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.

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  1. Ahmadi-Asl, Salman; Beik, Fatemeh Panjeh Ali: Iterative algorithms for least-squares solutions of a quaternion matrix equation (2017)
  2. Hnětynková, Iveta; Kubínová, Marie; Plešinger, Martin: Noise representation in residuals of LSQR, LSMR, and CRAIG regularization (2017)
  3. Ji, Hao; Li, Yaohang: Block conjugate gradient algorithms for least squares problems (2017)
  4. Mojarrab, M.; Toutounian, F.: Global LSMR(Gl-LSMR) method for solving general linear systems with several right-hand sides (2017)
  5. Renaut, Rosemary A.; Vatankhah, Saeed; Ardestani, Vahid E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems (2017)
  6. Scott, Jennifer: On using Cholesky-based factorizations and regularization for solving rank-deficient sparse linear least-squares problems (2017)
  7. Zwaan, Ian N.; Hochstenbach, Michiel E.: Multidirectional subspace expansion for one-parameter and multiparameter Tikhonov regularization (2017)
  8. Deadman, Edvin; Higham, Nicholas J.: Testing matrix function algorithms using identities (2016)
  9. Diamond, Steven; Boyd, Stephen: Matrix-free convex optimization modeling (2016)
  10. Greif, C.; Paige, C.C.; Titley-Peloquin, D.; Varah, J.M.: Numerical equivalences among Krylov subspace algorithms for skew-symmetric matrices (2016)
  11. van Leeuwen, T.; Herrmann, F.J.: A penalty method for PDE-constrained optimization in inverse problems (2016)
  12. Zhang, Xiaowei; Cheng, Li; Chu, Delin; Liao, Li-Zhi; Ng, Michael K.; Tan, Roger C.E.: Incremental regularized least squares for dimensionality reduction of large-scale data (2016)
  13. Chung, Julianne M.; Kilmer, Misha E.; O’Leary, Dianne P.: A framework for regularization via operator approximation (2015)
  14. Chung, Julianne; Palmer, Katrina: A hybrid LSMR algorithm for large-scale Tikhonov regularization (2015)
  15. Conder, James A.: Fitting multiple Bell curves stably and accurately to a time series as applied to Hubbert cycles or other phenomena (2015)
  16. Morikuni, Keiichi; Hayami, Ken: Convergence of inner-iteration GMRES methods for rank-deficient least squares problems (2015)
  17. Arridge, S.R.; Betcke, M.M.; Harhanen, L.: Iterated preconditioned LSQR method for inverse problems on unstructured grids (2014)
  18. Baglama, James; Richmond, Daniel J.: Implicitly restarting the LSQR algorithm (2014)
  19. Choi, Sou-Cheng T.; Saunders, Michael A.: Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems (2014)
  20. Gould, Nick; Orban, Dominique; Rees, Tyrone: Projected Krylov methods for saddle-point systems (2014)

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