TY - JOUR A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Bayesian Probabilistic Numerical Methods JF - SIAM Review N2 - 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. Y1 - 2019 U6 - https://doi.org/10.1137/17M1139357 VL - 61 IS - 4 SP - 756 EP - 789 ER - TY - CHAP A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems T2 - Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering N2 - This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model Y1 - 2017 U6 - https://doi.org/10.1063/1.4985359 VL - 1853 SP - 060001-1 EP - 060001-8 ER - TY - GEN A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems N2 - This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions. T3 - ZIB-Report - 16-31 KW - Probabilistic Numerics KW - Partial Differential Equations KW - Inverse Problems KW - Meshless Methods KW - Gaussian Processes KW - Pseudo-Marginal MCMC Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-59513 SN - 1438-0064 ER - TY - JOUR A1 - Girolami, Mark A. A1 - Ipsen, Ilse C. F. A1 - Oates, Chris A1 - Owen, Art B. A1 - Sullivan, T. J. T1 - Editorial: Special edition on probabilistic numerics JF - Statistics and Computing Y1 - 2019 U6 - https://doi.org/doi:10.1007/s11222-019-09892-y VL - 29 IS - 6 SP - 1181 EP - 1183 ER - TY - GEN A1 - Oates, Chris A1 - Cockayne, Jon A1 - Prangle, Dennis A1 - Sullivan, T. J. A1 - Girolami, Mark ED - Hickernell, F. J. ED - Kritzer, P. T1 - Optimality criteria for probabilistic numerical methods T2 - Multivariate Algorithms and Information-Based Complexity N2 - It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that the decision-theoretic framework is neither appropriate nor sufficient. To this end, we consider an alternative optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed. Y1 - 2020 U6 - https://doi.org/10.1515/9783110635461-005 VL - 27 SP - 65 EP - 88 PB - De Gruyter ER -