I am an applied mathematician specialising in developing theory and algorithms for solving inverse problems where the aim is to recover a hidden model parameter from indirect observations (data).
Solving inverse problems amounts to running a simulator `backwards', a procedure that is often ill-posed meaning that there can be multiple solutions consistent with data and/or solutions are sensitive to (small) variations in data (instability). A key component is the ability to simulate noise free data (forward operator), another is to handle the intrinsic instability in ill-posed problems. The latter cannot be addressed by abundantly sampling data even when the forward operator is formally invertible. Instead, one needs to go beyond controlling the discretisation error, which typically is the main concern in classical numerical analysis, and enforce stability through a prior model for the model parameter (regularisation). Computational feasibility is a further challenge for large scale inverse problems in time critical applications, like medical imaging.
Much of my research is therefore in the intersection of mathematical analysis, machine learning, statistics, and numerical analysis.