Partial Data and Hybrid Inverse Problems

In a typical inverse boundary problem, we have a differential equation with unknown coefficients on a fixed domain. We are given measurements of the boundary values of the solutions to the equation, and we want to determine the unknown coefficients. These problems arise in a number of practical contexts, from medical imaging to mineral exploration. My research concentrates on two aspects of these problems: partial data problems and hybrid problems. A partial data inverse problem is one in which we want to recover the coefficients from measurements restricted to a subset of the boundary. This is an important consideration in applications, since it is often impossible in practice to measure on the entire boundary. Partial data problems are still poorly understood in general; one focus of my research is to improve this state of affairs. Hybrid problems are ones in which we leverage the interaction of two physical phenomena (e.g. light and sound) to obtain better-posed inverse problems than we would get with either phenomenon alone. Several potential imaging methods in this area have yet to be fully understood mathematically, and a second focus of my research is to investigate problems in this area. Finally, in addition to these topics of research, I am open to further areas that might arise as applications of these. Currently I am also working on a pair of problems in control theory and unique continuation using techniques from the study of partial data inverse problems. I am optimistic that many more interesting problems like these will continue to arise as my research continues.