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

Abstract

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.

Towards Accuracy and Scalability: Combining Isogeometric Analysis with Deflation to Obtain Scalable Convergence for the Helmholtz Equation

“…While this improves the convergence significantly, the near-zero eigenvalues start reappearing for very large wave numbers . Consequently, it has been shown recently that the use of a quadratic interpolation scheme results in close to wave number independent convergence for the two-level deflation preconditioner 33 . In fact, the use of these higherorder deflation vectors results in a smaller projection error compared to the case where a linear interpolation schemes is used.…”

“…As a consequence, recent developments have led to a broad range of preconditioners such as domain decomposition based preconditioners 19,20,21,22,23,24,25 , sweeping preconditioners 26,27,28,29,30 and (multilevel) deflation based preconditioners 31,32,33 . One of these new preconditioners is the Adapted Deflation Preconditioner (ADP), which uses higher-order Bezier curves to construct the deflation space.…”

Inexact subdomain solves using deflated GMRES for Helmholtz problems

“…An interesting observation is that increasing ppwl leads to a higher iteration count. The opposite effect has been observed when using a two-level deflation preconditioner [7,4]. There, a finer mesh leads to a smaller number of iterations as the mapping of the eigenvectors from the fine-and coarse-grid becomes more accurate.…”

“…The choice of strongly dictates the overall convergence behaviour. Here, we use quadratic rational Bézier curves, as they have been shown to provide satisfactory convergence [4]. Consequently, if we let ˜ represent the -th degree of freedom (DOF) on subdomain Ω , then in 1D maps these nodal approximations onto their coarse-grid counterpart as follows…”

“…It is well known that the cost of GMRES increases with each iteration. Thus, in order to mitigate the number of iterations at the subdomain level, we use a two-level deflation preconditioner [7,4].…”