Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Abstract

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.

Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification

“…Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy was also central to the abstract theory in [70] of parametric operator equations with affine parameter dependence; this theory was then reviewed for the Helmholtz problem considered in [35] in [35, Appendix A] (although the bound (1.4) is proved in [35, section 4] by repeatedly differentiating the PDE with respect to \bfy and essentially applying the solution operator 2 ). This type of perturbation argument is also implicitly used in the Helmholtz context in [26,34,39].…”

“…UQ for the Helmholtz equation and k-explicit parametric regularity. While a large amount of initial UQ theory concerned Poisson's equation \nabla \cdot (A(\bfx , \bfy )\nabla u(\bfx , \bfy )) = - f (\bfx ), there has been increasing interest in UQ of the Helmholtz equation with (large) wavenumber k (see, e.g., [26,44,23,64,39,6,35]) and the time-harmonic Maxwell equations [46,47,27,1]. The Helmholtz equation with wavenumber k and random coefficients is…”

Wavenumber-explicit parametric holomorphy of Helmholtz solutions in the context of uncertainty quantification

“…This idea was also central to the abstract theory in [76] of parametric operator equations with affine parameter dependence; this theory was then reviewed for the Helmholtz problem considered in [39] in [39, Appendix A] (although the bound (1.4) is proved in [39, §4] by repeated differentiating the PDE with respect to y and essentially applying the solution operator 2 ). This type of perturbation argument is also implicitly used in the Helmholtz context in [28,37,43]. Applied to the set-up in [39] of (1.3) (and assuming further that y ∈ C N for N large), the conditions (1.11) and (1.13) are ensured if…”

“…UQ for the Helmholtz equation and k-explicit parametric regularity. Whilst a large amount of initial UQ theory concerned Poisson's equation ∇ • (A(x, y)∇u(x, y)) = −f (x), there has been increasing interest in UQ of Helmholtz equation with (large) wavenumber k [89,84,8,38,28,24,30,59,49,3,75,25,69,43,46,7,39,87] and the time-harmonic Maxwell equations [51,52,29,1]. The Helmholtz equation with wavenumber k and random coefficients is k −2 ∇ • (A(x, y)∇u(x, y)) + n(x, y)u(x, y) = −f (x) (1.2) where A and n depend on both the spatial variable x and the stochastic variable y.…”