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

Sheikh, A. H.; Lahaye, Domenico; Ramos, Luis García; Nabben, Reinhard; Vuik, C.

doi:10.1016/j.jcp.2016.06.025

Cited by 33 publications

(26 citation statements)

References 33 publications

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“…Based on the ideas by Nicolaides for deflated CG [109], several researchers, for example, Morgan [102,103], Karchenko and Yeremin [88], Erhel and collaborators [25,51], Baglama et al [12], and Frank and Vuik [61] proposed to use augmentation with (approximate) eigenvectors to maintain good convergence after restarting. Vuik and collaborators have also considered deflation preconditioned methods based on a priori considerations for good deflation spaces, based on algorithmic or application information [85,132]. These methods may use augmentation or a preconditioning approach to achieve the desired effect.…”

Section: Augmentation Based On Ideas Of Deflationmentioning

confidence: 99%

A survey of subspace recycling iterative methods

2020

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This survey concerns subspace recycling methods, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial vectors over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.

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Section: Augmentation Based On Ideas Of Deflationmentioning

confidence: 99%

A survey of subspace recycling iterative methods

2020

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“…The theoretical results in [15] help to explain this convergence rate. In [23,24] the shift is performed first, followed by the shifted Laplacian preconditioner. This approach can be seen as a preconditioned Multilevel Krylov method.…”

Section: Introduction We Consider the Iterative Solution Of Linear Smentioning

confidence: 99%

“…The numerical results in [8,9,23] demonstrate that the number of iterations on the second level is important. Typically 8 iterations on the second level are used, then 2 on the next level, while only 1 iteration on all other levels.…”

Section: Introduction We Consider the Iterative Solution Of Linear Smentioning

confidence: 99%

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Adaptive Multilevel Krylov Methods

Kehl¹,

Nabben²,

Szyld³

2019

etna

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Inexact (variable) preconditioning of Multilevel Krylov methods (MK methods) for the solution of linear systems of equations is considered. MK methods approximate the solution of the local systems on a subspace using a few, but fixed, number of iteration steps of a preconditioned flexible Krylov method. In this paper, using the philosophy of inexact Krylov subspace methods, we use a theoretically-derived criterion to choose the number of iterations needed on each level to achieve a desired tolerance. We use this criterion on one level and obtain an improved MK method. Inspired by these results, a second ad hoc method is also explored. Numerical experiments for the Poisson, Helmholtz, and the convection-diffusion equations illustrate the efficiency and robustness of this adaptive Multilevel Krylov method.

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“…While this resulted in a reduced number of iterations, the number of iterations still mildly grows with the constant wave number k. Balancing between wave number independent convergence and practical constraints created the opportunity to consider a deflation strategy, which was first proposed in [8]. Deflation, in essence, aims to move the unwanted eigenvalues to zero or one and has been studied widely; see [35,36,37]. While being able to improve the convergence and performance significantly, the near-zero eigenvalues keep reappearing for large wave numbers.…”

mentioning

confidence: 99%

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

Cited by 33 publications

References 33 publications

A survey of subspace recycling iterative methods

A survey of subspace recycling iterative methods

Adaptive Multilevel Krylov Methods

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

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