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 acceleration. An increase of the number of grid points per wavelength results in convergence acceleration. This contrasts to the use of the shifted Laplace preconditioner. Our analysis also shows that the use of deflation results in a spectrum more favorable to the convergence of the outer Krylov acceleration. The near-null space components are still insufficiently well resolved, and the number of iterations increases with the wavenumber. In the two-dimensional case, the number of near-zero eigenvalues is larger than in the one-dimensional case. We perform numerical computations with the two-level and multilevel versions of the algorithm on constant and nonconstant wavenumber problems. Our numerical results confirm our spectral analysis.

Applying GMRES to the Helmholtz equation with shifted Laplacian preconditioning: what is the largest shift for which wavenumber-independent convergence is guaranteed?

“…(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].…”

Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees

“…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.…”

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

“…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.…”

“…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.…”