"Compress and eliminate" solver for symmetric positive definite sparse matrices

TL;DR

This paper introduces a novel approximate factorization method for symmetric positive definite sparse matrices, leveraging hierarchical block Gaussian elimination and fill-in compression to improve efficiency in solving PDE discretization systems.

Contribution

It presents a new factorization approach that combines hierarchical elimination with fill-in compression, offering an efficient preconditioner for PDE-related sparse matrices.

Findings

Effective preconditioning for PDE discretization matrices

Competitive performance against state-of-the-art solvers

Reduced fill-in through compression improves computational efficiency

Abstract

We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in. The systems that have efficient compression of the fill-in mostly arise from discretization of partial differential equations. We show that the resulting factorization can be used as an efficient preconditioner and compare the proposed approach with state-of-art direct and iterative solvers.