An Algebraic Multilevel Schur Complement Preconditioner Based on Eigenvalue Deflation

ABSTRACT This article proposes a new multilevel Schur complement preconditioner for solving large sparse symmetric linear systems. We derive a low‐rank approximation formula for the Schur complement matrix within a multilevel framework by employing eigenvalue deflation. We apply hierarchical interface decomposition (HID) ordering to render the matrix in a multilevel recursive form, while using the reverse Cuthill‐McKee (RCM) method to process diagonal blocks and reduce fill‐ins. In numerical implementation, the construction of the low‐rank approximation ultimately transforms into solving a generalized eigenvalue problem, which can be accomplished through the implicit restart Lanczos algorithm. Numerical experiments demonstrate that, after fully utilizing instruction‐level and thread‐level parallelism and introducing GPU acceleration, this preconditioner not only achieves good computational efficiency on model problems but also exhibits excellent robustness on practical problems. The source codes are available at https://github.com/YidanDeng/PrisPack .