LSMR: An iterative algorithm for sparse least-squares problems
arXiv:1006.0758 · doi:10.1016/j.ijheatmasstransfer.2011.12.029
Abstract
An iterative method LSMR is presented for solving linear systems and least-squares problem $\min \norm{Ax-b}_2$, with 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 $\norm{A\T r_k}$ are monotonically decreasing (where is the residual for the current iterate ). In practice we observe that $\norm{r_k}$ also decreases monotonically. Compared to LSQR, for which only $\norm{r_k}$ is monotonic, it is safer to terminate LSMR early. Improvements for the new iterative method in the presence of extra available memory are also explored.
21 pages