# Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

This paper analyses the following question: let A_j, j=1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇· (A_j ∇ u_j) + k^2 n_j u_j= -f. How small must A_1 -A_2_L^q and n_1 - n_2_L^q be (in terms of k-dependence) for GMRES applied to either (A_1)^-1A_2 or A_2(A_1)^-1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A_1 to be a good left- or right-preconditioner for A_2?). 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.

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