### Welcome!

I am **Associate Professor in Predictive Modelling** in the Mathematics Institute and School of Engineering at the University of Warwick.
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.

### Order-theoretic perspectives on MAP estimation in SIAM/ASA JUQ

The final version of “An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems” by Hefin Lambley and myself has just appeared online in the *SIAM/ASA Journal on Uncertainty Quantification*.

On a heuristic level, modes and MAP estimators are intended to be the “most probable points” of a space \(X\) with respect to a probability measure \(\mu\). Thus, in some sense, they would seem to be the greatest elements of some order on \(X\), and a rigorous order-theoretic treatment is called for, especially for cases in which \(X\) is, say, an infinite-dimensional function space. Such an order-theoretic perspective opens up some attractive proof strategies for the existence of modes and MAP estimators but also leads to some interesting counterexamples. In particular, because the orders involved are not total, some pairs of points of \(X\) can be incomparable (i.e. neither is more nor less likely than the other). In fact we show that there are examples for which the collection of such mutually incomparable elements is dense in \(X\).

H. Lambley and T. J. Sullivan. “An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems.” *SIAM/ASA Journal on Uncertainty Quantification* 11(4):1195–1224, 2023.

Published on Friday 20 October 2023 at 09:00 UTC #publication #modes #order-theory #map-estimators #lambley #juq

### Error analysis for SParareal in SISC

The final version of “Error bound analysis of the stochastic parareal algorithm” by Kamran Pentland, Massimiliano Tamborrino, and myself has just appeared online in the *SIAM Journal on Scientific Computing* (SISC).

K. Pentland, M. Tamborrino, and T. J. Sullivan. “Error bound analysis of the stochastic parareal algorithm.” *SIAM Journal on Scientific Computing* 45(5):A2657–A2678, 2023.

**Abstract.**
Stochastic Parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as Parareal. Similarly to Parareal, it combines fine- and coarse-grained solutions to an ODE using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.

Published on Monday 9 October 2023 at 09:00 UTC #publication #prob-num #sparareal #pentland #tamborrino #sisc

### A periodic table of modes and MAP estimators

Ilja Klebanov and I have just uploaded a preprint of our paper “A ‘periodic table’ of modes and maximum a posteriori estimators” to the arXiv.

This paper forms part of the growing body of work on the ‘small balls’ theory of modes for probability measures on metric spaces, which is needed e.g. for the treatment of MAP estimation for Bayesian inverse problems with functional unknowns. There are already several versions in the literature: the strong mode, the weak mode, and the generalised strong mode. We take an axiomatic approach to the problem and identify a system of 17 essentially distinct notions of mode, proving implications between them and providing explicit counterexamples to distinguish them. From an axiomatic point of view, all these 17 seem to be ‘equally good’, suggesting that further research is needed in this area.

**Abstract.**
The last decade has seen many attempts to generalise the definition of modes, or MAP estimators, of a probability distribution \(\mu\) on a space \(X\) to the case that \(\mu\) has no continuous Lebesgue density, and in particular to infinite-dimensional Banach and Hilbert spaces \(X\). This paper examines the properties of and connections among these definitions. We construct a systematic taxonomy – or ‘periodic table’ – of modes that includes the established notions as well as large hitherto-unexplored classes. We establish implications between these definitions and provide counterexamples to distinguish them. We also distinguish those definitions that are merely ‘grammatically correct’ from those that are ‘meaningful’ in the sense of satisfying certain ‘common-sense’ axioms for a mode, among them the correct handling of discrete measures and those with continuous Lebesgue densities. However, despite there being 17 such ‘meaningful’ definitions of mode, we show that none of them satisfy the ‘merging property’, under which the modes of \(\mu|_{A}\), \(\mu|_{B}\), and \(\mu|_{A \cup B}\) enjoy a straightforward relationship for well-separated positive-mass events \( A, B \subseteq X\).

Published on Monday 17 July 2023 at 09:00 UTC #preprint #modes #map-estimators #klebanov

### Unbounded images of Gaussian and other stochastic processes

Tadashi Matsumoto and I have just uploaded a preprint of our note “Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators” to the arXiv.

The purpose of this note is to provide a self-contained rigorous proof of the well-known formula for the mean and covariance function of a stochastic process — in particular, a Gaussian process — when it is acted upon by an *unbounded* linear operator such as an ordinary or partial differential operator, as used in probabilistic approaches to the solution of ODEs and PDEs.
This result is easy to establish in the case of a bounded operator, but the unbounded case requires a careful application of Hille's theorem for the Bochner integral of a Banach-valued random variable.

**Abstract.**
Gaussian processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP \(v\) that is the image of another GP \(u\) under a linear transformation \(T\) acting on the sample paths of \(u\) are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when \(T\) is an unbounded operator such as a differential operator, which is common in several modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator \(T\) acting on the sample paths of a square-integrable stochastic process. Our proof technique relies upon Hille's theorem for the Bochner integral of a Banach-valued random variable.

Published on Monday 8 May 2023 at 13:00 UTC #preprint #prob-num #gp #matsumoto

### GParareal in Statistics and Computing

The article “GParareal: A time-parallel ODE solver using Gaussian process emulation” by Kamran Pentland, Massimiliano Tamborrino, James Buchanan, Lynton Appel and myself has just been published in its final form in *Statistics and Computing*.
In this paper, we show how a Gaussian process emulator for the difference between coarse/cheap and fine/expensive solvers for a dynamical system can be used to enable rapid and accurate solution of that dynamical system in a way that is parallel in time.
This approach extends the now-classical Parareal algorithm in a probabilistic way that allows for efficient use of both runtime and legacy data gathered about the coarse and fine solvers, which may be a critical performance advantage for complex dynamical systems for which the fine solver is too expensive to run in series over the full time domain.

K. Pentland, M. Tamborrino, T. J. Sullivan, J. Buchanan, and L. C. Appel. “GParareal: A time-parallel ODE solver using Gaussian process emulation.” *Statistics and Computing* 33(1):no. 20, 23pp., 2023.

**Abstract.**
Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, “predicting” the solution serially using a cheap (coarse) solver and “correcting” these values using an expensive (fine) solver that runs in parallel on a number of temporal subintervals.
In this work, we propose a time-parallel algorithm (GParareal) that solves IVPs by modelling the correction term, i.e. the difference between fine and coarse solutions, using a Gaussian process emulator.
This approach compares favourably with the classic parareal algorithm and we demonstrate, on a number of IVPs, that GParareal can converge in fewer iterations than parareal, leading to an increase in parallel speed-up.
GParareal also manages to locate solutions to certain IVPs where parareal fails and has the additional advantage of being able to use archives of legacy solutions, e.g. solutions from prior runs of the IVP for different initial conditions, to further accelerate convergence of the method - something that existing time-parallel methods do not do.

Published on Thursday 22 December 2022 at 12:00 UTC #publication #prob-num #pentland #tamborrino #buchanan #appel