Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Ivan Graham; Owen Pembery; Euan Spence

doi:10.1007/s10444-021-09889-0

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Ivan Graham, Owen Pembery, Euan Spence

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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 ∥A1−A2∥Lq and ∥n1−n2∥Lq 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.

Original language	English
Article number	68
Journal	Advances in Computational Mathematics
Volume	47
Issue number	5
Early online date	3 Sept 2021
DOIs	https://doi.org/10.1007/s10444-021-09889-0
Publication status	Published - 3 Sept 2021

Bibliographical note

Publisher Copyright:
© 2021, The Author(s).

Funding

We thank Th?ophile Chaumont-Frelet (INRIA, Nice), Stefan Sauter (Universit?t Z?rich), 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 Z?rich) and Robert Scheichl (Universit?t 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. 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.

Funders	Funder number
EPSRC Centre for Doctoral Training in Statistical	EP/R005591/1, EP/L015684/1
Robert Scheichl
Engineering and Physical Sciences Research Council	EP/S003975/1
University of Bath
Universität Heidelberg

Keywords

Helmholtz equation
Heterogeneous
High frequency
Preconditioning
Uncertainty quantification
Variable wave speed

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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2005.13390.PDF
This is a post-peer-review, pre-copyedit version of an article published in Advances in Computational Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s10444-021-09889-0
Accepted author manuscript, 572 KB

https://arxiv.org/pdf/2005.13390.pdf

Cite this

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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 ∥A1−A2∥Lq and ∥n1−n2∥Lq 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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Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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Fast solvers for frequency-domain wave-scattering problems and applications

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

Cite this

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Fast solvers for frequency-domain wave-scattering problems and applications

At the interface between semiclassical analysis and numerical analysis of Wave propogation problems

Cite this