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Due to its significance in terms of wave phenomena a considerable effort hasbeen put into the design of preconditioners for the Helmholtz equation. Oneoption to derive a preconditioner is to apply a multigrid method on a shiftedoperator. In such an approach, the wavenumber is shifted by some imaginaryvalue. This step is motivated by the observation that the shifted problem canbe more efficiently handled by iterative solvers when compared to the standardHelmholtz equation. However, up to now, it is not obvious what the beststrategy for the choice of the shift parameter is. It is well known that a goodshift parameter depends sensitively on the wavenumber and the discretizationparameters such as the order and the mesh size. Therefore, we study the choiceof a near optimal complex shift such that an FGMRES solver converges with feweriterations. Our goal is to provide a map which returns the near optimal shiftfor the preconditioner depending on the wavenumber and the mesh size. In orderto compute this map, a data driven approach is considered: We first generatemany samples, and in a second step, we perform a nonlinear regression on thisdata. With this representative map, the near optimal shift can be obtained by asimple evaluation. Our preconditioner is based on a twogrid V-cycle applied tothe shifted problem, allowing us to implement a semi matrix-free method. Theperformance of our preconditioned FGMRES solver is illustrated by severalbenchmark problems with heterogeneous wavenumbers in two and three spacedimensions.

Semi matrix-free twogrid shifted Laplacian preconditioner for the Helmholtz equation with near optimal shifts