An Algebraic Multilevel Preconditioner with Low-Rank Corrections for Sparse Symmetric Matrices
In this work, we describe a multilevel preconditioning technique for solving sparse symmetric linear systems of equations. This “Multilevel Schur Low-Rank” (MSLR) preconditioner first builds a tree structure $\mathcal{T}$ 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 $\mathcal{T}$ 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 two- and three-dimensional discretized PDE problems and with publicly available test problems.