**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**:

- François-Xavier Briol - University of Warwick, UK and Imperial College London, UK
- Oksana Chkrebtii - Ohio State University, USA
- Jon Cockayne - University of Warwick, UK
- Mark Girolami - Imperial College London, UK and the Alan Turing Institute, UK
- Philipp Hennig - Max Planck Institute, Tübingen, Germany
- Han Cheng Lie - Free University of Berlin, Germany and Zuse Institute Berlin, Germany
- Chris Oates - Newcastle University, UK and the Alan Turing Institute, UK
- Houman Owhadi - California Institute of Technology, USA
- Florian Schaefer - California Institute of Technology, USA
- Andrew Stuart - California Institute of Technology, USA
- Tim Sullivan - Free University of Berlin, Germany and Zuse Institute Berlin, Germany

**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.
- 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). [website]

**Presentations**:

- 23-01-2017: FX Briol to speak @ Stochastic Analysis Seminar, Mathematics Institute, University of Oxford, UK.
- 31-01-2017: FX Briol to speak @ Workshop on the Mathematics for Measurement, Edinburgh, UK.
- 16-02-2017: C Oates to speak @ the UNSW Workshop on High-Dimensional Approximation. Talk title: Bayesian Probabilistic Numerical Computation.
- 22-02-2017: J Cockayne to speak @ Computer Science Department, Imperial College London, UK.
- 01-03-2017: J Cockayne to speak @ SIAM Conference on Computation Science and Engineering. Talk title: Probabilistic Meshless Methods for Partial Differential Equations and Bayesian Inverse Problems. [Video]
- 24-03-2017: C Oates to give a tutorial on Probabilistic Numerics at the University of New South Wales, Australia.
- 05-05-2017: J Cockayne to speak @ Statistics Seminar Series, Imperial College London, UK.
- 15-05-2017: J Cockayne to present a poster @ Advances in Data Science, University of Manchester, UK.
- June 5-9, 2017: T Sullivan, J Cockayne, F Schaefer, H Owhadi, O Chkrebtii and C Oates to speak @ ICERM workshop on Probabilistic Scientific Computing: Statistical inference approaches to numerical analysis and algorithm design, organised by P Hennig, H Owhadi and others. [Videos]
- 13-06-2017: J Cockayne to speak @ Max Planck Institute for Intelligent Systems, Tuebingen, Germany.
- June 18-23, 2017: P Hennig and C Oates to run the Dobbiaco Summer School on Probabilistic Numerics in Bolzano, Italy.
- 05-07-2017: FX Briol to speak @ Statistical Data Science Workshop, Imperial College London and Winton Capital.
- July 10-14, 2017: FX Briol to speak @ SIAM Annual Meeting in Pittsburgh, PA, USA. [Abstract]
- July 29 - August 4, 2017: M Girolami to deliver Medallion Lecture @ Joint Statistical Meeting, Baltimore, USA. [Media]
- August 28 - Sept 1, 2017: T Sullivan and C Oates to speak at the SAMSI Program on Quasi Monte Carlo Opening Workshop in Duke, NC, USA.

**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.

**Publications**:

Barp A, Briol F-X, Kennedy A, Girolami M. (2018) Geometry and Dynamics for Markov Chain Monte Carlo. **Annual Reviews of Statistics and its Applications**, 5, to appear.

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]

Scharfer 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]

**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]