An alternating preconditioner for saddle point problems
Journal of Computational and Applied Mathematics
Preconditioning Saddle-Point Systems with Applications in Optimization
SIAM Journal on Scientific Computing
Preconditioning Iterative Methods for the Optimal Control of the Stokes Equations
SIAM Journal on Scientific Computing
Solving Hermitian positive definite systems using indefinite incomplete factorizations
Journal of Computational and Applied Mathematics
Computational Optimization and Applications
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It is widely appreciated that the iterative solution of linear systems of equations with large sparse matrices is much easier when the matrix is symmetric. It is equally advantageous to employ symmetric iterative methods when a nonsymmetric matrix is self-adjoint in a nonstandard inner product. Here, general conditions for such self-adjointness are considered. A number of known examples for saddle point systems are surveyed and combined to make new combination preconditioners which are self-adjoint in different inner products. In particular, a new method related to the Bramble-Pasciak CG method is introduced and it is shown that a combination of the two outperforms the widely used classical method on a number of examples. Furthermore, we combine Bramble and Pasciak's method with a recently introduced method by Schöberl and Zulehner. The result gives a new preconditioner and inner product that can outperform the original methods of Bramble-Pasciak and Schöberl-Zulehner.