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. Cerdán, J.; Guerrero, D.; Marín, J.; Mas, J.: Preconditioners for rank deficient least squares problems (2020)
  2. Chang, Xiao-Wen; Kang, Peng; Titley-Peloquin, David: Error bounds for computed least squares estimators (2020)
  3. Huang, Baohua; Ma, Changfeng: Global least squares methods based on tensor form to solve a class of generalized Sylvester tensor equations (2020)
  4. Jia, Zhongxiao: Approximation accuracy of the Krylov subspaces for linear discrete ill-posed problems (2020)
  5. 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)
  6. Schaub, Michael T.; Benson, Austin R.; Horn, Paul; Lippner, Gabor; Jadbabaie, Ali: Random walks on simplicial complexes and the normalized Hodge 1-Laplacian (2020)
  7. Asgari, Z.; Toutounian, F.; Babolian, E.; Tohidi, E.: LSMR iterative method for solving one- and two-dimensional linear Fredholm integral equations (2019)
  8. Buttari, Alfredo; Orban, Dominique; Ruiz, Daniel; Titley-Peloquin, David: A tridiagonalization method for symmetric saddle-point systems (2019)
  9. Chung, Julianne; Gazzola, Silvia: Flexible Krylov methods for (\ell_p) regularization (2019)
  10. Dahito, Marie-Ange; Orban, Dominique: The conjugate residual method in linesearch and trust-region methods (2019)
  11. Estrin, Ron; Orban, Dominique; Saunders, Michael A.: LSLQ: an iterative method for linear least-squares with an error minimization property (2019)
  12. Gazzola, Silvia; Sabaté Landman, Malena: Flexible GMRES for total variation regularization (2019)
  13. Karimi, Saeed; Jozi, Meisam: Weighted conjugate gradient-type methods for solving quadrature discretization of Fredholm integral equations of the first kind (2019)
  14. Liu, Xiaoxing; Morita, Koji; Zhang, Shuai: A pairwise-relaxing incompressible smoothed particle hydrodynamics scheme (2019)
  15. Paige, Christopher C.: Accuracy of the Lanczos process for the eigenproblem and solution of equations (2019)
  16. Arreckx, Sylvain; Orban, Dominique: A regularized factorization-free method for equality-constrained optimization (2018)
  17. Estrin, Ron; Greif, Chen: SPMR: A family of saddle-point minimum residual solvers (2018)
  18. Fong, Justin; Tan, Ying; Crocher, Vincent; Oetomo, Denny; Mareels, Iven: Dual-loop iterative optimal control for the finite horizon LQR problem with unknown dynamics (2018)
  19. Hallman, Eric; Gu, Ming: LSMB: minimizing the backward error for least-squares problems (2018)
  20. Holman, Sean; Monard, François; Stefanov, Plamen: The attenuated geodesic x-ray transform (2018)

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