具挑战性。对使用低精度算术兴趣的复苏使得对鲁棒性的探索更加紧迫和困难。本文聚焦于对称正定问题，探索了多种方法：用于尽可能提前预测中断的前瞻策略、全局平移的使用，以及对稠密矩阵完全Cholesky分解数值优化领域中一种思想的改进。我们的数值模拟针对高度病态的稀疏线性系统，目标是以半精度算术计算分解因子，然后通过混合精度精化实现双精度精度。