Double precision is not necessary for LSQR for solving discrete linear ill-posed problems

TL;DR

This paper demonstrates that using single precision in LSQR for solving discrete linear ill-posed problems does not compromise accuracy, enabling significant performance improvements in scientific computing.

Contribution

It shows that key parts of LSQR can be computed in single precision without losing accuracy, which is a novel insight for solving ill-posed problems efficiently.

Findings

Single precision suffices for Lanczos vector computation under mild conditions.

Iterative solution updates can be performed in single precision without accuracy loss.

Performance of LSQR can be significantly improved using single precision.

Abstract

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.