The method of polarized traces for the 3D Helmholtz equation

TL;DR

This paper introduces a fast, scalable solver for the 3D high-frequency Helmholtz equation using polarized traces, enabling efficient large-scale waveform inversion in frequency domain.

Contribution

The paper develops a novel polarized traces method combined with distributed linear algebra for efficient 3D Helmholtz equation solutions.

Findings

Achieves empirical runtime $\, ext{O}( ext{max}(1,R/n) N ext{log} N)$ for large problems.

Scalable to large degrees of freedom and multiple right-hand sides.

Facilitates large-scale full waveform inversion in frequency domain.

Abstract

We present a fast solver for the 3D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media. The solver is based on the method of polarized traces, coupled with distributed linear algebra libraries and pipelining to obtain an empirical online runtime $ \mathcal{O}(\max(1,R/n) N \log N)$ where $N = n^3$ is the total number of degrees of freedom and $R$ is the number of right-hand sides. Such a favorable scaling is a prerequisite for large-scale implementations of full waveform inversion (FWI) in frequency domain.