Computer-assisted proofs for partial differential equations in multiple spatial dimensions
From suspension bridges to fluid flows: solutions can be proven with the help of computers
Engineers rely heavily on computer calculations to predict physical behavior. The methods from this thesis help to strengthen the mathematical reliability behind such simulations. It is in line with a time in which we are increasingly dependent on computers.
Physical phenomena such as an undulating suspension bridge or a flow of fluid are described by complicated mathematical equations. Computer models can often approximate that behavior, but standard simulations never provide one hundred percent mathematical certainty. This is a big gap in science: we see a phenomenon on the screen, but cannot prove it definitively.
My research therefore focuses on the question: how can we use computers so that they provide rigorous evidence for the existence of these kinds of complex phenomena?
Too complex to calculate
Physical systems, such as those fluid flows or undulating bridges, are often too complex to calculate directly in real life, so we capture them in simplified mathematical models. Computer evidence has long been used to prove all kinds of models. In my research, I made these models more realistic by calculating in multiple spatial dimensions at the same time, and adapted the methods accordingly.
By combining computer power with classical pen-and-paper mathematics, we can show that specific patterns actually exist within these theoretical models. This removes the doubt whether an outcome on the screen is, for example, a calculation or rounding error.