标题： 低精度不完全Cholesky分解预条件器在稀疏线性最小二乘问题中的计算研究 显示英文标题

标题： A computational study of low precision incomplete Cholesky factorization preconditioners for sparse linear least-squares problems

摘要： 我们关注的是大规模稀疏线性最小二乘问题的鲁棒和高效求解。 近年来，硬件的发展促使人们重新关注在数值线性代数算法中使用混合精度算术，以利用内存需求、运行时间和能耗方面的潜在节省，同时仍达到所需的精度。 我们探讨在求解最小二乘问题时使用混合精度，重点在于使用低精度不完全Cholesky分解预条件器开发鲁棒方法的实用性。 较低精度相关的关键惩罚包括计算解的可靠性降低和准确性减少。 通过涉及实际应用问题的实验，我们研究使用低精度计算正规矩阵的不完全Cholesky分解，并使用这些分解以混合精度对LSQR进行预处理。 我们研究基于层次的和受内存限制的不完全分解预条件器。 我们发现前者对于最小二乘问题效果不佳，而后者可以提供高质量的预条件器。 特别是，如果解中不需要高精度或不完全因子的内存非常有限，则可以考虑使用半精度算术；否则，可以使用单精度，并在减少内存消耗的同时恢复双精度精度，即使对于病态问题也是如此。 显示英文摘要

摘要： Our interest lies in the robust and efficient solution of large sparse linear least-squares problems. In recent years, hardware developments have led to a surge in interest in exploiting mixed precision arithmetic within numerical linear algebra algorithms to take advantage of potential savings in memory requirements, runtime and energy use, whilst still achieving the requested accuracy. We explore employing mixed precision when solving least-squares problems, focusing on the practicalities of developing robust approaches using low precision incomplete Cholesky factorization preconditioners. Key penalties associated with lower precision include a loss of reliability and less accuracy in the computed solution. Through experiments involving problems from practical applications, we study computing incomplete Cholesky factorizations of the normal matrix using low precision and using the factors to precondition LSQR using mixed precision. We investigate level-based and memory-limited incomplete factorization preconditioners. We find that the former are not effective for least-squares problems while the latter can provide high-quality preconditioners. In particular, half precision arithmetic can be considered if high accuracy is not required in the solution or the memory for the incomplete factors is very restricted; otherwise, single precision can be used, and double precision accuracy recovered while reducing memory consumption, even for ill-conditioned problems.