Tim Sullivan

Welcome!

I am Junior Professor in Applied Mathematics with Specialism in Risk and Uncertainty Quantification at the Free University of 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.

SIAM UQ18

SIAM UQ18 in Garden Grove

The fourth SIAM Conference on Uncertainty Quantification (SIAM UQ18) will take place at the Hyatt Regency Orange County, Garden Grove, California, this week, 16–19 April 2018.

As part of this conference, Mark Girolami, Philipp Hennig, Chris Oates and I will organise a mini-symposium on “Probabilistic Numerical Methods for Quantification of Discretisation Error” (MS4, MS17 and MS32).

Published on Saturday 14 April 2018 at 08:00 UTC #event

British Library

ProbNum 2018

Next week Chris Oates and I will host the SAMSI–Lloyds–Turing Workshop on Probabilistic Numerical Methods at the Alan Turing Institute, London, housed in the British Library. The workshop is being held as part of the SAMSI Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics.

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 workshop aims, instead, to further the development of 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.

This workshop has been made possible by the generous support of SAMSI, the Alan Turing Institute, and the Lloyd's Register Foundation Data-Centric Engineering Programme.

Published on Friday 6 April 2018 at 07:00 UTC #event #prob-num #samsi

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

Dilations of Cauchy measures in Electron. Commun. Prob.

“Quasi-invariance of countable products of Cauchy measures under non-unitary dilations”, by Han Cheng Lie and myself, has just appeared online in Electronic Communications in Probability. This main result of this article can be understood as an analogue of the celebrated Cameron–Martin theorem, which characterises the directions in which an infinite-dimensional Gaussian measure can be translated while preserving equivalence of the original and translated measure; our result is a similar characterisation of equivalence of measures, but for infinite-dimensional Cauchy measures under dilations instead of translations.

H. C. Lie & T. J. Sullivan. “Quasi-invariance of countable products of Cauchy measures under non-unitary dilations.” Electronic Communications in Probability 23(8):1–6, 2018. doi:10.1214/18-ECP113

Abstract. Consider an infinite sequence \( (U_{n})_{n \in \mathbb{N}} \) of independent Cauchy random variables, defined by a sequence \( (\delta_{n})_{n \in \mathbb{N}} \) of location parameters and a sequence \( (\gamma_{n})_{n \in \mathbb{N}} \) of scale parameters. Let \( (W_{n})_{n \in \mathbb{N}} \) be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence \( (\sigma_{n} \gamma_{n})_{n \in \mathbb{N}} \) of scale parameters, with \( \sigma_{n} \neq 0 \) for all \( n \in \mathbb{N} \). Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of \( (U_{n})_{n \in \mathbb{N}} \) and \( (W_{n})_{n \in \mathbb{N}} \) are equivalent if and only if the sequence \( (| \sigma_{n}| - 1 )_{n \in \mathbb{N}} \) is square-summable.

Published on Wednesday 21 February 2018 at 09:30 UTC #publication #electron-commun-prob #cauchy-distribution

Random forward models and log-likelihoods in Bayesian inverse problems

Preprint: Random Bayesian inverse problems

Han Cheng Lie, Aretha Teckentrup, and I have just a preprint of our latest article, “Random forward models and log-likelihoods in Bayesian inverse problems”, to the arXiv. This paper considers the effect of approximating the likelihood in a Bayesian inverse problem by a random surrogate, as frequently happens in applications, with the aim of showing that the perturbed posterior distribution is close to the exact one in a suitable sense. This article considers general randomisation models, and thereby expands upon the previous investigations of Stuart and Teckentrup (2017) in the Gaussian setting.

Abstract. We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations.

Published on Tuesday 19 December 2017 at 08:30 UTC #publication #preprint #inverse-problems

Exact active subspace Metropolis-Hastings, with applications to the Lorenz-96 system

Preprint: Active subspace Metropolis-Hastings

Ingmar Schuster, Paul Constantine and I have just uploaded a preprint of our latest article, “Exact active subspace Metropolis–Hastings, with applications to the Lorenz-96 system”, to the arXiv. This paper reports on our first investigations into the acceleration of Markov chain Monte Carlo methods using active subspaces as compared to other adaptivity techniques, and is supported by the DFG through SFB 1114 Scaling Cascades in Complex Systems.

Abstract. We consider the application of active subspaces to inform a Metropolis–Hastings algorithm, thereby aggressively reducing the computational dimension of the sampling problem. We show that the original formulation, as proposed by Constantine, Kent, and Bui-Thanh (SIAM J. Sci. Comput., 38(5):A2779–A2805, 2016), possesses asymptotic bias. Using pseudo-marginal arguments, we develop an asymptotically unbiased variant. Our algorithm is applied to a synthetic multimodal target distribution as well as a Bayesian formulation of a parameter inference problem for a Lorenz-96 system.

Published on Friday 8 December 2017 at 08:00 UTC #publication #preprint #mcmc #sfb1114

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