The Helmholtz equation is the simplest model of acoustic wave propagation and as such has applications in areas such as seismic and ultrasound imaging. When seeking to model situations involving uncertainty the wavenumber of the Helmholtz equation becomes random. The matrices arising from discretisations of the Helmholtz equation with piecewise linear finite elements are large and non-Hermitian. Therefore solving the resulting linear systems using iterative methods necessitates the use of preconditioners.

We will present a preconditioning strategy for the Helmholtz equation with random wavenumber where we form a preconditioner and reuse it to precondition the different matrices arising from different realisations of the wavenumber. We will present rigorous convergence results for this strategy and corresponding numerical results. Finally, we will discuss how this preconditioning technique can be applied to Markov Chain Monte Carlo methods for the associated Bayesian inverse problem.