Sparse Approximate Multifrontal Factorization with Butterfly Compression for High Frequency Wave Equations

TL;DR

This paper introduces a fast approximate multifrontal solver for large sparse systems from high-frequency wave equations, utilizing butterfly algorithms for efficient compression and factorization.

Contribution

It develops a novel solver combining butterfly compression with hierarchical matrices, achieving near-linear complexity for high-frequency wave problems.

Findings

Achieves $ ext{O}(N ext{log}^2 N)$ computational complexity.

Uses hierarchical butterfly algorithms for matrix compression.

Demonstrates efficiency on 3D Helmholtz and Maxwell problems.

Abstract

We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.