# Tim Sullivan

### Welcome!

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.

### SIAM UQ20 in Munich

The 2020 SIAM conference on Uncertainty Quantification (UQ20) will take place from 24 to 27 March 2020, on the Garching campus (near Munich) of the Technical University of Munich (TUM), Germany. UQ20 is being organised in cooperation with the GAMM Activity Group on UQ.

The website for UQ20 is now live and the call for submissions is open.

Published on Monday 8 July 2019 at 12:00 UTC #event #siam

### Preprint: Geodesic analysis in Kendall’s shape space

Esfandiar Nava-Yazdani, Christoph von Tycowicz, Hans-Christian Hege, and I have just uploaded an updated preprint of our work “Geodesic analysis in Kendall's shape space with epidemiological applications” (previously entitled “A shape trajectories approach to longitudinal statistical analysis”) to the arXiv. This work is part of the ECMath / MATH+ project CH-15 “Analysis of Empirical Shape Trajectories”.

Abstract. We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall's shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone.

Published on Monday 1 July 2019 at 08:00 UTC #publication #preprint #ch15 #shape-trajectories #nava-yazdani #von-tycowicz #hege

I have just uploaded a preprint of “Comments on the article ‘A Bayesian conjugate gradient method’” to the arXiv. This note discusses the recent paper “A Bayesian conjugate gradient method” in Bayesian Analysis by Jon Cockayne, Chris Oates, Ilse Ipsen, and Mark Girolami, and is an invitation to a rejoinder from the authors.

Abstract. The recent article “A Bayesian conjugate gradient method” by Cockayne, Oates, Ipsen, and Girolami proposes an approximately Bayesian iterative procedure for the solution of a system of linear equations, based on the conjugate gradient method, that gives a sequence of Gaussian/normal estimates for the exact solution. The purpose of the probabilistic enrichment is that the covariance structure is intended to provide a posterior measure of uncertainty or confidence in the solution mean. This note gives some comments on the article, poses some questions, and suggests directions for further research.

Published on Wednesday 26 June 2019 at 08:00 UTC #publication #preprint #prob-num

### Course in Inverse Problems at FU Berlin

This Summer Semester 2019, I will offer a course in the mathematics of Inverse Problems at the Freie Universität Berlin. The course load will be 2+2 SWS and the course will be a valid selection for the Numerik IV and Stochastic IV modules.

Inverse problems, meaning the recovery of parameters or states in a mathematcial model that best match some observed data, are ubiquitous in applied sciences. This course will provide an introduction to the deterministic (variational) and stochastic (Bayesian) theories of inverse problems in function spaces.

#### Course Contents

1. Examples of inverse problems in mathematics and physical sciences
2. Preliminaries from functional analysis
3. Preliminaries from probability theory
4. Linear inverse problems and variational regularisation
5. Bayesian regularisation of inverse problems
6. Monte Carlo methods for Bayesian problems

For further details see the KVV page.

Published on Monday 8 April 2019 at 09:00 UTC #fub #inverse-problems

### Preprint: Computing with dense kernel matrices at near-linear cost

Florian Schäfer, Houman Owhadi, and I have just uploaded a revised and improved version of our preprint “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” to the arXiv. This paper shows how a surprisingly simple algorithm — the zero fill-in incomplete Cholesky factorisation — with respect to a cleverly-chosen sparsity pattern allows for near-linear complexity compression, inversion, and approximate PCA of square matrices of the form

$$\Theta = \begin{bmatrix} G(x_{1}, x_{1}) & \cdots & G(x_{1}, x_{N}) \\ \vdots & \ddots & \vdots \\ G(x_{N}, x_{1}) & \cdots & G(x_{N}, x_{N}) \end{bmatrix} \in \mathbb{R}^{N \times N} ,$$

where $$\{ x_{1}, \dots, x_{N} \} \subset \mathbb{R}^{d}$$ is a data set and $$G \colon \mathbb{R}^{d} \times \mathbb{R}^{d} \to \mathbb{R}$$ is a covariance kernel function. Such matrices play a key role in, for example, Gaussian process regression and RKHS-based machine learning techniques.

Abstract. Dense kernel matrices $$\Theta \in \mathbb{R}^{N \times N}$$ obtained from point evaluations of a covariance function $$G$$ at locations $$\{ x_{i} \}_{1 \leq i \leq N}$$ arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions of elliptic boundary value problems and homogeneously-distributed sampling points, we show how to identify a subset $$S \subset \{ 1 , \dots , N \}^2$$, with $$\# S = O ( N \log (N) \log^{d} ( N /\varepsilon ) )$$, such that the zero fill-in incomplete Cholesky factorisation of the sparse matrix $$\Theta_{ij} 1_{( i, j ) \in S}$$ is an $$\varepsilon$$-approximation of $$\Theta$$. This factorisation can provably be obtained in complexity $$O ( N \log( N ) \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2}( N ) \log^{2d}( N /\varepsilon) )$$ in time; we further present numerical evidence that $$d$$ can be taken to be the intrinsic dimension of the data set rather than that of the ambient space. The algorithm only needs to know the spatial configuration of the $$x_{i}$$ and does not require an analytic representation of $$G$$. Furthermore, this factorization straightforwardly provides an approximate sparse PCA with optimal rate of convergence in the operator norm. Hence, by using only subsampling and the incomplete Cholesky factorization, we obtain, at nearly linear complexity, the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky factorization we also obtain a solver for elliptic PDE with complexity $$O ( N \log^{d}( N /\varepsilon) )$$ in space and $$O ( N \log^{2d}( N /\varepsilon) )$$ in time.

Published on Tuesday 26 March 2019 at 12:00 UTC #publication #preprint #prob-num #schaefer #owhadi