The method of polarized traces for the 2D Helmholtz equation
Name
Demanet_The method of.pdf
Size
4.55 MB
Format
Adobe PDF
Checksum (MD5)
2a5ce7b0623cf5d4b8c33f9475b23be4
Author(s)
Zepeda Nunez, Leonardo Andres
Demanet, Laurent
Date Issued
December 2015
Journal
Journal of Computational Physics
Publisher
Elsevier BV
Citation
Zepeda-Núñez, Leonardo, and Laurent Demanet. “The Method of Polarized Traces for the 2D Helmholtz Equation.” Journal of Computational Physics 308 (March 2016): 347–388 © 2015 Elsevier Inc
Version
Original manuscript
Abstract
We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. While the parallelism in this paper stems from decomposing the domain, we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines. Keywords: Domain decomposition; Helmholtz equation; Integral equations; High-frequency; Fast methods
MIT Department
Massachusetts Institute of Technology. Department of Earth, Atmospheric, and Planetary Sciences
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/J.JCP.2015.11.040
