# Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy

@article{Darrigrand2022InexactIG, title={Inexact inner-outer Golub-Kahan bidiagonalization method: A relaxation strategy}, author={Vincent Darrigrand and A. Dumitrasc and Carola Kruse and Ulrich R{\"u}de}, journal={ArXiv}, year={2022}, volume={abs/2208.01079} }

We study an inexact inner-outer generalized Golub-Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer eﬃcient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the inﬂuence of the accuracy of an inner iterative solution…

## References

SHOWING 1-10 OF 31 REFERENCES

### Generalized Golub-Kahan Bidiagonalization and Stopping Criteria

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2013

It is shown how to transform augmented systems arising from the mixed finite-element approximation of partial differential equations in order to achieve a convergence rate independent of the finite dimensional problem size.

### Inexact Matrix-Vector Products in Krylov Methods for Solving Linear Systems: A Relaxation Strategy

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 2005

This paper experimentally shows that Krylov methods for solving linear systems can still perform very well in the presence of carefully monitored inexact matrix-vector products.

### Regularization by inexact Krylov methods with applications to blind deblurring

- MathematicsSIAM J. Matrix Anal. Appl.
- 2021

Two new inexact Krylov methods are derived that can be efficiently applied to unregularized or Tikhonov-regularized least squares problems, and their theoretical properties are studied, including links with their exact counterparts and strategies to monitor the amount of inexactness.

### Hierarchical Krylov and nested Krylov methods for extreme-scale computing

- Computer ScienceParallel Comput.
- 2014

### Application of an iterative Golub-Kahan algorithm to structural mechanics problems with multi-point constraints

- EngineeringAdv. Model. Simul. Eng. Sci.
- 2020

The Craig variant of the Golub-Kahan bidiagonalization algorithm is used as an iterative method to solve the arising linear system with a saddle point structure and the condition number of the preconditioned operator is shown to be close to unity and independent of the mesh size.

### A restarted Krylov method with inexact inversions

- Computer ScienceNumer. Linear Algebra Appl.
- 2019

A new type of restarted Krylov method for calculating the smallest eigenvalues of a symmetric positive definite matrix G is presented, which avoids the Lanczos tridiagonalization process and the use of polynomial filtering.

### A Flexible Inner-Outer Preconditioned GMRES Algorithm

- Computer ScienceSIAM J. Sci. Comput.
- 1993

A variant of the GMRES algorithm is presented that allows changes in the preconditioning at every step, resulting in a result of the flexibility of the new variant that any iterative method can be used as a preconditionser.

### Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner–outer iterations

- Computer Science, Mathematics
- 2000

### Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems

- Computer ScienceSIAM J. Sci. Comput.
- 2008

A multilevel projection-based method for acceleration of Krylov subspace methods that is insensitive to the inaccurate solve of the Galerkin matrix, which with some particular choice of deflation subspaces is related to the coarse-grid solve in multigrid or domain decomposition methods.