# 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.

### Implicit Probabilistic Integrators in NeurIPS

The paper “Implicit probabilistic integrators for ODEs” by Onur Teymur, Han Cheng Lie, Ben Calderhead and myself has now appeared in Advances in Neural Information Processing Systems 31 (NIPS 2018). This paper forms part of an expanding body of work that provides mathematical convergence analysis of probabilistic solvers for initial value problems, in this case implicit methods such as (probabilistic versions of) the multistep Adams–Moulton method.

O. Teymur, H. C. Lie, T. J. Sullivan & B. Calderhead. “Implicit probabilistic integrators for ODEs” in Advances in Neural Information Processing Systems 31 (NIPS 2018), ed. N. Cesa-Bianchi, K. Grauman, H. Larochelle & H. Wallach. 2018. http://papers.nips.cc/paper/7955-implicit-probabilistic-integrators-for-odes

Abstract. We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams–Moulton method. The implicit construction allows for dynamic feedback from the forthcoming time-step, in contrast to previous probabilistic integrators, all of which are based on explicit methods. We begin with a concise survey of the rapidly-expanding field of probabilistic ODE solvers. We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a rigorous proof of its well-definedness and convergence. We discuss the problem of the calibration of such integrators and suggest one approach. We give an illustrative example highlighting the effect of the use of probabilistic integrators — including our new method — in the setting of parameter inference within an inverse problem.

Published on Thursday 13 December 2018 at 12:00 UTC #publication #nips #neurips #prob-num

### Random Bayesian inverse problems in JUQ

The article “Random forward models and log-likelihoods in Bayesian inverse problems” by Han Cheng Lie, Aretha Teckentrup, and myself has now appeared in its final form in the SIAM/ASA Journal on Uncertainty Quantification, volume 6, issue 4. 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.

H. C. Lie, T. J. Sullivan & A. L. Teckentrup. “Random forward models and log-likelihoods in Bayesian inverse problems.” SIAM/ASA Journal on Uncertainty Quantification 6(4):1600–1629, 2018. doi:10.1137/18M1166523

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 Monday 10 December 2018 at 12:00 UTC #publication #inverse-problems #juq #prob-num

### Opening for Postdoctoral Researcher

There is an opening in my research group for a postdoctoral researcher in Uncertainty Quantification, for the period 01.01.2019 to 31.12.2020. The position will be associated to the research project TrU-2 “Demand modelling and control for e-commerce using RKHS transfer operator approaches” within the Berlin Mathematics Excellence Cluster MATH+; the project will be led by Stefan Klus and myself. Strong candidates with backgrounds in mathematics, statistics, or computational science are encouraged to apply. For details see the website of the Zuse Institute Berlin, reference number WA57/18. Informal enquiries may be directed to Stefan Klus or to me, and formal applications should be sent to jobs@zib.de by 13.12.2018.

Published on Friday 23 November 2018 at 12:34 UTC #group #job #tru2 #mathplus

### Weak and strong modes in Inverse Problems

The paper “Equivalence of weak and strong modes of measures on topological vector spaces” by Han Cheng Lie myself has now appeared in Inverse Problems. This paper 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.

H. C. Lie & T. J. Sullivan. “Equivalence of weak and strong modes of measures on topological vector spaces.” Inverse Problems 34(11):115013, 2018. doi:10.1088/1361-6420/aadef2

(See also H. C. Lie & T. J. Sullivan. “Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013).” Inverse Problems 34(12):129601, 2018. doi:10.1088/1361-6420/aae55b)

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 Saturday 22 September 2018 at 12:00 UTC #publication #inverse-problems #modes #map-estimator

### Kalman Lecture by Andrew Stuart at SFB1294

Andrew Stuart (Caltech) will give the inaugural Kalman Lecture of SFB 1294 Data Assimilation on the topic of “Large Graph Limits of Learning Algorithms”.

Time and Place. Friday 24 August 2018, 10:15–11:45, University of Potsdam, Campus Golm, Building 27, Lecture Hall 0.01

Abstract. Many problems in machine learning require the classification of high dimensional data. One methodology to approach such problems is to construct a graph whose vertices are identified with data points, with edges weighted according to some measure of affinity between the data points. Algorithms such as spectral clustering, probit classification and the Bayesian level set method can all be applied in this setting. The goal of the talk is to describe these algorithms for classification, and analyze them in the limit of large data sets. Doing so leads to interesting problems in the calculus of variations, in stochastic partial differential equations and in Monte Carlo Markov Chain, all of which will be highlighted in the talk. These limiting problems give insight into the structure of the classification problem, and algorithms for it.

Published on Friday 3 August 2018 at 11:00 UTC #event #sfb1294