Double Precision is not Necessary for LSQR for Solving Discrete Linear Ill-Posed Problems

The growing availability and usage of low precision floating point formats attracts many interests of developing lower or mixed precision algorithms for scientific computing problems. In this paper we investigate the possibility of exploiting mixed precision computing in LSQR for solving discrete linear ill-posed problems. Based on the commonly used regularization model for linear inverse problems, we analyze the choice of proper computing precision in the two main parts of LSQR, including the construction of Krylov subspace 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 the final regularized solution 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 several highly time consuming parts of the algorithm can be implemented using lower precisions, and provide a theoretical guideline for implementing a robust and efficient mixed precision variant of LSQR for solving discrete linear ill-posed problems. Numerical experiments are made to test two mixed precision variants of LSQR and confirming our results.