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. Chang, Xiao-Wen; Paige, Christopher C.; Titley-Peloquin, David: Structure in loss of orthogonality (2021)
  2. Montoison, Alexis; Orban, Dominique: TriCG and TriMR: two iterative methods for symmetric quasi-definite systems (2021)
  3. Morikuni, Keiichi: Projection method for eigenvalue problems of linear nonsquare matrix pencils (2021)
  4. Bock, Hans Georg; Gutekunst, Jürgen; Potschka, Andreas; Garcés, María Elena Suaréz: A flow perspective on nonlinear least-squares problems (2020)
  5. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  6. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  7. Coppé, Vincent; Huybrechs, Daan; Matthysen, Roel; Webb, Marcus: The AZ algorithm for least squares systems with a known incomplete generalized inverse (2020)
  8. Gazzola, Silvia; Kilmer, Misha E.; Nagy, James G.; Semerci, Oguz; Miller, Eric L.: An inner-outer iterative method for edge preservation in image restoration and reconstruction (2020)
  9. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  10. Jia, Zhongxiao: Regularization properties of LSQR for linear discrete ill-posed problems in the multiple singular value case and best, near best and general low rank approximations (2020)
  11. Jia, Zhongxiao: Regularization properties of Krylov iterative solvers CGME and LSMR for linear discrete ill-posed problems with an application to truncated randomized SVDs (2020)
  12. Jia, Zhongxiao: Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems (2020)
  13. Ranocha, Hendrik; Ostaszewski, Katharina; Heinisch, Philip: Discrete vector calculus and Helmholtz Hodge decomposition for classical finite difference summation by parts operators (2020)
  14. Schaub, Michael T.; Benson, Austin R.; Horn, Paul; Lippner, Gabor; Jadbabaie, Ali: Random walks on simplicial complexes and the normalized Hodge 1-Laplacian (2020)
  15. Asgari, Z.; Toutounian, F.; Babolian, E.; Tohidi, E.: LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations (2019)
  16. Avron, Haim; Druinsky, Alex; Toledo, Sivan: Spectral condition-number estimation of large sparse matrices. (2019)
  17. Buttari, Alfredo; Orban, Dominique; Ruiz, Daniel; Titley-Peloquin, David: A tridiagonalization method for symmetric saddle-point systems (2019)
  18. Chung, Julianne; Gazzola, Silvia: Flexible Krylov methods for (\ell_p) regularization (2019)
  19. Dahito, Marie-Ange; Orban, Dominique: The conjugate residual method in linesearch and trust-region methods (2019)
  20. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LSLQ: an iterative method for linear least-squares with an error minimization property (2019)

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