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 - http://dx.doi.org/10.1137/17M1139357
VL - 61
IS - 4
SP - 756
EP - 789
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 - GEN
A1 - Oates, Chris. J.
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 - 2019
PB - De Gruyter
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 - http://dx.doi.org/10.1063/1.4985359
VL - 1853
SP - 060001-1
EP - 060001-8
ER -
TY - JOUR
A1 - Girolami, Mark A.
A1 - Ipsen, Ilse C. F.
A1 - Oates, Chris J.
A1 - Owen, Art B.
A1 - Sullivan, T. J.
T1 - Editorial: Special edition on probabilistic numerics
JF - Statistics and Computing
Y1 - 2019
U6 - http://dx.doi.org/doi:10.1007/s11222-019-09892-y
VL - 29
IS - 6
SP - 1181
EP - 1183
ER -