SAMSI Working Group on Probabilistic Numerics

Duration: Until May 31st 2018, as part of the Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics.

Group Leaders: Tim Sullivan and Chris Oates

Description: The accuracy and robustness of numerical predictions that are based on mathematical models depend critically upon the construction of accurate discrete approximations to key quantities of interest. The exact error due to approximation will be unknown to the analyst, but worst-case upper bounds can often be obtained. This working group aims, instead, to develop Probabilistic Numerical Methods, which provide the analyst with a richer, probabilistic quantification of the numerical error in their output, thus providing better tools for reliable statistical inference.

Research Topics:

  • Reference priors for the probabilistic solution of differential equations.

  • Heavy-tailed stable distributions for robust uncertainty quantification.

  • Statistical estimation with multi-resolution operator decompositions.

  • Probabilistic numerical methods as Bayesian inversion methods.

Group Members

Workshops and Visits

  • July and August, 2017: F Schaefer to visit M Girolami and F-X Briol @ Alan Turing Institute and Imperial College London.

  • August and September, 2017: F-X Briol to visit H Owhadi, A Stuart and F Schaefer @ Caltech.

  • August 28 - Sept 1, 2017: Group meeting at the SAMSI Program on Quasi Monte Carlo Opening Workshop in Duke, NC, USA.

  • April 11-13, 2018: Meeting of the working group at the Alan Turing Institute, London. Supported by SAMSI (10,000 USD) and the Lloyds Register Foundation Programme on Data-Centric Engineering at the Alan Turing Institute (3,000 GBP). [web] [web2]

Reading Group: (organised by F-X Briol)

  • 24-01-2017: Louis Ellam - A statistical model of urban retail structure.

  • 07-02-2017: Jon Cockayne - Discussion of "A probabilistic model for the numerical solution of initial value problems" by Schober et al. [slides]

  • 21-02-2017: Chris Oates - Discussion of "Probabilistic interpretation of linear solvers" by P. Hennig. [slides]

  • 07-03-2017: Francois-Xavier Briol - Discussion of "An introduction to sampling via measure transport" by Marzouk et al. [slides]

  • 21-03-2017: Tim Sullivan - Discussion of "MAP estimators and their consistency in Bayesian nonparametric inverse problems" by Dashti et al and "Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems", by Helin and Burger.

  • 04-04-2017: Han Cheng Lie - Discussion of "Why does Monte Carlo Fail to Work Properly in High-Dimensional Optimization Problems?", by Polyak and Shcherbakov.

  • 18-04-2017: Jon Cockayne - Linear Algebra for Probabilistic Numerics. [slides]

  • 02-05-2017: Louis Ellam - Pre-conditioned Ensemble Monte Carlo.

  • 16-05-2017: Onur Teymur - Discussion of "Bayesian Inference of Log Determinants" by Fitzsimons et al.

  • 11-07-2017: Toni Karvonen - Discussion of "Fully symmetric kernel quadrature" by Karvonen and Särkkä. [slides]

  • 25-07-2017: Discussion of Mike Larkin's work:

    • Chris Oates to discuss "Estimation of a non-negative function". [slides]

    • Tim Sullivan to discuss "Optimal approximation in Hilbert spaces with reproducing kernel functions".

    • Han Cheng-Lie to discuss "Gaussian measure in Hilbert space and applications in numerical analysis".

    • Jon Cockayne to discuss "Weak probability distributions on reproducing kernel Hilbert spaces" [slides]

  • 08-08-2017: Tom Rainforth - Discussion of "Bayesian Optimization for Probabilistic Programs" by Rainforth et al.

  • 02-10-2017 (3pm UK): Tim Sullivan and Chris Oates - (Re)introduction to the SAMSI Working Group.

  • 16-10-2017 (3pm UK): Motonobu Kanagawa - Discussion of "Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings" by Kanagawa et al. [slides]

  • 30-10-2017 (3pm UK): Jon Cockayne - Discussion of "Bayesian Probabilistic Numerical Methods for Industrial Process Monitoring" by Oates et al. [slides]

  • 13-11-2017 (4pm UK): Jens Oettershagen - Discussion of "Construction of Optimal Cubature Algorithms with Applications to Econometrics and Uncertainty Quantification", PhD thesis. [slides]

  • 27-11-2017 (4pm UK): Florian Schaeffer - Discussion of "Compression, Inversion and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity", by Schaeffer et al. 

  • 11-12-2017 (4pm UK): Chris Oates - Discussion of "Better Together? Statistical Learning in Models Made of Modules", by Jacob et al.

Publications:

  • Oates CJ, Cockayne J, Robert GA (2017) Bayesian Probabilistic Numerical Methods for Industrial Process Monitoring. [arXiv]

  • Oates CJ, Niederer S, Lee A, Briol F-X, Girolami M. (2017) Probabilistic Models for Integration Error in Assessment of Functional Cardiac Models. [arXiv] [Blog]

  • Schaefer F, Sullivan TJ, Owhadi H (2017) Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity. [arXiv]

  • Lie HC, Stuart AM, Sullivan TJ (2017) Strong Convergence Rates of Probabilistic Integrators for Ordinary Differential Equations. [arXiv]

  • Cockayne J, Oates CJ, Sullivan T, Girolami M. (2017) Bayesian Probabilistic Numerical Methods. [arXiv]

  • Cockayne J, Oates CJ, Sullivan T, Girolami M (2017) Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems. Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Ed. Geert Verdoolaege, AIP Conference Proceedings, 1853:060001. [Journal] [arXiv]

  • Briol F-X, Cockayne J, Teymur, O. (2016) Contributed Discussion on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Analysis, 11(4):1285-1293. [Journal] [arXiv]

Presentations:

Crediting SAMSI:

“This material was based upon work partially supported by the National Science Foundation under Grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.” [details]