Generalized Golub-Kahan bidiagonalization for nonsymmetric saddle point systems

TL;DR

This paper extends the generalized Golub-Kahan bidiagonalization method to nonsymmetric saddle point systems, enhancing its applicability and efficiency by adapting it from the Conjugate Gradient framework and proposing new stopping criteria.

Contribution

The authors develop a novel extension of the Golub-Kahan bidiagonalization for nonsymmetric saddle point systems, incorporating a Full Orthogonalization approach and tailored stopping rules.

Findings

The new method shows lower memory usage compared to GMRES.

Numerical experiments demonstrate improved convergence properties.

The approach effectively handles nonsymmetric saddle point problems.

Abstract

The generalized Golub-Kahan bidiagonalization has been used to solve saddle-point systems where the leading block is symmetric and positive definite. We extend this iterative method for the case where the symmetry condition no longer holds. We do so by relying on the known connection the algorithm has with the Conjugate Gradient method and following the line of reasoning that adapts the latter into the Full Orthogonalization Method. We propose appropriate stopping criteria based on the residual and an estimate of the energy norm for the error associated with the primal variable. Numerical comparison with GMRES highlights the advantages of our proposed strategy regarding its low memory requirements and the associated implications.