We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇ · (A∇u) + k 2 nu = −f where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are L ∞ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be C 0 and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for C ∞ convex interfaces with strictly positive curvature.Recap of existing well-posedness results. In 2-d, the unique continuation principle (UCP) holds (and gives uniqueness) when A is L ∞ and n ∈ L p for some p > 1 [2]. In 3-d, the UCP holds when A is Lipschitz [38,49] and n ∈ L 3/2 [48, 94]; see [39] for these results applied specifically to Helmholtz problems. Fredholm theory then gives existence and an a priori bound on the solution; this bound, however, is not explicit in k, A, or n.An example of an A ∈ C 0,α for all α < 1 for which the UCP fails in 3-d is given in [34]. Nevertheless, the UCP can be extended from Lipschitz A to piecewise-Lipschitz A by the Bairecategory argument in [4] (see also [52, Proposition 2.11]), with well-posedness then following by Fredholm theory as before -we discuss this argument of [4] further in §2.4. Recap of existing a priori bounds on the EDP in trapping and nontrapping situations.In this overview discussion, for simplicity, we consider the case of zero Dirichlet boundary conditions on ∂Ω − , where Ω − denotes the obstacle.When A, n, and Ω − are all C ∞ and such that the problem is nontrapping (i.e. all billiard trajectories starting in an exterior neighbourhood of Ω + := R d \ Ω − and evolving according to the Hamiltonian flow defined by the symbol of (1.1) escape from that neighbourhood after some uniform time), then either (i) the propagation of singularities results of [56] combined with either the paramatrix argument of [90] or Lax-Phillips theory [50], or (ii) the defect-measure argument of [15] 1 proves the estimate that, given k 0 > 0 and R > 0,for all k ≥ k 0 , where Ω R := Ω + ∩ B R (0), and C 1 (A, n, Ω − , R, k 0 ) is some (unknown) function of A, n, Ω − , R, and k 0 , but is independent of k. Without the nontrapping assumption, and assuming 1 The arguments in [15] actually require that, additionally, ∂Ω − has no points where the tangent vector makes infinite-order contact with ∂Ω − .

We prove well-posedness results and a priori bounds on the solution of the Helmholtz equation ∇ · (A∇u) + k 2 nu = −f , posed either in R d or in the exterior of a star-shaped Lipschitz obstacle, for a class of random A and n, random data f , and for all k > 0. The particular class of A and n and the conditions on the obstacle ensure that the problem is nontrapping almost surely. These are the first well-posedness results and a priori bounds for the stochastic Helmholtz equation for arbitrarily large k and for A and n varying independently of k. These results are obtained by combining recent bounds on the Helmholtz equation for deterministic A and n and general arguments (i.e. not specific to the Helmholtz equation) presented in this paper for proving a priori bounds and well-posedness of variational formulations of linear elliptic stochastic PDEs. We emphasise that these general results do not rely on either the Lax-Milgram theorem or Fredholm theory, since neither are applicable to the stochastic variational formulation of the Helmholtz equation.

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 $\|A_{1} -A_{2}\|_{L^{q}}$ ∥ A 1 − A 2 ∥ L q and $\|{n_{1}} - {n_{2}}\|_{L^{q}}$ ∥ n 1 − n 2 ∥ L q be (in terms of k-dependence) for GMRES applied to either $(\mathbf {A}_1)^{-1}\mathbf {A}_2$ ( A 1 ) − 1 A 2 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.

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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) + k 2 nj uj = −f . How small must A1 − A2 L q and n1 − n2 L q be (in terms of k-dependence) for GMRES applied to either (A1) −1 A2 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.