Papers › Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
Ivan G. Graham, Owen R. Pembery, Euan A. Spence
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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) + k² 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)⁻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.
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