LSMR: An iterative algorithm for sparse least-squares problems

Abstract. An iterative method LSMR is presented for solving linear systems Ax = b and leastsquares problem 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). In practice we observe that rk also decreases monotonically. Compared to LSQR, for which only rk 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. Key words. least-squares problem, sparse matrix, LSQR, MINRES, Krylov subspace method, Golub-Kahan process, conjugate-gradient method, minimum-residual method, iterative method AMS subject classifications. 15A06, 65F10, 65F20, 65F22, 65F25, 65F35, 65F50, 93E24 DOI. xxx/xxxxxxxxx

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: David Fong, Michael Saunders
• Keywords: CORR 2010, Education, Iterative Method, Iterative Method Lsmr, Linear Systems Ax