A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

@article{Liao2022ASG,
  title={A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations},
  author={Zeyu Liao and Ken Hayami and Keiichi Morikuni and Junfeng Yin},
  journal={Japan Journal of Industrial and Applied Mathematics},
  year={2022}
}
Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized… 

GMRES using pseudo-inverse for range symmetric singular systems

TLDR
This work derived the necessary and sufficient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix.

References

SHOWING 1-10 OF 34 REFERENCES

GMRES using pseudo-inverse for range symmetric singular systems

TLDR
This work derived the necessary and sufficient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix.

GMRES On (Nearly) Singular Systems

TLDR
Conditions under which the GMRES iterates safely to a least-squares solution or to the pseudoinverse solution are given, which apply to any residual minimizing Krylov subspace method that is mathematically equivalent to GMRES.

Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES

TLDR
This work shows that MGS-GMRES is backward stable, and uses other new results on MGS and its loss of orthogonality, together with an important but neglected condition number, and a relation between residual norms and certain singular values.

GMRES-type methods for inconsistent systems

Error estimates for the regularization of least squares problems

TLDR
A class of error estimates previously introduced by the authors are extended to the least squares solution of consistent and inconsistent linear systems, and their application to various direct and iterative regularization methods is discussed.

LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

TLDR
Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I ~QR is the most reliable algorithm when A is ill-conditioned.

Algorithms for range restricted iterative methods for linear discrete ill-posed problems

TLDR
It is described MATLAB codes for the best of these implementations of range restricted iterative methods based on the Arnoldi process for symmetric linear discrete ill-posed problems with error-contaminated data.

Breakdown-free GMRES for Singular Systems

TLDR
Property of GMRES solutions at breakdown are discussed and a modification of GM RES to overcome the breakdown is presented.

Inner-Iteration Krylov Subspace Methods for Least Squares Problems

TLDR
Numerical experiments on overdetermined sparse least squares problems show that the proposed methods outperform previous methods, especially for ill-conditioned and rank-deficient problems.