A fast direct solver for network matrices

TL;DR

This paper introduces a fast direct inversion method for large sparse matrices from elliptic PDE discretizations, significantly reducing computational complexity and storage requirements, enabling efficient solutions for very large systems.

Contribution

The paper presents a novel $O(N ext{log}^2 N)$ scheme for directly inverting large sparse matrices from 2D elliptic PDE discretizations, with practical efficiency demonstrated on million-scale matrices.

Findings

Requires $O(N ext{log}^2 N)$ operations for approximate inversion.

Achieves $O(N ext{log} N)$ storage, or $O( ext{sqrt}(N) ext{log} N)$ for single solve.

Inverts a $10^6 imes 10^6$ matrix to seven digits in four minutes.

Abstract

A fast direct inversion scheme for the large sparse systems of linear equations resulting from the discretization of elliptic partial differential equations in two dimensions is given. The scheme is described for the particular case of a discretization on a uniform square grid, but can be generalized to more general geometries. For a grid containing $N$ points, the scheme requires $O(N \log^{2}N)$ arithmetic operations and $O(N \log N)$ storage to compute an approximate inverse. If only a single solve is required, then the scheme requires only $O(\sqrt{N} \log N)$ storage; the same storage is sufficient for computing the Dirichlet-to-Neumann operator as well as other boundary-to-boundary operators. The scheme is illustrated with several numerical examples. For instance, a matrix of size $10^6 \times 10^6$ is inverted to seven digits accuracy in four minutes on a 2.8GHz P4 desktop PC.