PhD Blog: Tackling a Real-World Problem with an Integrative Think Tank (ITT) – A reflection from Alisdair Crossley
Hi, my name’s Alastair and I am an incoming PhD student at the University of Bath where I will be working under the MaThRad grant. Over the summer I was given the opportunity to attend an Integrative Think Tank (ITT), to gain experience ahead of my PhD. During an ITT, partners present “real life” problems, in the hope that these can then be formulated into mathematical problems, enabling sound solutions to be sought. The ITT I attended was focused on tackling challenges in global energy provision. Proposed solutions could then go on to receive funding from partners for short-term or long-term projects, conducted as summer projects, or picked up as part of a PhD . One of the partners at the ITT was the French Alternative Energies and Atomic Energy Commission (CEA), a French government-funded technological research organisation focusing on low-carbon energies, defence and security, information technologies and health technologies . They gave a presentation on sensitivity analysis in Monte Carlo neutron transport, and one of their problems pertained to understanding the effects of geometrical perturbations in the nuclear reactor on the leading eigenvalue of the system.
When approaching this complex problem, we first decided to tackle a much simpler, one-dimensional, one-speed slab reactor. In doing this, we found that if the system was perturbed by increasing the fission rate of reaction by a small amount, the perturbed system could be viewed as being “close” to the unperturbed system. In contrast, if the size of the ”area of fission” in the perturbed system was increased by a small amount, the perturbed system could not be viewed as being “close” to the unperturbed system in the same way. Let [−L,L] be the size of the reactor. Let Fϵ,1 be the perturbed fission operator by increasing its rate by ϵ on the interval [−x, x] and let Fϵ,2 be the perturbed fission operator by increasing the interval [−x, x] by ϵ. See Figure 1for a pictorial description of the fission operators Fε,1and Fε,2 on the slab reactor.
Figure 1: The fission operators Fε,1 (pert. in nuclear data) and Fε,2 (pert. in geometry).  Observing how these perturbed systems behaved was interesting, potentially providing a clue as to why it is harder to model geometrical perturbations in comparison to perturbing the fission rate. It was at this point I was offered the chance to continue this research as a summer project. We wanted to find a concrete example to show that, in the right topology, the (geometrically) perturbed system is close to the unperturbed system with small perturbations. To further simplify the problem, we looked at branching Brownian motion. This involved learning a lot about operator theory, perturbation theory, and PDE theory including Sobolev spaces. A highlight of the project was the opportunity for me to travel to Bordeaux to visit another MaThRad partner, Inria, a French institute for research in computer science and automation . I really enjoyed the opportunity to continue our research out in Bordeaux and spend some time exploring the city (not to mention the crepes with cider were delicious!).
I found my time on this project really insightful. I have learnt a vast amount, and have found the project to be a very good introduction to my PhD. I am excited to see how this research develops over time and look forward to seeing its future applications.
I would like to thank Dr Pryer in giving me the opportunity to undertake this project and Dr Horton and Professor Cox for supervising me.
 Integrative Think Tanks. 2022. https://www.bath.ac.uk/campaigns/integrative-think-tanks/. 06/10/2022.
 French Alternative Energies and Atomic Energy Commission. 2022. https://en.wikipedia.org/wiki/French Alternative Energies and Atomic Energy Commission. 06/10/2022.
 Lim, Y. S. ITT16: A PDE approach to sensitivity analysis for geometrical perturbations in nuclear reactors. Working Document. 2022.
 French Institute for Research in Computer Science and Automation. 2022. https://en.wikipedia.org/wiki/French Alternative Energies and Atomic Energy Commission. 06/10/2022.