Research Interests

What happens when a solution exists, but computation reaches its limit?

My research sits where analysis, computation, and partial differential equations meet. Generally speaking, a pure mathematician is interested in the minimum number of assumptions on the initial data of a PDE to have a unique solution that only changed slightly if the boundary conditions change slightly. This is known as well-posedness. An applied mathematician is interested in getting an algorithm to approximate a solution to a PDE and, in order to get a faster algorithm, will make larger assumptions on the data. For example, to solve the heat equation, there may be an assumpton on the boundedness of the third derivative of the solution - even though the equation is only a second order one. This is known heuristically but the question is whether it is necessary in order to actually compute the solution. When higher assumptions on regularity than necessary for existence are not present, uncertainty principles can form about the location of the solution. This is how we can embed a halting problem in order to show undecidability of the norm of the solution.