Smoothed aggregation for Helmholtz problems

Olson, Luke N.; Schroder, Jacob B.

doi:10.1002/nla.686

Cited by 25 publications

(19 citation statements)

References 45 publications

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“…QA * AQ-CGNR variant. The QA * AQ-CGNR variant was previously presented in [18], but we now provide a more thorough investigation. Consider using CG to solve…”

Section: Qaq-cg Variantmentioning

confidence: 92%

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A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization

Olson¹,

Schroder²,

Tuminaro³

2011

SIAM J. Sci. Comput.

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Algebraic multigrid methods solve sparse linear systems Ax = b by automatic construction of a multilevel hierarchy. This hierarchy is defined by grid transfer operators that must accurately capture algebraically smooth error relative to the relaxation method. We propose a methodology to improve grid transfers through energy minimization. The proposed strategy is applicable to Hermitian, non-Hermitian, definite, and indefinite problems. Each column of the grid transfer operator P is minimized in an energy-based norm while enforcing two types of constraints: a defined sparsity pattern and preservation of specified modes in the range of P. A Krylov-based strategy is used to minimize energy, which is equivalent to solving AP j = 0 for each column j of P , with the constraints ensuring a nontrivial solution. For the Hermitian positive definite case, a conjugate gradient (CG-)based method is utilized to construct grid transfers, while methods based on generalized minimum residual (GMRES) and CG on the normal equations (CGNR) are explored for the general case. The approach is flexible, allowing for arbitrary coarsenings, unrestricted sparsity patterns, straightforward long-distance interpolation, and general use of constraints, either user-defined or auto-generated. We conclude with numerical evidence in support of the proposed framework.

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“…QA * AQ-CGNR variant. The QA * AQ-CGNR variant was previously presented in [18], but we now provide a more thorough investigation. Consider using CG to solve…”

Section: Qaq-cg Variantmentioning

confidence: 92%

“…The process results in standard SA prolongation smoothing when only one iteration is used [16]; this is also related to an extension of [16] to a conjugate gradient (CG-)based approach by Vaněk (unpublished). Additionally, [18] developed a conjugate gradient normal residual (CGNR-)based prolongation smoothing scheme, which is further developed here.…”

mentioning

confidence: 99%

A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization

Olson¹,

Schroder²,

Tuminaro³

2011

SIAM J. Sci. Comput.

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“…This can impose challenges if the solution of (1) is required for multiple frequencies. Other iterative approaches include those in the works of Gordon et al 26,27 as well as Haber and Maclachlan, 15 Brandt and Livshits, 28 Olson and Schroder, 29 and Livshits, 30 which are multigrid based.…”

Section: Introductionmentioning

confidence: 99%

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Treister

Haber

2018

Numerical Linear Algebra App

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The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex-valued advection-diffusion-reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second-order advection-diffusion-reaction discretization is more accurate than the standard second-order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.

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“…An important contribution is the publication of the wave-ray method first published in [3] and later further elaborated in [4]. Alternative techniques are the use of multigrid methods with Krylov smoothers [5], sweeping preconditioners [6], domain decomposition [7,8], adaptive [9] and smoothed aggregation multigrid [10] methods. All these techniques have a limited range of applicability and no standard method exists at the moment.…”

Section: Introductionmentioning

confidence: 99%

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

Sheikh

Lahaye

Ramos

et al. 2016

Journal of Computational Physics

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Smoothed aggregation for Helmholtz problems

Cited by 25 publications

References 45 publications

A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization

A General Interpolation Strategy for Algebraic Multigrid Using Energy Minimization

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

Accelerating the shifted Laplace preconditioner for the Helmholtz equation by multilevel deflation

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