@article{WangCockayneChkrebtiietal., author = {Wang, Junyang and Cockayne, Jon and Chkrebtii, Oksana and Sullivan, T. J. and Oates, Chris}, title = {Bayesian numerical methods for nonlinear partial differential equations}, series = {Statistics and Computing}, volume = {31}, journal = {Statistics and Computing}, number = {5}, doi = {10.1007/s11222-021-10030-w}, language = {en} } @article{OatesSullivan, author = {Oates, Chris and Sullivan, T. J.}, title = {A modern retrospective on probabilistic numerics}, series = {Statistics and Computing}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {10.1007/s11222-019-09902-z}, pages = {1335 -- 1351}, abstract = {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.}, language = {en} } @article{GirolamiIpsenOatesetal., author = {Girolami, Mark A. and Ipsen, Ilse C. F. and Oates, Chris and Owen, Art B. and Sullivan, T. J.}, title = {Editorial: Special edition on probabilistic numerics}, series = {Statistics and Computing}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {doi:10.1007/s11222-019-09892-y}, pages = {1181 -- 1183}, language = {en} } @article{CockayneOatesSullivanetal., author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Bayesian Probabilistic Numerical Methods}, series = {SIAM Review}, volume = {61}, journal = {SIAM Review}, number = {4}, doi = {10.1137/17M1139357}, pages = {756 -- 789}, 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 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.}, language = {en} }