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. Greif, C.; Paige, C.C.; Titley-Peloquin, D.; Varah, J.M.: Numerical equivalences among Krylov subspace algorithms for skew-symmetric matrices (2016)
  2. van Leeuwen, T.; Herrmann, F.J.: A penalty method for PDE-constrained optimization in inverse problems (2016)
  3. 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)
  4. Chung, Julianne M.; Kilmer, Misha E.; O’Leary, Dianne P.: A framework for regularization via operator approximation (2015)
  5. Chung, Julianne; Palmer, Katrina: A hybrid LSMR algorithm for large-scale Tikhonov regularization (2015)
  6. Conder, James A.: Fitting multiple Bell curves stably and accurately to a time series as applied to Hubbert cycles or other phenomena (2015)
  7. Arridge, S.R.; Betcke, M.M.; Harhanen, L.: Iterated preconditioned LSQR method for inverse problems on unstructured grids (2014)
  8. Baglama, James; Richmond, Daniel J.: Implicitly restarting the LSQR algorithm (2014)
  9. Choi, Sou-Cheng T.; Saunders, Michael A.: Algorithm 937: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems (2014)
  10. Gould, Nick; Orban, Dominique; Rees, Tyrone: Projected Krylov methods for saddle-point systems (2014)
  11. Schiavazzi, Daniele; Coletti, Filippo; Iaccarino, Gianluca; Eaton, John K.: A matching pursuit approach to solenoidal filtering of three-dimensional velocity measurements (2014)
  12. Baglama, J.; Reichel, L.; Richmond, D.: An augmented LSQR method (2013)
  13. Bellavia, Stefania; Gondzio, Jacek; Morini, Benedetta: A matrix-free preconditioner for sparse symmetric positive definite systems and least-squares problems (2013)
  14. Foster, Leslie V.; Davis, Timothy A.: Algorithm 933, reliable calculation of numerical rank, null space bases, pseudoinverse solutions, and basic solutions using SuiteSparseQR (2013)
  15. Gratton, Serge; Jiránek, Pavel; Titley-Peloquin, David: Simple backward error bounds for linear least-squares problems (2013)
  16. Li, Zhengming; Zhu, Qi; Xie, Binglei; Cao, Jian; Zhang, Jin: A collaborative neighbor representation based face recognition algorithm (2013)
  17. Zhao, Chao; Huang, Ting-Zhu; Zhao, Xi-Le; Deng, Liang-Jian: Two new efficient iterative regularization methods for image restoration problems (2013)
  18. Doan, Xuan Vinh; Kruk, Serge; Wolkowicz, Henry: A robust algorithm for semidefinite programming (2012)
  19. Fong, David Chin-Lung; Saunders, Michael: LSMR: an iterative algorithm for sparse least-squares problems (2011)