Probing Methods for Generalized Saddle-Point Problems

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen ’00], [de Sturler and Liesen ’03] and [Siefert and de Sturler ’04]. This approach can be applied in similar preconditioners as well. We discuss the implementation of probing-based approximations to Schur complements. We consider the application of those approximations in preconditioners for Navier-Stokes problems and metal deformation problems. Finally, we present eigenvalue clustering for the preconditioned matrices, and convergence and timing results. These demonstrate the effectiveness of the proposed preconditioners with probing-based approximate Schur complements.