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Bayesian numerical methods for nonlinear partial differential equations

Lookup NU author(s): Dr Junyang Wang, Professor Chris Oates

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This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

Abstract

© 2021, The Author(s). 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.

Publication metadata

Author(s): Wang J, Cockayne J, Chkrebtii O, Sullivan TJ, Oates CJ

Publication type: Article

Publication status: Published

Journal: Statistics and Computing

Year: 2021

Volume: 31

Issue: 5

Online publication date: 27/07/2021

Acceptance date: 11/07/2021

Date deposited: 17/08/2021

ISSN (print): 0960-3174

ISSN (electronic): 1573-1375

Publisher: Springer Nature

URL: https://doi.org/10.1007/s11222-021-10030-w

DOI: 10.1007/s11222-021-10030-w

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