# 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.

### Preprint: Weak and strong modes

Han Cheng Lie and I have just uploaded a revised preprint of our paper, “Equivalence of weak and strong modes of measures on topological vector spaces”, to the arXiv. This addresses a natural question in the theory of modes (or maximum a posteriori estimators, in the case of posterior measure for a Bayesian inverse problem) in an infinite-dimensional space $$X$$. Such modes can be defined either strongly (a la Dashti et al. (2013), via a global maximisation) or weakly (a la Helin & Burger (2015), via a dense subspace $$E \subset X$$). The question is, when are strong and weak modes equivalent? The answer turns out to be rather subtle: under reasonable uniformity conditions, the two kinds of modes are indeed equivalent, but finite-dimensional counterexamples exist when the uniformity conditions fail.

Abstract. A strong mode of a probability measure on a normed space $$X$$ can be defined as a point $$u \in X$$ such that the mass of the ball centred at $$u$$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $$E$$ of $$X$$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with non-atomic measures that are finite on bounded sets, the density of $$E$$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.

Published on Monday 9 July 2018 at 08:00 UTC #publication #preprint #inverse-problems #modes #map-estimator

### ECMath Colloquium

This week's colloquium at the Einstein Center for Mathematics Berlin will be on the topic of “Stochastics meets PDE.” The speakers will be:

• Antoine Gloria (Sorbonne): Stochastic homogenization: regularity, oscillations, and fluctuations
• Peter Friz (TU Berlin and WIAS Berlin): Rough Paths, Stochastics and PDEs
• Nicholas Dirr (Cardiff): Interacting Particle Systems and Gradient Flows

Time and Place. Friday 6 July 2018, 14:00–17:00, Humboldt-Universität zu Berlin, Main Building Room 2094, Unter den Linden 6, 10099 Berlin.

Published on Monday 2 July 2018 at 12:00 UTC #event

### 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

### 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

### 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