In this edition, we profile Jon Cockayne, PhD Student at the University of Warwick, who is focusing on probabilistic numerical methods.
I completed an MSc in Mathematics at Imperial College London in 2009. Upon finishing university, I worked for a number of financial organisations including RBS and Barclays Capital. I joined the startup Warwick Analytics in 2014 as a data scientist, where I met my PhD supervisor Mark Girolami. Through Mark, I became interested in probabilistic numerics and he subsequently invited me to start a PhD in the area at the University of Warwick.
Probabilistic numerical methods is my area of study. Unlike classical numerical methods, in probabilistic numerics the focus is to construct a probabilistic estimate of the solution to an intractable problem, such as ordinary differential equations (ODEs), partial differential equations (PDEs) and high dimensional integrals. This probabilistic solution quantifies the uncertainty that arises from the discretisation assumptions required to solve the problem.
This is not an area that I have always been interested in. Though it is something that has been proposed several times over the years by people such as Larkin, Diaconis and Skilling, it is only really recently that people have started to research probabilistic numerics further. My interest in the area comes from the fact that it combines my interest in statistics, which is an area that I studied quite heavily in my undergraduate degree, with computer science and software development, areas that I focused on when in industry. My research is mainly on probabilistic numerics for PDEs, which is challenging both from a computational perspective and a theoretical perspective.
Having worked in industry before, my aspirations are to now remain in academia for the foreseeable future. I am getting to the point in my career where I am starting to look at Post-Doctoral opportunities, to continue studying probabilistic numerics if possible.
I think the field of probabilistic numerics has the potential to have quite a big impact on applied problems. There are still some huge challenges to be addressed before we can get to this stage. The complexity of probabilistic numerical methods tends to be very high compared to standard numerical solvers, and there is still a disconnect between the kind of uncertainty quantification that probabilistic numerics provides, and the kind of uncertainty quantification that numerical analysts would like when they are working on those problems for which discretization error is a real factor.
One of the areas where the impact is most visible is in chaotic systems. Some researchers have applied probabilistic numerical methods to chaotic ODEs. The advantage here becomes clear, in that it is visible that the accumulated error from solving the chaotic ODE causes the numerical solution to effectively become meaningless. Probabilistic numerical solvers convey this, where a standard numerical solver is unable to. One of the applications that I am working towards is turbulent combustion. In these kinds of computational fluid dynamics problems, turbulence has a similar chaotic effect.
“An application of probabilistic numerics to the Lorenz system, at t=50. The red dot is the deterministic solution, while the blue dots are samples from a probabilistic numerical method. As a result of accumulated numerical error, there is a huge amount of variability in the probabilistic solution, which the deterministic solution does not capture.”
My most recent paper has very recently been submitted to the arxiv and is called ‘Bayesian Probabilistic Numerical Methods’. I think that it is a project that thus far I have been the most proud of. It is a piece of work that aims to provide a theoretical basis for probabilistic numerical methods, providing a definition of what it means for probabilistic numerical methods to be “Bayesian”, which is important from a statistical point of view and had not been addressed in the literature prior to this paper. I hope that this paper will pave the way for more tractable probabilistic numerical methods, which seek to approximate the measure of uncertainty produced rather than characterize it exactly.
Bayesian Probabilistic Numerical Methods
Authors: Jon Cockayne, Chris Oates, Tim Sullivan, Mark Girolami
Academically, the co-authors of the two papers I have worked on (Chris Oates, Tim Sullivan and Mark Girolami). They have all been hugely supportive in the first year and a half of my PhD. I found them to be a pleasure to work with, and they are all very positive academics who have set fantastic examples for me as I look forward in my career.
Non-academically, I have always drawn a lot of inspiration from golden age British comedians; Douglas Adams, John Cleese and Michael Palin, because I think it is really easy to take work too seriously. I think their work sets an example for me, of how to enjoy my work and see the funny side in it.