Probabilistic numerical methods for Inverse Problems

This project is concerned with numerical methods for function spaces that would possibly include applications to inverse problems. In particular, probabilistic methods for function spaces provide us with a mathematical framework to define relationships between quantities of interest and our observed data. Because the relationships are probabilistic, it also enables us to make inferences on the quantities in a probabilistic manner. A good example of an inverse problem is determining material properties of a structure by looking at a satellite image. In its nature, the problem is very ill-defined, e.g. the solutions are unstable with respect to the data or not unique. For such problems, we need to find ways to define and characterize function spaces and to be able to learn from noisy and limited data. This mathematical theory is fundamental for the ongoing development of methods to solve such ill-posed problems.