A short note on a pipelined polarized-trace algorithm for 3D Helmholtz

Zepeda-Núñez, Leonardo; Demanet, Laurent; Hewett, Russell J.; Scheuer, Adrien

doi:10.1190/segam2016-13867211.1

Cited by 3 publications

(1 citation statement)

References 28 publications

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“…5.1 (right), with the same set of frequencies and incoming waves as before. We use the implementation of the sparsifying preconditioner in [65]. The average execution times of the on-line stage are reported in Fig.…”

Section: Wave Scatteringmentioning

confidence: 99%

Fast Alternating BiDirectional Preconditioner for the 2D High-Frequency Lippmann--Schwinger Equation

Zepeda-Núñez¹,

Zhao²

2016

SIAM J. Sci. Comput.

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Abstract. This paper presents a fast iterative solver for Lippmann-Schwinger equation for highfrequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner [63] and a domain decomposition approach similar to the method of polarized traces [64]. The iterative solver has two levels, the outer level in which a sparsifying preconditioner for the Lippmann-Schwinger equation is constructed, and the inner level, in which the resulting sparsified system is solved fast using an iterative solver preconditioned with a bidirectional matrix-free variant of the method of polarized traces. The complexity of the construction and application of the preconditioner is O (N ) and O(N log N ) respectively, where N is the number of degrees of freedom. Numerical experiments in 2D indicate that the number of iterations in both levels depends weakly on the frequency resulting in a method with an overall O(N log N ) complexity.

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Section: Wave Scatteringmentioning

confidence: 99%

Fast Alternating BiDirectional Preconditioner for the 2D High-Frequency Lippmann--Schwinger Equation

Zepeda-Núñez¹,

Zhao²

2016

SIAM J. Sci. Comput.

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Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation

Taus

Demanet

Nunez

et al. 2017

SEG Technical Program Expanded Abstracts 2017

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In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where the polynomial degree is chosen such that p = O(log ω). These results hold even if the discontinuities in the wave speed are not resolved by the hdG mesh, as long as the integration procedure used in the assembly of the stiffness matrix respects the discontinuities. Further, we show that the associated linear systems can be solved using a modification of the method of polarized traces resulting in a method with linear complexity up to a poly-logarithmic factor, or sub-linear complexity in a parallel environment. To our knowledge and surprise, this note contains the first instance of a numerical method that is at the same time fast (O(N) runtime) and accurate (second-order, pollution-free) in the context of models of geophysical interest.

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Diffraction Imaging and Modeling and Seismic Modeling Complete Session

2017

SEG Technical Program Expanded Abstracts 2017

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A short note on a pipelined polarized-trace algorithm for 3D Helmholtz

Cited by 3 publications

References 28 publications

Fast Alternating BiDirectional Preconditioner for the 2D High-Frequency Lippmann--Schwinger Equation

Fast Alternating BiDirectional Preconditioner for the 2D High-Frequency Lippmann--Schwinger Equation

Pollution-free and fast hybridizable discontinuous Galerkin solvers for the high-frequency Helmholtz equation

Diffraction Imaging and Modeling and Seismic Modeling Complete Session

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