Tim Sullivan


I am Junior Professor in Applied Mathematics with Specialism in Risk and Uncertainty Quantification at the Freie Universität Berlin and Research Group Leader for Uncertainty Quantification at the Zuse Institute Berlin. I have wide interests in uncertainty quantification the broad sense, understood as the meeting point of numerical analysis, applied probability and statistics, and scientific computation. On this site you will find information about how to contact me, my research, publications, and teaching activities.

Bayesian probabilistic numerical methods

Bayesian probabilistic numerical methods in SIAM Review

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. doi:10.1137/17M1139357

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.

Published on Thursday 7 November 2019 at 07:00 UTC #publication #bayesian #siam-review #prob-num #cockayne #girolami #oates

A modern retrospective on probabilistic numerics

Probabilistic numerics retrospective in Statistics and Computing

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. doi:10.1007/s11222-019-09902-z

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.

Published on Tuesday 5 November 2019 at 12:30 UTC #publication #stat-comp #prob-num #oates

Strong convergence rates of probabilistic integrators for ordinary differential equations

Strong convergence rates of probabilistic integrators for ODEs in Statistics and Computing

The special issue of Statistics and Computing dedicated to probabilistic numerics carries the article “Strong convergence rates of probabilistic integrators for ODEs” by Han Cheng Lie, Andrew Stuart, and myself on the convergence analysis of randomised perturbation-based time-stepping methods for the solution of ODE initial value problems.

H. C. Lie, A. M. Stuart, and T. J. Sullivan. “Strong convergence rates of probabilistic integrators for ordinary differential equations.” Statistics and Computing 29(6):1265–1283, 2019. doi:10.1007/s11222-019-09898-6

Abstract. Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.

Published on Tuesday 5 November 2019 at 12:00 UTC #publication #stat-comp #prob-num #lie #stuart

Editorial: Special edition on probabilistic numerics

Statistics and Computing special issue on Probabilistic Numerics

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:

  1. “De-noising by thresholding operator adapted wavelets” by G. R. Yoo and H. Owhadi
  2. “Optimal Monte Carlo Integration on Closed Manifolds” by M. Ehler, M. Gräf, and C. J. Oates
  3. “Fast automatic Bayesian cubature using lattice sampling” by R. Jagadeeswaran and F. J. Hickernell
  4. “Symmetry exploits for Bayesian cubature methods” by T. Karvonen, S. Särkkä, and C. J. Oates
  5. “Probabilistic linear solvers: a unifying view” by S. Bartels, J. Cockayne, I. C. F. Ipsen, and P. Hennig
  6. “Strong convergence rates of probabilistic integrators for ordinary differential equations” by H. C. Lie, A. M. Stuart, and T. J. Sullivan
  7. “Adaptive step-size selection for state-space based probabilistic differential equation solvers” by O. A. Chkrebtii and D A. Campbell
  8. “Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective” by F. Tronarp, H. Kersting, S. Särkkä, and P. Hennig
  9. “On the positivity and magnitudes of Bayesian quadrature weights” by T. Karvonen, M. Kanagawa, and S. Särkkä
  10. “A modern retrospective on probabilistic numerics” by C. J. Oates and T. J. Sullivan

Published on Wednesday 30 October 2019 at 12:00 UTC #stat-comp #prob-num #girolami #ipsen #oates #owen

Freie Universität Berlin

Postdoc position: Analysis of MAP estimators

Within a few weeks a full-time two-year position for a postdoctoral researcher in the UQ group will be advertised by the Freie Universität Berlin. This position will be associated to the project “Analysis of maximum a posteriori estimators: Common convergence theories for Bayesian and variational inverse problems” funded by the DFG.

This project aims to advance the state of the art in rigorous mathematical understanding of MAP estimators in infinite-dimensional statistical inverse problems. In particular, the research in this project will connect the “small balls” approach of Dashti, Law, Stuart, and Voss to the calculus of variations and hence properly link the variational and fully Bayesian points of view on inverse problems.

Precise details of the application process will appear soon, at which point this news item will be updated accordingly. In the meantime, prospective candidates are welcome to contact me with informal enquiries.

Published on Monday 21 October 2019 at 08:00 UTC #group #job #fu-berlin #inverse-problems #dfg #map-estimators