Using partial spectral information for block diagonal preconditioning of saddle-point systems

Considering saddle-point systems of the Karush–Kuhn–Tucker (KKT) form, we propose approximations of the “ideal” block diagonal preconditioner based on the exact Schur complement proposed by Murphy et al. (SIAM J Sci Comput 21(6):1969–1972, 2000). We focus on the case where the (1,1) block is symmetric and positive definite, but with a few very small eigenvalues that possibly affect the convergence of Krylov subspace methods like Minres. Assuming that these eigenvalues and their associated eigenvectors are available, we first propose a Schur complement preconditioner based on this knowledge and establish lower and upper bounds on the preconditioned Schur complement. We next analyse theoretically the spectral properties of the preconditioned KKT systems using this Schur complement approximation in two spectral preconditioners of block diagonal forms. In addition, we derive a condensed “two in one” formulation of the proposed preconditioners in combination with a preliminary level of preconditioning on the KKT system. Finally, we illustrate on a PDE test case how, in the context of a geometric multigrid framework, it is possible to construct practical block preconditioners that help to improve on the convergence of Minres.