A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation
Abstract
Accurate simulation of three-dimensional time-harmonic wave propagation over many wavelengths requires control of phase error and efficient solution of large indefinite systems. We develop a three-grid solver based on the compact 27-point interpolated optimized finite-difference (IOFD) discretization. Its wavenumber-dependent stencil supports a fine-grid resolution of six points per shortest wavelength and an unshifted physical correction on the 2h grid at only three points per shortest wavelength. The method retains unshifted IOFD operators on the h and 2h grids, while a complex-shifted 2h--4h auxiliary cycle preconditions a factorization-free iterative approximation of the coarse inverse. Restricting the shift to this auxiliary cycle preserves the propagative character of the coarse correction. Comparison with the outgoing Green function confirms phase and relative-amplitude accuracy on a sequence of meshes up to \revision{102403}, \revision{exceeding one trillion unknowns,} with the largest problem spanning approximately \revision{1704} wavelengths per coordinate. The same fixed solver configuration retains robust convergence across smooth, discontinuous, high-contrast, and geophysical velocity models and exhibits scalable parallel performance. In particular, a \revision{25603} problem spanning approximately \revision{425} wavelengths in each coordinate direction is solved in \revision{43.8 seconds} on just 64 NVIDIA A100 GPUs.
Related
- Massively parallel Schwarz methods for the high-frequency Helmholtz equation with O(k) convergence
- Robust edge multiscale space based hybrid Schwarz preconditioner for high-frequency Helmholtz problems
- Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
- Cartesian PML truncation for the Helmholtz equation is exponentially accurate at high frequency
- Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
BibTeX
@misc{fu2026massively,
title = {{A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation}},
author = {Shubin Fu and Yitong Wang and Zixiao Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22946},
eprint = {2608.22946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22946},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Massively Parallel Three-Grid Preconditioner for the High-Frequency Helmholtz Equation
AU - Shubin Fu
AU - Yitong Wang
AU - Zixiao Zhao
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.22946
UR - https://arxiv.org/abs/2608.22946
AB - Accurate simulation of three-dimensional time-harmonic wave propagation over many wavelengths requires control of phase error and efficient solution of large indefinite systems. We develop a three-grid solver based on the compact 27-point interpolated optimized finite-difference (IOFD) discretization. Its wavenumber-dependent stencil supports a fine-grid resolution of six points per shortest wavelength and an unshifted physical correction on the 2h grid at only three points per shortest wavelength. The method retains unshifted IOFD operators on the h and 2h grids, while a complex-shifted 2h--4h auxiliary cycle preconditions a factorization-free iterative approximation of the coarse inverse. Restricting the shift to this auxiliary cycle preserves the propagative character of the coarse correction. Comparison with the outgoing Green function confirms phase and relative-amplitude accuracy on a sequence of meshes up to \revision{10240^3}, \revision{exceeding one trillion unknowns,} with the largest problem spanning approximately \revision{1704} wavelengths per coordinate. The same fixed solver configuration retains robust convergence across smooth, discontinuous, high-contrast, and geophysical velocity models and exhibits scalable parallel performance. In particular, a \revision{2560^3} problem spanning approximately \revision{425} wavelengths in each coordinate direction is solved in \revision{43.8 seconds} on just 64 NVIDIA A100 GPUs.
ER -