An Algebraic Multilevel Preconditioner with Low-rank Corrections for General Sparse Symmetric Matrices

This paper describes a multilevel preconditioning technique for solving linear systems with general sparse symmetric coefficient matrices. This “multilevel Schur low rank” (MSLR) preconditioner first builds a tree structure T based on a hierarchical decomposition of the matrix and then computes an approximate inverse of the original matrix level by level. Unlike classical direct solvers, the construction of the MSLR preconditioner follows a top-down traversal of T and exploits a low-rank property that is satisfied by the difference between the inverses of the local Schur complements and specific blocks of the original matrix. A few steps of the generalized Lanczos tridiagonalization procedure are applied to capture most of this difference. Numerical results are reported to illustrate the efficiency and robustness of the MSLR preconditioner with both twoand three-dimensional discretized PDE problems as well as some publicly available test problems.