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 parallel to Ax-b parallel to(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 parallel to A(T)r(k)parallel to are monotonically decreasing (where r(k) = b - Ax(k) is the residual for the current iterate x(k)). We observe in practice that parallel to r(k)parallel to also decreases monotonically, so that compared to LSQR (for which only parallel to r(k)parallel to is monotonic) it is safer to terminate LSMR early. We also report some experiments with reorthogonalization.