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:低精度顶点格式的日益普及和使用吸引了许多利益,为科学计算问题开发低精密算法或混合精密算法。在本文件中,我们调查了利用LSQR中低精密计算来解决离散线性问题的可能性。我们分析了LSQR两个主要部分的适当计算精确度的选择,包括建造Lanczos矢量以及更新迭代解决方案的程序。我们表明,在某些温和条件下,只要噪音水平不极小,就可使用单一精确度来计算朗克佐斯矢量,而不会丧失最终正规化解决方案的任何精确度。我们还表明,更新迭代解决方案最耗费时间的部分可以使用单一精确度,而不会牺牲任何精确度。结果显示,该算法中最耗时的部分可以使用单一精确度来实施,从而可以大大提高LSQR在解决离线性线性问题方面的性能。在测试LSQR的单一精度变方和确认我们的结果方面进行了纳米实验。