A Preconditioner for Generalized Saddle Point Problems

Abstract

Abstract. In this paper we consider the solution of linear systems of saddle point type by preconditioned Krylov subspace methods. A preconditioning strategy based on the symmetric/ skew-symmetric splitting of the coefficient matrix is proposed, and some useful properties of the preconditioned matrix are established. The potential of this approach is illustrated by numerical experiments with matrices from various application areas.

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

“…This leads to the preconditioned Hermitian and skew-Hermitian splitting (PHSS) iteration method as follows. See also [2,3,4] and [11,12].…”

“…From Theorem 3.1 and Proposition 3.3, one can easily deduce the convergence results on the HSS iteration method for positive definite matrices in [3] and on the PHSS iteration methods for special positive semidefinite saddle-point matrices (1.2) in [2,4,11]. In the former case, it is clear that no eigenvalue has the form iξ with ξ ∈ R. In the lattter case, under the assumption that E has full column rank, condition (d) in Proposition 3.3 cannot hold.…”

“…with B ∈ C p×p being positive definite (i.e., H(B) is Hermitian positive definite), C ∈ C q×q being Hermitian positive semidefinite, and E ∈ C p×q being of full column rank, Benzi, and Golub further proved in [11] that the HSS iteration method for the corresponding saddle-point problem…”

“…Hence, in a practical computation, it is a crucial but difficult problem to determine a good preconditioner P and choose an optimal iteration parameter α. For some discussions on this aspect, we refer the readers to [2,4,11,12].…”

“…the relation between the null spaces of the matrices H 1 , H 2 , E, and E * may not be too useful in determining whether ρ(L(α, P )) < 1; see [11]. However, it holds that null(…”