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Bayesian numerical methods for nonlinear partial differential equations
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Wang, Junyang, Cockayne, Jon, Chkrebtii, Oksana, Sullivan, Tim J. and Oates, Chris. J. (2021) Bayesian numerical methods for nonlinear partial differential equations. Statistics and Computing, 31 (5). 55. doi:10.1007/s11222-021-10030-w ISSN 1573-1375.
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Official URL: https://doi.org/10.1007/s11222-021-10030-w
Abstract
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| SWORD Depositor: | Library Publications Router | ||||||
| Library of Congress Subject Headings (LCSH): | Differential equations, Differential equations -- Numerical solutions, Nonlinear partial differential operators , Differential equations, Partial, Differential equations, Nonlinear, Approximation theory , Inverse problems (Differential equations) , Analysis of covariance, Probabilities | ||||||
| Journal or Publication Title: | Statistics and Computing | ||||||
| Publisher: | Springer US | ||||||
| ISSN: | 1573-1375 | ||||||
| Official Date: | 27 July 2021 | ||||||
| Dates: |
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| Volume: | 31 | ||||||
| Number: | 5 | ||||||
| Article Number: | 55 | ||||||
| DOI: | 10.1007/s11222-021-10030-w | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | ** From Springer Nature via Jisc Publications Router ** History: received 07-07-2021; accepted 11-07-2021; registration 13-07-2021; pub-electronic 27-07-2021; online 27-07-2021; pub-print 09-2021. ** Licence for this article: http://creativecommons.org/licenses/by/4.0/ | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Date of first compliant deposit: | 26 August 2021 | ||||||
| Date of first compliant Open Access: | 27 August 2021 | ||||||
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