Flexible Modified LSMR Method for Least Squares Problems
Mei Yang, Gul Karaduman, Ren-Cang Li
TL;DR
The paper addresses efficient solution of least-squares problems with right preconditioning by introducing Flexible Modified LSMR (FMLSMR), a method that merges two inner solves into a single system via $M = L^{\mathsf{T}}L$ and leverages flexible GMRES ideas to support dynamic inner preconditioning. It combines the MLSMR framework with a flexible inner-solver strategy (e.g., MINRES) to form a flexible variant that maintains a Krylov subspace $\mathcal{K}_k(M^{-1}A^{\mathsf{T}}A, M^{-1}A^{\mathsf{T}}b)$ while allowing varying preconditioners. The paper provides theoretical comparisons among MLSMR, FLSMR, and FMLSMR, and demonstrates that FMLSMR achieves faster convergence and lower storage requirements than its counterparts on challenging, large-scale problems. Numerical experiments on eight sparse matrices show that FMLSMR often reduces iterations and CPU time and remains robust where FLSMR and LSMR struggle, underscoring its practical value for suitable preconditioning scenarios. Overall, FMLSMR offers a robust, scalable approach for right-preconditioned least-squares problems with flexible inner iterations and reduced computational cost.
Abstract
LSMR is a widely recognized method for solving least squares problems via the double QR decomposition. Various preconditioning techniques have been explored to improve its efficiency. One issue that arises when implementing these preconditioning techniques is the need to solve two linear systems per iterative step. In this paper, to tackle this issue, among others, a modified LSMR method (MLSMR), in which only one linear system per iterative step needs to be solved instead of two, is introduced, and then it is integrated with the idea of flexible GMRES to yield a flexible MLSMR method (FMLSMR). Numerical examples are presented to demonstrate the efficiency of the proposed FMLSMR method.