Bayesian Probabilistic Numerical Methods
Abstract
Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the nonlinear and non-Gaussian contexts. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented.
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Supplementary Material
Index of Supplementary Materials
Title of paper: Bayesian Probabilistic Numerical Methods
Authors: Jon Cockayne, Chris J. Oates, T. J. Sullivan and Mark Girolami
File: M113935SupMat.pdf
Type: PDF
Contents:
SM1: Proofs
SM2: Philosophical Status of the Belief Distribution
SM3: Dichotomy of Existing PNMs
SM4: Decision-Theoretic Treatment
SM5: Monte Carlo Methods for Numerical Disintegration
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© 2019, Society for Industrial and Applied Mathematics.
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Submitted: 18 July 2017
Accepted: 14 February 2019
Published online: 6 November 2019
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Funding Information
Freie Universität Berlin https://doi.org/10.13039/501100007537 : 1114
Lloyd's Register Foundation https://doi.org/10.13039/100008885
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1127914
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/R034710/1, EP/R018413/1, EP/R004889/1, EP/P020720/1, EP/J016934/3
Royal Academy of Engineering https://doi.org/10.13039/501100000287
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