A short note on the nested-sweep polarized traces method for the 2D Helmholtz equation

Zepeda-Núñez, Leonardo; Demanet, Laurent

doi:10.1190/segam2015-5838886.1

Cited by 3 publications

(3 citation statements)

References 16 publications

(16 reference statements)

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“…[20]). Closely related to the content of this paper, we find the method of polarized traces [99], and an earlier version of this work [98].…”

Section: Related Worksupporting

confidence: 58%

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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“…[20]). Closely related to the content of this paper, we find the method of polarized traces [99], and an earlier version of this work [98].…”

Section: Related Worksupporting

confidence: 58%

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

Zepeda-Núñez¹,

Demanet²

2018

SIAM J. Sci. Comput.

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“…Depending on the discretization, change in the selection of the sets Γ 1 and Γ 2 may be required. Similar schemes have been applied for many different discretizations such as high-order finite difference methods [89], finite element methods [84], enriched finite element methods [30], discontinuous Galerkin methods [74], and integral representations [87]. For example, for higher-order finite difference methods the stencils centered at the points in Ω 2 (respecting Γ 2 , Γ 1 , or Ω 1 ) cannot involve discretization points in Γ 1 (respecting Ω 1 , Ω 2 , Γ 2 ).…”

Section: Discrete Polarizationmentioning

confidence: 99%

L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation

Taus,

Zepeda-Núñez,

Hewett

et al. 2019

Preprint

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We present the first fast solver for the high-frequency Helmholtz equation that scales optimally in parallel, for a single right-hand side. The L-sweeps approach achieves this scalability by departing from the usual propagation pattern, in which information flows in a 180 • degree cone from interfaces in a layered decomposition. Instead, with L-sweeps, information propagates in 90 • cones induced by a checkerboard domain decomposition (CDD). We extend the notion of accurate transmission conditions to CDDs and introduce a new sweeping strategy to efficiently track the wave fronts as they propagate through the CDD. The new approach decouples the subdomains at each wave front, so that they can be processed in parallel, resulting in better parallel scalability than previously demonstrated in the literature. The method has an overall O (N/p) log ω empirical run-time for N = n d total degrees-of-freedom in a d-dimensional problem, frequency ω, and p = O(n) processors. We introduce the algorithm and provide a complexity analysis for our parallel implementation of the solver. We corroborate all claims in several two-and three-dimensional numerical examples involving constant, smooth, and discontinuous wave speeds.

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“…Even though the amount of literature dealing with both issues separately is vast [33,79,12,41,55,83,40,66,58,31,29,65,23,72], only a few references deal with both issues simultaneously. We refer to, for example, [68] in which a hybridizable discontinuous Galerkin methods is coupled with the method of polarized traces, [81,45] in which an integral version of the Helmholtz equation is coupled with sparsification and a fast preconditioner, and [34] in which an adaptive discretization is built by learning the dominant wave directions.…”

Section: Introductionmentioning

confidence: 99%

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

Fang

Qian

Zepeda-Núñez

et al. 2018

Journal of Computational Physics

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We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM) [34], whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly capture the singularity of the solution in the near-field. The method requires a fixed number of grid points per wavelength to accurately represent the wave field with an asymptotic convergence rate of O(ω −1/2 ), where ω is the frequency parameter in the Helmholtz equation. In addition, a fast sweepingtype preconditioner is used to solve the resulting linear system.We present numerical examples in 2D to show both accuracy and efficiency of our method as the frequency increases. In particular, we provide numerical evidence of the convergence rate, and we show empirically that the overall complexity is O(ω 2 ) up to a poly-logarithmic factor.1 The ratio between numerical error and best approximation error from a discrete finite element space is ω dependent.

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A short note on the nested-sweep polarized traces method for the 2D Helmholtz equation

Cited by 3 publications

References 16 publications

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

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

L-Sweeps: A scalable, parallel preconditioner for the high-frequency Helmholtz equation

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

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