The Helmholtz equation models waves propagating with a fixed frequency. Discretising the Helmholtz equation for high frequencies via standard finite-elements results in linear systems that are large, non-Hermitian, and indefinite. Therefore, when solving these linear systems, one uses preconditioned iterative methods. When one considers uncertainty quantification for the Helmholtz equation, one will typically need to solve many (thousands) of linear systems corresponding to different realisations of the coefficients. At face value, this will require the computation of many preconditioners, a potentially expensive task.
Therefore, we investigate how well a preconditioner for one realisation of the Helmholtz equation works as a preconditioner for another realisation. We prove that if the two realisations are 'nearby' (with a precise meaning of 'nearby'), then the preconditioner is robust (that is, preconditioned GMRES converges in a number of iterations that is independent of frequency). We also give some preliminary computational results indicating the speedup one obtains in uncertainty quantification calculations.