# Student Research Article - Jonna Roden

**PhD Student Jonna Roden has written the following article as part of our series of Student Research Articles!**

What do bird flocking, printing, beer brewing and nano-filtration have in common? At first glance not much. However, if you look closely, all of these processes can be described as particles in a fluid. For example, birds (particles) move around in the air (fluid), pigment particles are mixed into water to make ink, and yeast particles are suspended in beer during brewing.

We can build a mathematical model of particles submerged in a fluid and use this to describe all the things mentioned above! Such models are called partial differential equation (PDE) models, and they capture how the particles move under the influence of different forces, such as gravity, and how they interact with each other.

Once we have this model, we can ask further questions: How do we get the yeast to sediment quicker so the brewing process is sped up? How do we get the ink to dry uniformly on the paper when we print? Mathematically, these are questions about optimization, or more specifically, about optimal control of PDE models. Optimal control is all about the question: What is the optimal problem setup (the thing I can control) that will get me the closest to my desired outcome?

In my research project I am working on exactly these questions, i.e. the optimal control of the PDEs that describe particle dynamics. In particular, I want to apply this to industrial processes, such as brewing and nano-filtration, to save time, energy and resources where possible.

What does my day-to-day research look like?

First, I need to derive the PDE model and optimality conditions that describe the problem I want to solve. This involves reading relevant literature and doing a lot of 'pen and paper' calculations. Once I have all the information I need, I want to run computational simulations of the model itself and its optimization to better understand the processes that I aim to model. The numerical implementation requires researching and testing of different numerical algorithms for efficiency and accuracy, implementing the relevant equations in Matlab, and troubleshooting any issues that come up throughout the process. Finally, we will have some results that we can talk about. I presented my current results at different occasions, such as an autumn school and a workshop and we have just finished writing a paper on this topic.

What's next? Extending the work to different models, which describe different applications and starting the cycle all over!

That much about my research. So, who am I?

I am a third year PhD student in applied and computational mathematics and I am originally from Germany. I moved to Scotland to study for a BSc in Mathematics at the University of Strathclyde, Glasgow and then joined the MIGSAA Centre for Doctoral Training in Edinburgh. In the first year of the programme different classes and group projects are completed, before choosing a PhD research topic for year two and beyond.

Apart from doing my research, I go to seminars, take classes, tutor undergraduate courses and organise events. For example, last year I organised the weekly PG Colloquium where PhD students talk about anything fun in maths (including, but not limited to, their own research), and this year I am part of the Edinburgh SIAM-IMA Student Committee, which is currently planning all kinds of fun events for the upcoming semester, so keep an eye out for our emails!