GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
A taxonomy for conjugate gradient methods
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific and Statistical Computing
An implementation of the QMR method based on coupled two-term recurrences
SIAM Journal on Scientific Computing
Iterative methods for solving linear systems
Iterative methods for solving linear systems
Preconditioning techniques for large linear systems: a survey
Journal of Computational Physics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
On nonsymmetric saddle point matrices that allow conjugate gradient iterations
Numerische Mathematik
Combination Preconditioning and the Bramble-Pasciak$^{+}$ Preconditioner
SIAM Journal on Matrix Analysis and Applications
Using Perturbed $QR$ Factorizations to Solve Linear Least-Squares Problems
SIAM Journal on Matrix Analysis and Applications
Adaptive Techniques for Improving the Performance of Incomplete Factorization Preconditioning
SIAM Journal on Scientific Computing
Communication-avoiding krylov subspace methods
Communication-avoiding krylov subspace methods
The university of Florida sparse matrix collection
ACM Transactions on Mathematical Software (TOMS)
Hi-index | 7.29 |
Incomplete LDL^* factorizations sometimes produce an indefinite preconditioner even when the input matrix is Hermitian positive definite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive definite preconditioner. One approach, that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process. We explore a different approach: use the incomplete factorization with a Krylov method that can accept an indefinite preconditioner. The conventional wisdom has been that long recurrence methods (like GMRES), or alternatively non-optimal short recurrence methods (like symmetric QMR and BiCGStab) must be used if the preconditioner is indefinite. We explore the performance of these methods when used with an incomplete factorization, but also explore a less known Krylov method called PCG-ODIR that is both optimal and uses a short recurrence and can use an indefinite preconditioner. Furthermore, we propose another optimal short recurrence method called IP-MINRES that can use an indefinite preconditioner, and a variant of PCG-ODIR, which we call IP-CG, that is more numerically stable and usually requires fewer iterations.