Like many international conferences, the SIAM Conference on Uncertainty Quantification planned for 24–27 March 2020 had to be postponed indefinitely in view of the Covid-19 pandemic. Undeterred by this, the speakers of four minisymposia on the theme of Probabilistic Numerical Methods have generously taken the time to adapt their talks for a new medium and record them for general distribution. The talks can be found at http://probabilistic-numerics.org/meetings/SIAMUQ2020/.
We hope that these talks will be of general interest. Furthermore, the speakers have declared themselves ready to answer questions in written form. If you would like to ask any questions or contribute to the discussion, then please submit your question via this form by 10 May 2020.
The 2019 Q4 issue of SIAM Review will carry an article by Jon Cockayne, Chris Oates, Mark Girolami, and myself on the Bayesian formulation of probabilistic numerical methods, i.e. the interpretation of deterministic numerical tasks such as quadrature and the solution of ordinary and partial differential equations as (Bayesian) statistical inference tasks.
J. Cockayne, C. J. Oates, T. J. Sullivan, and M. Girolami. “Bayesian probabilistic numerical methods.” SIAM Review 61(4):756–789, 2019.
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.
The special issue of Statistics and Computing dedicated to probabilistic numerics carries the article “A modern retrospective on probabilistic numerics” by Chris Oates and myself, which gives a historical overview of the field and some perspectives on future challenges.
C. J. Oates and T. J. Sullivan. “A modern retrospective on probabilistic numerics.” Statistics and Computing 29(6):1335–1351, 2019.
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 both to applications and to a modern formal treatment. 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.
It is a great pleasure to announce that a special issue of Statistics and Computing (vol. 29, no. 6) dedicated to the theme of probabilistic numerics is now fully available online in print. This special issue, edited by Mark Girolami, Ilse Ipsen, Chris Oates, Art Owen, and myself, accompanies the 2018 Workshop on Probabilistic Numerics held at the Alan Turing Institute in London.
The special issue consists of a short editorial and ten full-length peer-reviewed research articles:
- “De-noising by thresholding operator adapted wavelets” by G. R. Yoo and H. Owhadi
- “Optimal Monte Carlo integration on closed manifolds” by M. Ehler, M. Gräf, and C. J. Oates
- “Fast automatic Bayesian cubature using lattice sampling” by R. Jagadeeswaran and F. J. Hickernell
- “Symmetry exploits for Bayesian cubature methods” by T. Karvonen, S. Särkkä, and C. J. Oates
- “Probabilistic linear solvers: a unifying view” by S. Bartels, J. Cockayne, I. C. F. Ipsen, and P. Hennig
- “Strong convergence rates of probabilistic integrators for ordinary differential equations” by H. C. Lie, A. M. Stuart, and T. J. Sullivan
- “Adaptive step-size selection for state-space based probabilistic differential equation solvers” by O. A. Chkrebtii and D. A. Campbell
- “Probabilistic solutions to ordinary differential equations as non-linear Bayesian filtering: A new perspective” by F. Tronarp, H. Kersting, S. Särkkä, and P. Hennig
- “On the positivity and magnitudes of Bayesian quadrature weights” by T. Karvonen, M. Kanagawa, and S. Särkkä
- “A modern retrospective on probabilistic numerics” by C. J. Oates and T. J. Sullivan
Chris Oates and I have just uploaded a preprint of our paper “A modern retrospective on probabilistic numerics” to the arXiv.
Abstract. This article attempts to cast 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.