A Semi Matrix-Free Twogrid Preconditioner for the Helmholtz Equation with Near Optimal Shifts

Due to its significance in terms of wave phenomena a considerable effort has been put into the design of preconditioners for the Helmholtz equation. One option to design a preconditioner is to apply a multigrid method on a shifted operator. In such an approach, the wavenumber is shifted by some imaginary value. This step is motivated by the observation that the shifted problem can be more efficiently handled by iterative solvers when compared to the standard Helmholtz equation. However, up to now, it is not obvious what the best strategy for the choice of the shift parameter is. It is well known that a good shift parameter depends sensitively on the wavenumber and the discretization parameters such as the order and the mesh size. Therefore, we study the choice of a near optimal complex shift such that a flexible generalized minimal residual (FGMRES) solver converges with fewer iterations. Our goal is to provide a map which returns the near optimal shift for the preconditioner depending on the wavenumber and the mesh size. In order to compute this map, a data driven approach is considered: We first generate many samples, and in a second step, we perform a nonlinear regression on this data. With this representative map, the near optimal shift can be obtained by a simple evaluation. Our preconditioner is based on a twogrid V-cycle applied to the shifted problem, allowing us to implement a semi matrix-free method in which only the coarse grid and boundary matrices need to be stored in memory. On the fine grid, only the action of the matrix applied to a vector is computed, without assembling the global matrix. This enables efficient use of computing resources and allows problems to be solved at scales that were previously limited by the available memory. The performance of our preconditioned FGMRES solver is illustrated by several benchmark problems with heterogeneous wavenumbers in two and three space dimensions.