The growing availability and usage of low precision foating point formats has attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. In this paper we investigate the possibility of exploiting lower precision computing in LSQR for solving discrete linear ill-posed problems. We analyze the choice of proper computing precisions in the two main parts of LSQR, including the construction of Lanczos vectors and updating procedure of iterative solutions. We show that, under some mild conditions, the Lanczos vectors can be computed using single precision without loss of any accuracy of final regularized solutions as long as the noise level is not extremely small. We also show that the most time consuming part for updating iterative solutions can be performed using single precision without sacrificing any accuracy. The results indicate that the most time consuming parts of the algorithm can be implemented using single precision, and thus the performance of LSQR for solving discrete linear ill-posed problems can be significantly enhanced. Numerical experiments are made for testing the single precision variants of LSQR and confirming our results.

翻译：低精度浮点格式的日益普及和应用，激发了人们开发科学计算问题中低精度或混合精度算法的浓厚兴趣。本文研究了在求解离散线性不适定问题的LSQR算法中利用低精度计算的可行性。我们分析了LSQR两个主要部分（包括Lanczos向量的构建和迭代解的更新过程）中合理计算精度的选择。研究表明，在温和条件下，只要噪声水平不是极低，Lanczos向量可以使用单精度计算，且不会损失最终正则化解的任何精度。我们还证明，迭代解更新过程中最耗时的部分可以在不牺牲任何精度的前提下使用单精度执行。这些结果表明，算法中最耗时的部分可以以单精度实现，从而显著提升LSQR求解离散线性不适定问题的性能。通过数值实验对LSQR的单精度变体进行了测试，验证了我们的结论。