Graph-Induced Rank Structures and their Representations

Abstract: A new framework is proposed to study rank-structured matrices arising from discretizations of 2D and 3D elliptic operators. In particular, we introduce the notion of a graph-induced rank structure (GIRS) which aims to capture the fine low rank structures which appear in sparse matrices and their inverses in terms of the adjacency graph $\mathbb{G}$. We show that the GIRS property is invariant under inversion, and hence any effective representation of the inverse of GIRS matrices would lead to effective solvers. Starting with the observation that sequentially semi-separable (SSS) matrices form a good candidate for representing GIRS matrices on the line graph, we propose two extensions of SSS matrices to arbitrary graphs: Dewilde-van der Veen (DV) representations and $\mathbb{G}$-semi-separable ($\mathbb{G}$-SS) representations. It is shown that both these representations come naturally equipped with fast solvers where the solve complexity is commensurate to fast sparse Gaussian elimination on the graph $\mathbb{G}$, and $\mathbb{G}$-SS representations have a linear time multiplication algorithm. We show the construction of these representations to be highly nontrivial by determining the minimal $\mathbb{G}$-SS representation for the cycle graph $\mathbb{G}$. To obtain a minimal representation, we solve an exotic variant of a low-rank completion problem.