6616
2019
eng
756
789
4
61
article
0
--
--
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Bayesian Probabilistic Numerical Methods
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 non-linear and non-Gaussian context. 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.
SIAM Review
1702.03673
10.1137/17M1139357
Yes
2019
Jon Cockayne
T. J. Sullivan
Chris Oates
T. J. Sullivan
Mark Girolami
Numerical Mathematics
no-project
6657
2019
eng
1265
1283
6
29
article
0
--
--
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Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator.
Statistics and Computing
1703.03680
10.1007/s11222-019-09898-6
yes
Han Cheng Lie
T. J. Sullivan
T. J. Sullivan
Andrew Stuart
Numerical Mathematics
no-project
Lie, Han
7501
2019
eng
1181
1183
6
29
article
0
--
--
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Editorial: Special edition on probabilistic numerics
Statistics and Computing
doi:10.1007/s11222-019-09892-y
no
Mark A. Girolami
T. J. Sullivan
Ilse C. F. Ipsen
Chris Oates
Art B. Owen
T. J. Sullivan
Numerical Mathematics
no-project
7380
2019
eng
985
989
3
14
article
0
--
--
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Contributed discussion on the article "A Bayesian conjugate gradient method"
The recent article "A Bayesian conjugate gradient method" by Cockayne, Oates, Ipsen, and Girolami proposes an approximately Bayesian iterative procedure for the solution of a system of linear equations, based on the conjugate gradient method, that gives a sequence of Gaussian/normal estimates for the exact solution. The purpose of the probabilistic enrichment is that the covariance structure is intended to provide a posterior measure of uncertainty or confidence in the solution mean. This note gives some comments on the article, poses some questions, and suggests directions for further research.
Bayesian Analysis
1906.10240
10.1214/19-BA1145
yes
T. J. Sullivan
T. J. Sullivan
Numerical Mathematics
no-project
7144
2019
eng
1335
1351
6
29
article
0
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--
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A modern retrospective on probabilistic numerics
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related to modern formal treatments and applications. We highlight in particular the parallel contributions of Sul'din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works.
Statistics and Computing
1901.04457
10.1007/s11222-019-09902-z
2019
Yes
Chris Oates
T. J. Sullivan
T. J. Sullivan
Numerical Mathematics
no-project