  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 derive 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 an 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. The
performance of our preconditioned FGMRES solver is illustrated by several
benchmark problems with heterogeneous wavenumbers in two and three space
dimensions.
