A sweeping preconditioner for Yee’s finite difference approximation of time-harmonic Maxwell’s equations

Tsuji, Paul H.; Ying, Lexing

doi:10.1007/s11464-012-0191-8

Cited by 14 publications

(11 citation statements)

References 7 publications

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“…Though this paper is focused on the parallel solution of Helmholtz equations, which are the time-harmonic form of acoustic wave equations, Tsuji et al have shown that the moving PML sweeping preconditioner is equally effective for time-harmonic Maxwell's equations [38,39], and we believe that the same will hold true for timeharmonic linear elasticity. The rest of the paper will be presented in the context of the Helmholtz equation, but we emphasize that the parallelization should carry over to more general wave equations in a conceptually trivial way.…”

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confidence: 85%

A Parallel Sweeping Preconditioner for Heterogeneous 3D Helmholtz Equations

Poulson¹,

Engquist²,

Li³

et al. 2013

SIAM J. Sci. Comput.

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A parallelization of a sweeping preconditioner for three-dimensional Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be O(γ 2 N 4/3 ) and O(γN log N ), where γ(ω) denotes the modestly frequency-dependent number of grid points per perfectly matched layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.

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confidence: 85%

A Parallel Sweeping Preconditioner for Heterogeneous 3D Helmholtz Equations

Poulson¹,

Engquist²,

Li³

et al. 2013

SIAM J. Sci. Comput.

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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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“…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%