Scalable multi-level deflation preconditioning for the highly indefinite Helmholtz equation

Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method in [6] to a multilevel deflation method. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the fine-grid operator keeping the spectrum of the preconditioned system fixed away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver with theoretical linear complexity for the highly indefinite Helmholtz equation. This can be attributed to a fixed number of iterations independent of the wave number and an optimal use of the CSLP-preconditioner. We approximate the CSLP-preconditioner, while allowing the complex shift to be small. The proposed configuration additionally shows very promising results for the more challenging Marmousi problem.