The method of polarized traces for the 3D Helmholtz equation


Title

The method of polarized traces for the 3D Helmholtz equation

Publication Type
Journal Article
Year of Publication
2019
Journal
GEOPHYSICS
Volume
84
Issue
4
Pagination
T313 – T333
Date Published
Mar-07-2020
Publication Language
eng
Citation Key
3367
ISSN
0016-8033
Abstract

We have developed a fast solver for the 3D Helmholtz equation, in heterogeneous, constant density, acoustic media, in the high-frequency regime. The solver is based on the method of polarized traces, a layered domain-decomposition method, where the subdomains are connected via transmission conditions prescribed by the discrete Green’s representation formula and artificial reflections are avoided by enforcing nonreflecting boundary conditions between layers. The method of polarized traces allows us to consider only unknowns at the layer interfaces, reducing the overall cost and memory footprint of the solver. We determine that polarizing the wavefields in this manner yields an efficient preconditioner for the reduced system, whose rate of convergence is independent of the problem frequency. The resulting preconditioned system is solved iteratively using generalized minimum residual, where we never assemble the reduced system or preconditioner; rather, we implement them via solving the Helmholtz equation locally within the subdomains. This scaling is favorable for regimes in which the number of sources (distinct RHS) is large, for example, enabling large-scale implementations of frequency-domain full-waveform inversion.

Short Title
GEOPHYSICS