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Cockayne, Jon, Oates, Chris J., Sullivan, T. J. and Girolami, Mark (2019) Bayesian probabilistic numerical methods. SIAM Review, 61 (4). pp. 756-789. doi:10.1137/17M1139357 ISSN 0036-1445.
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WRAP-Bayesian-probabilistic-numerical-methods-Sullivan-2019.pdf - Accepted Version - Requires a PDF viewer. Download (2974Kb) | Preview |
Official URL: http://dx.doi.org/10.1137/17M1139357
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.
| Item Type: | Journal Article | ||||||||||||||||||
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| Subjects: | Q Science > QA Mathematics T Technology > TA Engineering (General). Civil engineering (General) |
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| Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering Faculty of Science, Engineering and Medicine > Science > Mathematics |
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| Library of Congress Subject Headings (LCSH): | Bayesian statistical decision theory, Probabilities, Gaussian measures | ||||||||||||||||||
| Journal or Publication Title: | SIAM Review | ||||||||||||||||||
| Publisher: | Society for Industrial and Applied Mathematics | ||||||||||||||||||
| ISSN: | 0036-1445 | ||||||||||||||||||
| Official Date: | 6 November 2019 | ||||||||||||||||||
| Dates: |
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| Volume: | 61 | ||||||||||||||||||
| Number: | 4 | ||||||||||||||||||
| Page Range: | pp. 756-789 | ||||||||||||||||||
| DOI: | 10.1137/17M1139357 | ||||||||||||||||||
| Status: | Peer Reviewed | ||||||||||||||||||
| Publication Status: | Published | ||||||||||||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||||||||||||
| Copyright Holders: | © 2019, Society for Industrial and Applied Mathematics | ||||||||||||||||||
| Date of first compliant deposit: | 15 April 2020 | ||||||||||||||||||
| Date of first compliant Open Access: | 15 April 2020 | ||||||||||||||||||
| RIOXX Funder/Project Grant: |
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