Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices

Morgan, Ronald B.; Yang, Zhao; Zhong, Baojiang

doi:10.1002/nla.2067

Cited by 3 publications

(4 citation statements)

References 35 publications

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“…While the literature does not quantify terms as a "small" condition number or "not too far from normality/unitary" for this particular application, there exists vast numerical evidence showing that altering the spectrum leads to better GMRES convergence. This corroborates the acceleration of GMRES convergence using deflation preconditioning techniques [19,5,29,31]. In fact, in [31] the authors state that "deflated GMRES can be effective even when the eigenvectors are poorly defined .…”

Section: A904supporting

confidence: 72%

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Scalable Convergence Using Two-Level Deflation Preconditioning for the Helmholtz Equation

Dwarka¹,

Vuik²

2020

SIAM J. Sci. Comput.

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Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subpsace methods. The requisite for these efforts lies in the fact that the widely used and well-acknowledged complex shifted Laplacian preconditioner (CSLP) shifts the eigenvalues of the preconditioned system towards the origin as the wave number increases. The two-level-deflation preconditioner combined with CSLP showed encouraging results in moderating the rate at which the eigenvalues approach the origin. However, for large wave numbers the initial problem resurfaces and the near-zero eigenvalues reappear. Our findings reveal that the reappearance of these near-zero eigenvalues occurs if the near-singular eigenmodes of the fine-grid operator and the coarse-grid operator are not properly aligned. This misalignment is caused by accumulating approximation errors during the inter-grid transfer operations. We propose the use of higher-order approximation schemes to construct the deflation vectors. The results from rigorous Fourier analysis and numerical experiments confirm that our newly proposed scheme outperforms any other deflation-based preconditioner for the Helmholtz problem. In particular, the spectrum of the adjusted preconditioned operator stays fixed near one. These results can be generalized to general shifted indefinite systems with random right-hand sides. For the first time, the convergence properties for very large wave numbers (k = 10 6 in one dimension and k = 10 3 in two dimensions) have been studied, and the convergence is close to wave number independence. Wave number independence for three dimensions has been obtained for wave numbers up to k = 75. The new scheme additionally shows very promising results for the more challenging Marmousi problem. Irrespective of the strongly varying wave number, we obtain a constant number of iterations and a reduction in computational time as the results remain robust without the use of the CSLP preconditioner.

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Section: A904supporting

confidence: 72%

“…This corroborates the acceleration of GMRES convergence using deflation preconditioning techniques [19,5,29,31]. In fact, in [31] the authors state that "deflated GMRES can be effective even when the eigenvectors are poorly defined . .…”

Section: A904supporting

confidence: 71%

Scalable Convergence Using Two-Level Deflation Preconditioning for the Helmholtz Equation

Dwarka¹,

Vuik²

2020

SIAM J. Sci. Comput.

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show abstract

“…We point here to the more recent work of Liu et al [174] who applied polynomial preconditioning to GMRES-DR. In addition, the effectiveness of GMRES-DR was studied by Morgan et al [195]. The main idea is that the approximate eigenvectors generated by GMRES-DR can be seen as pseudoeigenvectors.…”

Section: Deflation and Augmentationmentioning

confidence: 99%

“…It was shown in [195] by examples that GMRES-DR can also work for highly nonnormal matrices. Some deflation strategies are highly related to preconditioning.…”

Section: Deflation and Augmentationmentioning

confidence: 99%