Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

This paper analyses the following question: let Aj, j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (Aj∇uj) + k2njuj = −f. How small must ∥A1−A2∥Lq and ∥n1−n2∥Lq be (in terms of k-dependence) for GMRES applied to either (A1)−1A2 or A2(A1)− 1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A1 to be a good left or right preconditioner for A2?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients A and n. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different A and n, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.