LSMR: An Iterative Algorithm for Sparse Least-Squares Problems

An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares 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 $A^T\! Ax = A^T\! b$, so that the quantities $\|A^T\! r_k\|$ are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). We observe in practice that $\|r_k\|$ also decreases monotonically, so that compared to LSQR (for which only $\|r_k\|$ is monotonic) it is safer to terminate LSMR early. We also report some experiments with reorthogonalization.