A Bayesian conjugate gradient method (with discussion)
A Bayesian conjugate gradient method (with discussion)
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about, for example, the magnitude of the error. In this paper we propose a novel statistical model for this error, set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.
Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Oates, Chris J.
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Ipsen, Ilse C.F.
83eae4c2-19d4-4f74-9d16-4146a63d2c4c
Girolami, Mark
4feb7248-7beb-4edc-8509-139b4049d23b
1 September 2019
Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Oates, Chris J.
3af13c56-dc47-4d2c-867f-e4e933e74619
Ipsen, Ilse C.F.
83eae4c2-19d4-4f74-9d16-4146a63d2c4c
Girolami, Mark
4feb7248-7beb-4edc-8509-139b4049d23b
Cockayne, Jonathan, Oates, Chris J., Ipsen, Ilse C.F. and Girolami, Mark
(2019)
A Bayesian conjugate gradient method (with discussion).
Bayesian Analysis, 14 (3).
(doi:10.1214/19-ba1145).
Abstract
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about, for example, the magnitude of the error. In this paper we propose a novel statistical model for this error, set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.
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19-BA1145
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Published date: 1 September 2019
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Local EPrints ID: 451592
URI: http://eprints.soton.ac.uk/id/eprint/451592
ISSN: 1931-6690
PURE UUID: b27e5acd-d90a-48f7-95b3-22c51b45bd0e
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Date deposited: 12 Oct 2021 16:34
Last modified: 17 Mar 2024 04:09
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Author:
Chris J. Oates
Author:
Ilse C.F. Ipsen
Author:
Mark Girolami
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