  Solving the indefinite Helmholtz equation is not only crucial for the
understanding of many physical phenomena but also represents an
outstandingly-difficult benchmark problem for the successful application of
numerical methods. Here we introduce a new approach for evolving efficient
preconditioned iterative solvers for Helmholtz problems with multi-objective
grammar-guided genetic programming. Our approach is based on a novel
context-free grammar, which enables the construction of multigrid
preconditioners that employ a tailored sequence of operations on each
discretization level. To find solvers that generalize well over the given
domain, we propose a custom method of successive problem difficulty adaption,
in which we evaluate a preconditioner's efficiency on increasingly
ill-conditioned problem instances. We demonstrate our approach's effectiveness
by evolving multigrid-based preconditioners for a two-dimensional indefinite
Helmholtz problem that outperform several human-designed methods for different
wavenumbers up to systems of linear equations with more than a million
unknowns.
