The Multilevel Krylov-Multigrid Method for the Helmholtz Equation Preconditioned by the Shifted Laplacian

Erlangga, Yogi A.; Ramos, Luis García; Nabben, Reinhard

doi:10.1007/978-3-319-28832-1_5

Cited by 20 publications

(45 citation statements)

References 33 publications

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“…Proof-of-concept studies from other application areas exist that use the same software infrastructure [Weinzierl et al 2014] to embed small regular Cartesian grids into each spacetree cell. These small grids, patches, allow for improved robustness due to stronger smoothers resulting from Chebyshev iterations, higher-order smoothing schemes on embedded regular grids or a multilevel Krylov solver based on recursive coarse grid deflation [Sheikh et al 2013;Erlangga and Nabben 2008]. Deflation is a particular interesting feature for highly heterogeneous Helmholtz problems where bound states emerge as isolated eigenvalues near the origin.…”

Section: Discussionmentioning

confidence: 99%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations arising from Schrödinger equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling to the indefinite Helmholtz operator. Our implementations are based upon the notion of a spacetree and work exclusively with a finite number of precomputed local element matrices. They are globally matrix-free.Combining various relaxation factors with two grid transfer operators allows us to switch from additive multigrid over a hierarchical basis method into a Bramble-Pasciak-Xu (BPX)-type solver, with several multiscale smoothing variants within one code base. Pipelining allows us to realise full approximation storage (FAS) within the additive environment where, amortised, each grid vertex carrying degrees of freedom is read/written only once per iteration. The codes realise a single-touch policy. Among the features facilitated by matrix-free FAS is arbitrary dynamic mesh refinement (AMR) for all solver variants. AMR as enabler for full multigrid (FMG) cycling-the grid unfolds throughout the computation-allows us to reduce the cost per unknown per order of accuracy.The present paper primary contributes towards software realisation and design questions. Our experiments show that the consolidation of single-touch FAS, dynamic AMR and vectorisation-friendly, complex scaled, matrix-free FMG cycles delivers a mature implementation blueprint for solvers of Helmholtz equations in general. For this blueprint, we put particular emphasis on a strict implementation formalism as well as some implementation correctness proofs.

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Section: Discussionmentioning

confidence: 99%

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Reps

Weinzierl

2017

ACM Trans. Math. Softw.

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“…In this paper we further develop ideas initially established by Erlangga and Nabben in their recent paper [20]. In this paper the authors propose to deploy a deflation procedure [26,27,28] to remove the eigenmodes that hamper the fast convergence of the CSLP preconditioner.…”

Section: Introductionmentioning

confidence: 97%

“…On each level a Krylov subspace method accelerates the CSLP preconditioner. Spectral analysis and numerical results in [20] show that this so-called multilevel Krylov method significantly reduces the required number of iterations. The required deflated preconditioned operator is too difficult to construct and some form of approximation is mandatory.…”

Section: Introductionmentioning

confidence: 97%

“…The required deflated preconditioned operator is too difficult to construct and some form of approximation is mandatory. Such an approximation results in a computationally feasible multilevel method in [20]. An alternative multilevel Krylov approach in which the original Helmholtz operator is deflated instead was proposed in [29].…”

Section: Introductionmentioning

confidence: 99%

“…The idea of adding weight to the diagonal of the ILU preconditioner forms the basis of [14]. The CSLP preconditioners were further developed in [15,16] and later generalized in [17,18,19,20,21]. This lead to a breakthrough in industrial applications [22,23,24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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On the convergence of two‐level Krylov methods for singular symmetric systems

Erlangga

Nabben

2017

Numerical Linear Algebra App

Self Cite

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We discuss the convergence of a two-level version of the multilevel Krylov method for solving linear systems of equations with symmetric positive semidefinite matrix of coefficients. The analysis is based on the convergence result of Brown and Walker for the Generalized Minimal Residual method (GMRES), with the left-and right-preconditioning implementation of the method. Numerical results based on diffusion problems are presented to show the convergence.

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The Multilevel Krylov-Multigrid Method for the Helmholtz Equation Preconditioned by the Shifted Laplacian

Cited by 20 publications

References 33 publications

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

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

On the convergence of two‐level Krylov methods for singular symmetric systems

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