# Regularization Properties of the Krylov Iterative Solvers CGME and LSMR For Linear Discrete Ill-Posed Problems with an Application to Truncated Randomized SVDs

## TL;DR

This paper analyzes the regularization properties of Krylov solvers CGME and LSMR for large-scale ill-posed problems, comparing their effectiveness with LSQR and improving understanding of their solutions and truncation effects.

## Contribution

It establishes the regularization properties of CGME and LSMR, including filtered SVD expansions and accuracy comparisons with LSQR, and improves fundamental results on randomized truncated SVDs.

## Key findings

- CGME and LSMR have regularization properties similar to LSQR.
- The solutions obtained by CGME and LSMR are less accurate than those by LSQR.
- Truncation in randomized SVDs can reduce accuracy, as analyzed in the paper.

## Abstract

For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or $Ax=b$ with $b$ contaminated by Gaussian white noise, there are four commonly used Krylov solvers: LSQR and its mathematically equivalent CGLS, the Conjugate Gradient (CG) method applied to $A^TAx=A^Tb$, CGME, the CG method applied to $\min\|AA^Ty-b\|$ or $AA^Ty=b$ with $x=A^Ty$, and LSMR, the minimal residual (MINRES) method applied to $A^TAx=A^Tb$. These methods have intrinsic regularizing effects, where the number $k$ of iterations plays the role of the regularization parameter. In this paper, we establish a number of regularization properties of CGME and LSMR, including the filtered SVD expansion of CGME iterates, and prove that the 2-norm filtering best regularized solutions by CGME and LSMR are less accurate than and at least as accurate as those by LSQR, respectively. We also prove that the semi-convergence of CGME and LSMR always occurs no later and sooner than that of LSQR, respectively. As a byproduct, using the analysis approach for CGME, we improve a fundamental result on the accuracy of the truncated rank $k$ approximate SVD of $A$ generated by randomized algorithms, and reveal how the truncation step damages the accuracy. Numerical experiments justify our results on CGME and LSMR.