# Inner-Iteration Krylov Subspace Methods for Least Squares Problems

@article{Morikuni2013InnerIterationKS, title={Inner-Iteration Krylov Subspace Methods for Least Squares Problems}, author={Keiichi Morikuni and Ken Hayami}, journal={SIAM J. Matrix Anal. Appl.}, year={2013}, volume={34}, pages={1-22} }

Stationary inner iterations in combination with Krylov subspace methods are proposed for overdetermined least squares problems. The inner iterations are efficient in terms of computational work and memory and also serve as powerful preconditioners for ill-conditioned and rank-deficient problems. Theoretical justifications for using the inner iterations as preconditioners are presented. Numerical experiments on overdetermined sparse least squares problems show that the proposed methods…

## 35 Citations

### Convergence of Inner-Iteration GMRES Methods for Least Squares Problems

- Computer Science, Mathematics
- 2012

A general convergence theory for the generalized minimal residual method for least squares problems preconditioned with inner iterations is developed and improved particularly in the rankdeficient case.

### Convergence of Inner-Iteration GMRES Methods for Rank-Deficient Least Squares Problems

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2015

A general convergence theory for the generalized minimal residual method preconditioned by inner iterations for solving least squares problems is developed and numerical experiments show that the proposed methods are more robust and efficient compared to previous methods for some rank-deficient problems.

### The State-of-the-Art of Preconditioners for Sparse Linear Least-Squares Problems

- Computer ScienceACM Trans. Math. Softw.
- 2017

This study briefly reviews preconditioners for which software has been made available, then presents a numerical evaluation of them using performance profiles and a large set of problems arising from practical applications.

### Symmetric inner-iteration preconditioning for rank-deficient least squares problems

- Computer Science
- 2015

The CG and MR-type methods such as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations justify using these methods for solving least squares and minimum-norm solution problems those coefficient matrices are not necessarily of full rank.

### Inner-iteration preconditioning with a symmetric splitting matrix for rank-deficient least squares problems.

- Computer Science
- 2019

Results are applied to the CG and MINRES-type methods such as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations, and justify using these methods for solving least squares and minimum-norm solution problems whose coefficient matrices are not necessarily of full rank.

### Implementation of interior-point methods for LP based on Krylov subspace iterative solvers with inner-iteration preconditioning

- Computer ScienceComput. Optim. Appl.
- 2019

The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria and presents a fairly extensive benchmark test for several renowned solvers including direct and iterativesolvers.

### Multistep matrix splitting iteration preconditioning for singular linear systems

- Computer ScienceNumerical Algorithms
- 2017

Numerical experiments show that the multistep generalized shifted splitting and Hermitian and skew-Hermitian splitting iteration preconditionsing are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.

### Kaczmarz-type inner-iteration preconditioned flexible GMRES methods for consistent linear systems

- Computer ScienceSIAM J. Sci. Comput.
- 2021

Numerical experiments on overdetermined and underdetermined linear systems show that the proposed method is superior to the GMRES method preconditioned by NE-SOR inner iterations in terms of total CPU time.

### Modulus-Type Inner Outer Iteration Methods for Nonnegative Constrained Least Squares Problems

- MathematicsSIAM J. Matrix Anal. Appl.
- 2016

For the solution of large sparse nonnegative constrained linear least squares (NNLS) problems, a new iterative method is proposed which uses the CGLS method for the inner iterations and the modulus…

### General-purpose Preconditioners for the Conjugate Gradient (cg) and Gen

- Mathematics

eralized minimal residual (GMRES) type methods are proposed for solving the linear least squares problem min x∈R n b − Ax 2 and the general least squares problem min x∈S x 2 , S = {x ∈ R n : b − Ax 2…

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