LSMR: An iterative algorithm for sparse least-squares problems

Abstract

An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problems min ||Ax-b||2, with A being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation ATAx = ATb, so that the quantities ||ATrk|| are monotonically decreasing (where rk = b-Axk is the residual for the current iterate xk). We observe in practice that ||rk|| also decreases monotonically, so that compared to LSQR (for which only ||rk|| is monotonic) it is safer to terminate LSMR early. We also report some experiments with reorthogonalization. © 2011 Society for Industrial and Applied Mathematics.