Convergence analysis of sweeping preconditioner with sequentially semiseparable Schur complement approximation

We investigate the convergence properties of the sweeping preconditioner with sequentially semiseparable (SSS) Schur complement approximation proposed by Qiu et al. (2015b, Efficient preconditioners for PDE-constrained optimization problems with a multilevel sequentially semiseparable matrix structure. Electron. Trans. Numer. Anal., 44, 367–400). The sweeping preconditioner is obtained by replacing the Schur complements in the block LU factorization of the global linear system by SSS matrix approximations with low off-diagonal rank. We show that the spectrum of the preconditioned system is confined within a circle centred at $(1, 0)$ and derive an analytic estimate of the radius of this circle. By appropriately tuning a parameter in the sweeping SSS preconditioner the radius can be made arbitrarily small. Our analysis results hold for a broad class of linear systems arising from discretizations of two-dimensional partial differential equations with uniform structured grids, irrespective of whether the system matrix is symmetric or unsymmetric, definite or indefinite and even with saddle-point structure. We demonstrate the efficacy of our analysis by numerical experiments.