Mr. König, in your bachelor thesis you deal with the quantification of uncertainties. What exactly does that mean?
We mathematicians generally live in a world in which all the information needed to solve a problem is known exactly. Yet this is almost never the case in the real world. For example, measured values are always subject to measurement errors.
Therefore, we should not only ask ourselves what the solution to a problem is, but also how
confident we are about this solution and how much we can trust it. This is usually done by
specifying confidence regions, i.e., instead of a single solution that could be falsified due to
measurement errors or other disturbances, one calculates a region that contains the solution with
high probability. The size of the region can then be interpreted as (in)certainty.
The applications for this are extremely diverse. Uncertainty and risk have always played a major role in financial markets, but uncertainties are also of immense importance in the context of e.g. artificial intelligence.
The thesis is titled "About Bayesian Inversion Theory for Parabolic Partial Differential Equations" - can you decode that a little bit for us?
In my bachelor's thesis, I looked at stochastic methods for inverse problems. Inverse problems usually arise when you want to infer a quantity that you cannot observe directly based on another quantity. A famous example of this is computed tomography. Since one should not simply cut open every patient, their insides cannot be observed directly. Instead, we measure how the patient's body absorbs radiation, and based on this absorption, we can then infer the internal structure.
Such inverse problems bring with them some exciting mathematical properties that make them perfect candidates for uncertainty analysis. Indeed, inverse problems usually have no or infinitely many solutions and are also susceptible to perturbations such as measurement inaccuracies. As a result, they are usually difficult to solve and the computed solutions can quickly become inaccurate - which is where uncertainty computation comes in.
Specifically, I was concerned with the time inversion of a diffusion process. This is easy to illustrate. For example, if you put a drop of ink into a glass of water, you can imagine the spreading of the ink quite well. Such a spreading process can be modeled by a parabolic PDE and then simulated. However, if we let the glass rest for a while and look at it only at a later time, it is extremely difficult to calculate where exactly in the water the drop once was or what shape it had. Looking into the past is therefore much more difficult with diffusion processes than looking into the future. The situation is similar, for example, with a melting ice cube.
Since we can no longer say with certainty what the drop of ink or the melted ice cube once looked like, the question here is particularly how certain we can still be about the results of our calculations. To calculate uncertainties in inverse problems, Bayesian statistics has proven to be particularly suitable, also called Bayesian inversion in this context.
What is it about this topic that particularly appeals to you?
On the one hand, it is exciting to study inverse problems with stochastic methods, since we can make more statements than with classical methods, which "only" search for a solution of the inverse problem without taking uncertainties into account. On the other hand, this approach combines a wide range of mathematical disciplines. Knowledge about the underlying process, i.e. in my case about the modeling of the physical process by means of PDEs, approximation methods for the simulation of the process as well as stochastic methods for the quantification of uncertainties are required. So you should be mathematically broad.
Did you already focus on stochastics during your studies?
Actually, I had had only sparse contact with stochastics until the first meeting with Prof. Barth, who supervised my bachelor thesis. I actually heard the lecture on probability theory in the same semester that I wrote the thesis. In the bachelor's program, I had first immersed myself in differential geometry and mathematical physics, but towards the end of the bachelor's program I also heard lectures from the field of numerics. In retrospect, I would definitely recommend that first-year students do not just concentrate on one subject area. It is worthwhile to gather overview knowledge and build up knowledge from different areas and only specialize later.
The end of your bachelor's degree was already a year ago. What do you think about when you think back to your bachelor studies? Are there any experiences or advice that you would like to pass on to others?
I think very fondly of my time as a student, which is slowly coming to an end. I always say that it is hard to study mathematics, but that it is completely impossible to study mathematics alone. I am very grateful for the people who have accompanied me along the way: Fellow students, friends, and family. The simple fact is that studying is hard. It requires a high degree of frustration tolerance and stamina. That's why it's all the more important to help each other out and build each other up.
And that's the best advice I can give to prospective students: Walk the path together.
Oliver König B.Sc.
Graduatea award winner of th Robert Bosch GmbH for outstanding B.Sc. degree in the Department of Mathematics