论文标题
LSQR不需要双重精度解决离散线性不良问题
Double precision is not necessary for LSQR for solving discrete linear ill-posed problems
论文作者
论文摘要
越来越多的可用性和低精度富集点格式的使用引起了许多兴趣，即为科学计算问题开发较低或混合的精度算法。在本文中，我们研究了利用LSQR中较低精度计算来解决离散线性不良问题的可能性。我们分析了LSQR两个主要部分的适当计算精度的选择，包括构建Lanczos向量以及迭代解决方案的更新过程。我们表明，在某些温和的条件下，只要噪声水平不小，就可以使用单个精度来计算兰努佐载体，而不会损失最终正则化解决方案的任何精度。我们还表明，可以使用单个精度执行更新迭代解决方案的最耗时的零件，而无需牺牲任何准确性。结果表明，算法中最耗时的部分可以使用单个精度实现，因此可以显着提高LSQR用于求解离散线性不良问题的LSQR。进行数值实验，用于测试LSQR的单个精度变体并确认我们的结果。
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.
