Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 33, Issue 5, Pages 2950-2971Publisher
SIAM PUBLICATIONS
DOI: 10.1137/10079687X
Keywords
least-squares problem; sparse matrix; LSQR; MINRES; Krylov subspace method; Golub-Kahan process; conjugate-gradient method; minimum-residual method; iterative method
Categories
Funding
- Office of Naval Research [N00014-08-1-0191]
- U.S. Army Research Laboratory through the Army High Performance Computing Research Center [W911NF-07-0027]
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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.
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