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 \(\|A_{1} -A_{2}\|_{L^{q}}\) and \(\|{n_{1}} - {n_{2}}\|_{L^{q}}\) be (in terms of k-dependence) for GMRES applied to either \((\mathbf {A}_1)^{-1}\mathbf {A}_2\) 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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Acknowledgements
We thank Théophile Chaumont-Frelet (INRIA, Nice), Stefan Sauter (Universität Zürich), and Nilima Nigam (Simon Fraser University) for useful comments and discussions about this work at the conference MAFELAP 2019. We thank the referees for their constructive comments and insightful suggestions. Finally, we thank Ralf Hiptmair (ETH Zürich) and Robert Scheichl (Universität Heidelberg) for useful comments on this work in the course of examining ORP’s PhD thesis [53].
IGG acknowledges support from EPSRC grant EP/S003975/1. ORP acknowledges support by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the EPSRC grant EP/L015684/1. EAS acknowledges support from EPSRC grant EP/R005591/1. This research made use of the Balena High Performance Computing (HPC) Service at the University of Bath.
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Communicated by: Ilaria Perugia
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Graham, I.G., Pembery, O.R. & Spence, E.A. Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification. Adv Comput Math 47, 68 (2021). https://doi.org/10.1007/s10444-021-09889-0
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DOI: https://doi.org/10.1007/s10444-021-09889-0
Keywords
- Helmholtz equation
- Preconditioning
- Heterogeneous
- Variable wave speed
- High frequency
- Uncertainty quantification