Multigrid based preconditioners for the numerical solution of two-dimensional heterogeneous problems in geophysics

We study methods for the numerical solution of the Helmholtz equation for twodimensional applications in geophysics. The common framework of the iterative methods in our study is a combination of an inner iteration with a geometric multigrid method used as a preconditioner and an outer iteration with a Krylov subspace method. The preconditioning system is based on either a pure or shifted Helmholtz operator. A multigrid iteration is used to approximate the inverse of this operator. The proposed solution methods are evaluated on a complex benchmark in geophysics involving highly variable coefficients and high wavenumbers. We compare this preconditioned iterative method with a direct method and a hybrid method that combines our iterative approach with a direct method on a reduced problem. We see that the hybrid method outperforms both the iterative and the direct approach.