A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements

Tsuji, Paul H.; Engquist, Björn; Ying, Lexing

doi:10.1016/j.jcp.2012.01.025

Cited by 35 publications

(16 citation statements)

References 15 publications

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“…• Poulson et al parallelized the sweeping preconditioners in 3D to deal with very large scale problems in geophysics [77]. Tsuji et al designed a spectrally accurate sweeping preconditioner for time-harmonic elastic waves [92], and time-harmonic Maxwell equations [91,93].…”

Section: Additional Related Workmentioning

confidence: 99%

The method of polarized traces for the 2D Helmholtz equation

Zepeda-Núñez

Demanet

2016

Journal of Computational Physics

129

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We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O( N L ), 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 secondorder 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.

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Section: Additional Related Workmentioning

confidence: 99%

The method of polarized traces for the 2D Helmholtz equation

Zepeda-Núñez

Demanet

2016

Journal of Computational Physics

129

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“…There are several previous works indicating the validity of this approach. For example, [19,20] applied the sweeping preconditioner to the time-harmonic Maxwell's equations. [15] combined the sweeping preconditioner with the sparsifying preconditioner and formed an efficient preconditioner that solves the Lippmann-Schwinger equation in quasi-linear time.…”

Section: Discussionmentioning

confidence: 99%

Sparsifying preconditioner for the time-harmonic Maxwell's equations

Liu

Ying

2019

Journal of Computational Physics

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“…The application of such ideas to the Helmholtz problem can be traced back, to great extent, to the AILU preconditioner of Gander and Nataf [43], in which a layered domain decomposition was used; and to Plessix and Mulder [76] in which a similar idea is used using separation of variables. However, it was Engquist and Ying who showed in [33,32] that such ideas could yield fast methods to solve the high-frequency Helmholtz equation, by introducing the sweeping preconditioner, which was then extended by Tsuji and collaborators to different discretizations and physics [89,90,91]. Since then, many other papers have proposed methods with similar claims.…”

Section: Related Workmentioning

confidence: 99%

Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

Zepeda-Núñez¹,

Demanet²

2018

SIAM J. Sci. Comput.

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We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous, constant density, acoustic media, with online parallel complexity that scales empirically as O( N P ), where N is the number of volume unknowns, and P is the number of processors, as long as P = O(N 1/5 ). This sublinear scaling is achieved by domain decomposition, not distributed linear algebra, and improves on the P = O(N 1/8 ) scaling reported earlier in [99]. The solver relies on a two-level nested domain decomposition: a layered partition on the outer level, and a further decomposition of each layer in cells at the inner level. The Helmholtz equation is reduced to a surface integral equation (SIE) posed at the interfaces between layers, efficiently solved via a nested version of the polarized traces preconditioner [99]. The favorable complexity is achieved via an efficient application of the integral operators involved in the SIE.

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A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements

Cited by 35 publications

References 15 publications

The method of polarized traces for the 2D Helmholtz equation

The method of polarized traces for the 2D Helmholtz equation

Sparsifying preconditioner for the time-harmonic Maxwell's equations

Nested Domain Decomposition with Polarized Traces for the 2D Helmholtz Equation

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