2013
DOI: 10.1002/nla.1882
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On the convergence of shifted Laplace preconditioner combined with multilevel deflation

Abstract: SUMMARYDeflating the shifted Laplacian with geometric multigrid vectors yields speedup. To verify this claim, we investigate a simplified variant of Erlangga and Nabben presented in [Erlangga and Nabben, ETNA, 2008;31:403-424]. We derive expressions for the eigenvalues of the two-level preconditioner for the onedimensional problem. These expressions show that the algorithm analyzed is not scalable. They also show that the imaginary shift can be increased without delaying the convergence of the outer Krylov ac… Show more

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Cited by 57 publications
(56 citation statements)
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“…(Using the notation above, preconditioning with the second operator corresponds to choosing ε ∼ k 2 and constructing B −1 ε using a multigrid V-cycle.) Preconditioning with ( + k 2 + iε) −1 and ε ∼ k 2 was then further investigated in the context of multigrid in [8] and [49].…”
Section: Previous Work On the Shifted Laplacian Preconditionermentioning
confidence: 99%
“…(Using the notation above, preconditioning with the second operator corresponds to choosing ε ∼ k 2 and constructing B −1 ε using a multigrid V-cycle.) Preconditioning with ( + k 2 + iε) −1 and ε ∼ k 2 was then further investigated in the context of multigrid in [8] and [49].…”
Section: Previous Work On the Shifted Laplacian Preconditionermentioning
confidence: 99%
“…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%
“…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]. This approach circumvents the necessity of constructing expensive operators.…”
Section: Introductionmentioning
confidence: 99%
“…In this paper we give a unified presentation of the methods proposed in [20] and in [29]. The method in [20] deflates the CSLP preconditioned system and requires no further preconditioning.…”
Section: Introductionmentioning
confidence: 99%