The University of Southampton
University of Southampton Institutional Repository

Probabilistic linear solvers: a unifying view

Probabilistic linear solvers: a unifying view
Probabilistic linear solvers: a unifying view
Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work,surprisingly general conditions for equivalence of these disparate methods arepresented. We also describe connections between probabilistic linear solvers andprojection methods for linear systems, providing a probabilistic interpretation of afar more general class of iterative methods. In particular, this provides such aninterpretation of the generalised minimum residual method. A probabilistic view ofpreconditioning is also introduced. These developments unify the literature onprobabilistic linear solvers and provide foundational connections to the literatureon iterative solvers for linear systems.
0960-3174
1249-1263
Bartels, Simon
c7c4c2dd-b284-4612-8b7a-7479cdc00cd7
Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Ipsen, Ilse C.F.
83eae4c2-19d4-4f74-9d16-4146a63d2c4c
Hennig, Philipp
de8f803a-be5c-409e-be48-9b0f3b2309dd
Bartels, Simon
c7c4c2dd-b284-4612-8b7a-7479cdc00cd7
Cockayne, Jonathan
da87c8b2-fafb-4856-938d-50be8f0e4a5b
Ipsen, Ilse C.F.
83eae4c2-19d4-4f74-9d16-4146a63d2c4c
Hennig, Philipp
de8f803a-be5c-409e-be48-9b0f3b2309dd

Bartels, Simon, Cockayne, Jonathan, Ipsen, Ilse C.F. and Hennig, Philipp (2019) Probabilistic linear solvers: a unifying view. Statistics and Computing, 29, 1249-1263. (doi:10.1007/s11222-019-09897-7).

Record type: Article

Abstract

Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work,surprisingly general conditions for equivalence of these disparate methods arepresented. We also describe connections between probabilistic linear solvers andprojection methods for linear systems, providing a probabilistic interpretation of afar more general class of iterative methods. In particular, this provides such aninterpretation of the generalised minimum residual method. A probabilistic view ofpreconditioning is also introduced. These developments unify the literature onprobabilistic linear solvers and provide foundational connections to the literatureon iterative solvers for linear systems.

Text
Bartels2019_Article_ProbabilisticLinearSolversAUni - Version of Record
Available under License Creative Commons Attribution.
Download (2MB)

More information

Accepted/In Press date: 17 October 2018
Published date: 10 September 2019

Identifiers

Local EPrints ID: 453617
URI: http://eprints.soton.ac.uk/id/eprint/453617
ISSN: 0960-3174
PURE UUID: 9b3d6b83-bfcb-485c-b1c5-6cc226fb8ec2
ORCID for Jonathan Cockayne: ORCID iD orcid.org/0000-0002-3287-199X

Catalogue record

Date deposited: 20 Jan 2022 17:39
Last modified: 17 Mar 2024 04:09

Export record

Altmetrics

Contributors

Author: Simon Bartels
Author: Ilse C.F. Ipsen
Author: Philipp Hennig

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×